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The

Origin

of

Music

A Theory of the Universal Development of Music

by ROBERT FINK Bach Music

ILLUSTRATED BY THE AUTHOR EXCEPT AS NOTED

Qze Qfeeqwtch

(^Meridiari Co., Publishers "This One

Copyright (c) 1981 by Grecnwich-Meridiaa 516 Ave. K South Saskatoon* Saskatchewan Canada Ltd. Edition of 100 Copies.

Published originally under the title: THE UNIVERSALITY OF MUSIC, (1970.) ISBN 0-912424-06-0 Library of Congress #81-670095

This book dedicated to the memory of

my Mother

and without her, this book may never have seen the light of day — Deborah Ensign.

INTRODUCTION

7

1 PRESENTATION OF THE PROBLEM C i 1 A PTE R I .

What's R]ght With Cul t lira] Theories of the Origin of Music? Origin of music based on two elements Cultural theories based on only one element What's right with cultural theories Notions of beauty What's Wrong With Cultural Theories of the Origin of Music? Noise and music not explained The octave. 4th and 5th not explained Scales not explained

13 14 15 16 -1

C H A PT E R 1 1 .

24 24 26 31

II THE THEORY CHAPTER III. - Development of the Scale & Melody ... 43 Overtones and intervals 43

Relativity of consonances Consonance and beauty A cultural theory of the scale Noise and music Completion of the scale The cycle of 5ths Tonality and the cycle of 5ths The fourth and cadence

*9 53 55 57 63 -74 76 M

CHAPTER IV. - Harmony Changing concepts of tonality The 3rd - How it spurred the discovery of harmony Other preconditions to the discovery of harmony The role of discord & melodicism Some other aspects of harmony Some final definitions Temperament

9* 99

128 129 131 133 135

CHAPTER V. - Rhythm and Emotion Rhythm - cultural or natural? The esthetics of rhythm

137 1 37 139

106

HHffl

CONCLUSIONS CHAPTER VI . - Modern Music The abandonment of tradition Why tradition is smashed The failure of modern music "Getting used to" theory The human ear

WS H6 151 157 161 167

CHAPTER VII.

169

- Originality

The duality of music and the arts compared The who and when of art Freedom of the artist Music and the future APPENDIX 1

Materialism & Music

1 70 181 202 204 209

APPENDIX II - Misc. Notes 224 The slendro: A study in equal & unequal division of pitch 225 Two arguments against overtone theories one old, one new 230 Method and Ethnomusicology .239 harmony in ancient and primitive music 248 The Greek tetrachord - origin .252 SUMMARY 257

IVlusic is intangible: It exists literally in the air, beyond grasp, sight, taste or smell. We perceive it only by ear. Unlike words, it can represent nothing exact. The idea of a man, a woman, a tree or a car cannot be expressed by music so that all hearing it think of any of these things. Music may appear to provoke thoughts of such concrete things, but the same music may call up entirely different images for other listeners. Yet the effect on the listener is not lessened by lack of concrete meaning in music: We hear it only, but we say, too, that we understand it, feel it, perhaps even see it in the mind's eye. Why can mere sounds, progressing as words do in a story, affect us as strongly as though they were a real story; as if they were a dramatic and meaningful series of real events? For example, if we rearrange the notes of a melody, even a melody new to us, the original form makes "sense," but the altered version often loses it. All these have been thousand-year-old unsolved mysteries. To solve these mysteries, the origin of music, another mystery, must be understood. We can understand a thing only by knowing how that thing started and what forces shaped its development. This book proposes a complete theory of the laws and forces governing 7

the origin and development of music. The complexity of the subject is such that no adequate synopsis of it can be given in few words. Instead, the reader is urged to consult the organization (given in the contents) of the parts and chapters, while reading, to help him maintain the order of the ideas. No theory has much value unless it resolves problems, explains phenom ena or predicts events. To acquaint the reader with the value of theory, Part I first illustrates some of the problems which the theory attempts to resolve. Not ail theories can answer all the questions. Following from this, a general discussion of various other theories surrounding music development, unfortunately, cannot be avoided. The reader will nevertheless be rewarded for the patience given to mastering Part I of the book, because he will better understand the rest of it. Equally impossible to avoid is a degree of polemics arising from the gulf which lies between some established current views and the theory given in this book. For example, were early astronomers to have projected the theory of a round world without cognizance and refutation of the prevailing flat-earth view, the strength of their case would have been limited. To answer the "facts" used as objections to their theory (such asKpeople on the bottom side would fall off, the rain would fall sideways or even upward in places"), we can see that the round earth theory had to extend into the nature of gravity, motion, etc. Against the theory in this work, the "facts" used as objections have been so many and they so persevere that some of them are of the order of fixations. They recur continually because the prejudices on which they are based are widespread. They must be anticipated and shown to be better accounted for by the new theory than by the established views, and so the text will often enter into whatever related fields of knowledge are necessary to accomplish this. The reader, who may view these digressions as un necessary or repetitious, fails to appreciate that a theory, which can successfully involve several angles of view, which can involve mutually supporting facts originating from different and independent pursuits, is a theory which is being presented to him in its fullest texture, providing its own test of confirmation and consistency. Such a theory can satisfy, if not the demands of literary esthetics, at least the needs of the serious scholar. The author intended to use a style which could reach the layman. But this work is not an easy narration. For one to whom the beauty in the integration of complex ideas is not apparent, this book may hold little interest. Expect no moving novel, no dramatic characterizations, out side of that drama inherent in the discovery of truth. 8

On the other hand, for the diligent reader, the search for the origin of music can read like an exciting mystery story: The actual origin of music is forever lost to our view, but it has left its mark: Everywhere that there is music there will exist a set of conditions which are the inevitable result of the original forces which shaped them. The conditions of music are the clues, and from them emanates, if we can find the key, the full story we seek to reconstruct. Each theoretical step proposed in this work is a way of playing back part of the recorded story. Each step reveals new clues, although some often seem to defy explanation. Then we must branch out, sometimes into other fields, but we read on, confident this mystery, authored by reality itself, can have no flaw in logic; that there must reside in each new fact, however puzzling at first, part of an integral and consistent pattern; and we can feel awe and wonder at the complete and beautiful lawfulness which interpenetrates nature as we unfold her secret. Part III of the book comprises the author's conclusions about the serious music of our own era. The reader who has avoided Parts I and II can neither accept nor reject Part III and still lay claim to having intelligent opinions. Much of the material in Part II is not new nor contested today. Its inclusion is mostly for the sake of completeness. However, some aspects of this part are, although not new, rejected today. Modern musicologists have rejected, wholesale, ideas which should have been only corrected or modified. What is new in this book is the correction and updating of these ideas, and the conscious application of a school of thought, materialism, to a subject till now untouched by that school. Discussion of materialism is found throughout the book and especially in Appendix I. Materialism, for the author, is the key which has unlocked the mysteries and provided the new relationships of ideas which finally make a theory of the origin of music whole and unified. Certain terms in this book will not be applied as they are used in current texts. Most notable is the term homophonic , because quotations taken from various sources use this term, as the French do, to mean monophonic music. Only recently has the term come to mean a form of polyphonic music. In this book the older meaning will apply and homophonic and monophonic will be identical. To substitute for the current meaning of homophonic, the term harmonic music will be used. Any other unusual uses of terms probably have the same reason, but should be clear in context. All terms part of the nomenclature of music will be defined as the book progresses, except those so commonly understood as to need no further definition. This book was greatly inspired by my friends, with whom I have often had long and sometimes heated discussions on art and music. They are not all students of music, but they have an interest in society, history and mankind. It is because of them, too, that parts of the book are in great 9

detail, even belabored on points which to music students may seem axiomatic. I have therefore assumed a minimum of musical knowledge on the part of the reader, so, if given to the diligence and patience mentioned earlier, anyone who can count should be able to follow this book. Many people love music and should know what it is - because they will love it more - and because it is a fascinating subject as well as a beautiful art. Finally, special thanks to Sue Swopeand MikeHeideman. Robert Fink REFERENCE TABLE FOR SYMBOLS & TERMS USED IN BOOK (Terms not shown here will be defined in book. See Index.) Interval:

Distance between any two notes.

Semitone:

Smallest interval on the piano, from a white-key note to the very next blackkey note, or from a black -key note to the very next white-key note. Also called "^-tone" & "Minor 2nd" (Music which uses many semitones, or a string of several semitones, is called "chromatic")

12-tone series:

All the semitones within scale.

Whole- tone:

An interval of two semitones.

Major interval:

Any interval from Do in the major scale (the "Do, Re, Mi scale"). Any interval may be changed from major to minor by lowering its top note a semi tone, & vice versa. For ex., C to E is a major 3rd; C to Eb is a minor 3rd.

Minor interval:

Any interval from Do in the minor scale not also in major scale. Means "flat" such as in B-flat: (Bb); To lower by a semitone.

jj —

Means "sharp", opposite of flat. Means "natural" Neither sharp nor flat. Means to return note from sharp or flat to its "natural" state, such as D# to D. 10

3PA3RX

Presentation the

of

Problem

What's Right with

Cultural

Theories on the Origin of Music?

Different types of theories are the principal characters in this chapter, beginning with a broad description of the author's own type, and following with a description of several types of cultural theories. These theories are necessarily generalized according to their common elements, because within each type, there are too many varieties of sub-theories, each differing in purpose and scope, to detail them all. In other chapters, some theories will be seen more concretely. To fail to first understand schools of thought and the types of theories within, is to later fail to understand the meaning of more concrete facts and ideas associated with them. For example, Romeo and Juliet were, first of all, members of rival clans headed by Montague and Capulet. In a sense, these clans are really the principal characters. To fail to understand the rivalry between them is to fail to understand the tragedy between Romeo and Juliet. So first, let's become familiar with the larger picture, distin guish only the principal characters within, and see what they can tell us about the origin of music. *3

Origin of Music Based on Two Elements In antiquity, before civilization and written history, people lived in small, wandering tribes, gathering food where they found it. They had the most rudimentary knowledge of the real world and few tools. Each day's needs were barely provided when the next day's needs were at hand. Because of this, survival was a full-time occupation, and all had to work together to perform tasks that one alone could not, and for protection against natural, animal and human enemies. (Needless to say, nobody had any time to fiddle around.) Art, music and other activities

like them could not exist except at the risk of life itself, and certainly not in the independent and well-developed form we know now. The origin of the arts, including music, was, therefore, based on two elements in human life. One was the development of tools, better methods of food gathering or production, learning how to harness some of the powers of nature; and this includes the relationships of men to each other, and their ideas, which

derived from their methods of producing the necessities of life. In a word: Culture. The development of these things began to free man from his animal-like existence and made possible many things, including the arts, which were not solely related to survival. The second element was the nature of human senses; sight, hearing, taste, touch and smell, and the nature of sound, color, etc., the material of the arts. In music we are concerned only with the nature of hearing and of sound. Our senses, evolved by the process of natural selection, as Darwin explained, allowed us to perceive the world in such a way that we could learn to make an artificial, protecting environment and survive against a hostile nature. But these senses also allowed us to perceive things which were not necessary only for the sake of survival. Primitives could see color, for example, even when they weren't trying to spot their enemies among the trees and brush. They could hear song birds as well as the nearness of a rattlesnake. We have the ability to smell perfume as well as smoke which warns us of fire. There fore, as these examples show, man had the potential to see colors, hear sounds, etc., and get a variety of pleasant physio logical effects from them, however unrelated some of these effects were to his everyday functional needs. Without these physiological capacities, few arts would have developed. Cultural Theories Based on Only One Element In many theories of the origin of music, only culture, the first element, in whole or in part, is their basis. Some theories are based on that part of culture dealing only with the social rela tionships and ideas of men, and this is done without under standing that economic forces, which caused these social relations and ideas themselves to come into being, also play a role in the shaping of musical art. Ideas do not drop into men's heads from the sky. They are caused, and both, ideas and the causes of them, can act upon the development of music and art. Those who try to explain music by omitting or minimizing the economic and material forces are greatly limited in their ability to explain everything. Often theories like this will include notions of accident, of the unknown, or of mystical qualities possessed by men ("genius"), to "explain" things which are otherwise left unexplained by their theories. Other theories, based also on culture, and which recognize the economic as well as the other forces that act on musical develop l5

ment, fail to appreciate that the economic aspects are more important or powerful than others. Holders of these theories may easily find the effect of economic forces they believe are acting in one place, but fail to see that they're acting, too, in a second place. They don't know - or can't explain why - that in this second place, other factors may have hidden or distorted the appearance of essentially economic impulses. (More concrete explanations to be found in Appendix II, "Method and Ethnomusicology") Finally, there are theorists such as Plekhanov, whose approach is based on all these acting forces and who also gives proper weight to the economic aspect. The basis on which his ideas rest is the best basis for cultural type theories. But it still remains a theory which is based only on culture, the first element earlier mentioned. Plekhanov claims the second element (man's relation to nature, the nature of his senses) has been made impotent or irrelevant by social and economic forces, and that this element does not play an important role in the shaping of music and art. In later parts of this book I will try to show that this is an incorrect claim. The impulse of nature is a very material factor, and one which cannot be treated as if it were no longer operative. On the other side of the coin, there are theories which rest on ideas of "natural law" alone, and which have been called "mechanistic" theories. Most of these theories are unaccepted in today's academic society. In this book, I do not put forward or defend such mechanistic theories, even though most of the book is devoted to this "natural law" element. This is because theories accepted today all underestimate this element, and I am trying to make up for that and correct the fault. This fault accounts for the failure of cultural theories to explain adequately the develop ment of music. I'll take up the limits of cultural theories in the next chapter. First, let's see what Plekhanov's type of theory is able to explain about the origin of music, and of the arts in general. All of the following serves to prove the value of a cultural theory which accepts economic, as well as social forces, in the shaping of things as apparently "non-economic" and "non-materialistic" as music. What's Right With Cultural Theories? The development of man's productive powers allowed him to produce what was needed in less than a full day, and in this first leisure, the arts began. The development of the state, with ruling 16

Types of Eastern Hemisphere Pan* s Pipes

Pan1 a Pipes » East & West (VERTICAL FLUTES) SEC P. IB

27

classes and oppressed classes, gave a great monopoly of leisure to a section of the population, and some of the members of the ruling classes became the first consciously "full-time" artists. The instruments of production, when they, too, found rest from constant use in production, were then used as objects of the arts, and some of them developed into musical instruments. The drum is the earliest kind of instrument. The striking of an object by another object, in order to shape it, to make some useful product, is an act of production which led to the development of the drum and stick. Originally, the drum was a method of signaling, helping in hunting, etc., and this was carried over to the enjoyment of the drum for the non-utilitarian rhythms which could be made on it. The flute is an instrument of the hunter, used by him to make it easier to communicate with fellow hunters without scaring the hunted animals, and to make it easier to lure them into traps. When not used in hunting, the sound of the flute lured the human animal too, and it slowly developed from a tool into a musical instrument. The drum and the flute have been widely found, among the oldest of them being Pans-Pipes. The development of stringed instruments was more rare, because the discovery of the resonator necessary to add to the hunter's bow was also rare among primitives. Hermann Smith, in

Primitive harp with skull as resonator

The World's Earliest Music, points out that the "murmur" of the bowl intrigued the primitive archer, and eventually, along 18

with the resonator, a second, third and more strings were added to the bow. The bow is the great-grandfather of modern stringed instruments. Looking at the picture below, we can see the development of the bow in historic stages as it was turned more and more into a musical instrument. The earliest instrument in the picture, on the top left, is merely a bow with a simple resonator. Examples (a), (b) and (c), below it are refinements made from the bow itself. The next two, the Arab 4-string lyre and the African harp, are later instruments, which, while still similar in shape to the bow, are successively less like the shape of the bow, until we come to the two Arabian fiddles. These last 4 are not made from the bow itself, as are the first 4, but are deliberately fashioned as instru ments only for music. If we were to continue along the same lines of thought, we would learn that all the instruments of music come from the productive side of life in the past. Of course, when music is a

African musical

*9

Arabian fiddles

20

permanent and full-time part of a culture, musical instruments are invented, rather than evolved, but many of these are still inspired from other spheres of human life. Even the content, speed and melodies of song are often derived from the economic and social processes of life, such as the rhythm in the Anvil Chorus, I've Been Workin' On the Railroad, etc. Notions of Beauty How do concepts of beauty originate? Do they have a material cause? If we examine a few concepts of beauty, we will be able to answer these questions. While these examples are not directly about music, they illustrate the point about the value which cultural theories do have. Most of us will agree that man (in most cases) considers him self to be the best and "highest" form of life on earth, that he tends to underline this by exaggerating what is different between himself and the rest of the animal kingdom, and that he tends to under-emphasize what is the same. We don't crawl, we walk up right, and this difference is not simply noted in passing - a big thing is made of it: The word "upright" has come to be a word with which we praise another person. When we say: "He is an upright fellow," we are not simply noting that one walks on two legs. We are praising the person. "Stand on your feet like a man!" etc., are statements which show that we are proud of the differences between ourselves and other animals. We hide and avoid the similarities. Even though man is an animal, the word "animal" has a connotation which makes its use an insult. How come this standard isn't true of savages? The Batokas, a tribe in the upper reaches of the Zambezi, according to Plekhanov, knock out their upper incisors in order to look beautiful! Why do they do such a "strange" thing in the name of beauty?* The Batokas can only be understood as products of *This is a "strange" thing only by our standards. But some of our own accepted practices are no more reasonable. The woman who wears rings through her ears today has less reason than the savage who wore them through her nose. The primitive wore rings through her nose as a sign of wealth which really repre sented wealth! In those days of early iron smelting, the iron in the rings was very valuable because the effort involved in smelting even a small amount of iron was considerable. Only a person whose family, clan or tribe had command of a labor force 21

their society, of the hunter's way of life. Attempting to resemble their cows and oxen, they knock out their upper incisors. So much of the experience of these peoples' lives is associated with these ruminating animals, that they are worshipped by the tribes men. Other primitives wear the skins, claws and teeth of wild animals in order to hint at their own strength and speed. If you can catch a tiger, you must be pretty fast yourself, and strong, too. If you clothe yourself in your conquest, you are parading your abilities. The sight of such garments is to the hunter beauti ful and admirable. Nowadays, we hang our conquests a little further away from our bodies, on the wall, because we have less experience with animals than did the primitive hunter. The Australian bushman hardly ever drew pictures of plants. Yet Australia is not a desert - in fact, plant life is all around. In one exceptional case, a picture was discovered of a bushman hiding behind a bush! But the bush itself is very poorly and inattentively drawn. The bushman is carefully and exactly drawn. Why? Because, again, the bushman is a hunter, and his range of vision doesn't include that which isn't necessary to his life. On the other hand, an agricultural society produces an art which does reflect plant life. The great landscape paintings of all time are the art-product of European agricultural society. Landscape painting diminished when European industrial and trading cities began, and surely enough, was revived in agricultural America in the colonies and in the states afterward. America imported its manufactured goods and was basically an agrarian country and this was reflected then in early American landscape painting. Today, again, landscape painting has diminished in somewhat the same degree as American agrarianism has been replaced by industrialism. In this chapter I have tried to show that a concept of history which is based on material forces in society can explain a great deal about the origins and forms of art. Basing a theory on these material forces is the best basis for a cultural theory. But it omits one important material force: the effects of nature and the nature of man's senses. Theories with this omission have this limitation: They can explain, as we have briefly seen, the different forms of art, music, ideas, etc., among nations, and great enough to supply the iron could afford to wear such rings. The modern woman with rings through her ears can make no such claims. 22

show that cultural differences explain why the arts are different. But in most cases, as we will see next, they cannot explain the similarities - especially of elements of music - among nations. It is to explain this and to deal with the effects of nature in the shaping of music that this book is written. Therefore, let's take up the problems and limitations of currently accepted cultural theories on the origin of music in more detail.

^5

What's Wrong with

Cultural

Theories on the Origin of Music?

Noise and Music Not Explained The first limitation of cultural theories on the origin of music is their inability to explain the difference between noise and music and why peoples have universally made a separation between them. A quote from Edgerly shows this separation. The primitive Omaha Indian (North America) "holds the bow, which is whitened, in his left hand and the rattle and arrow in his right. He strikes the arrow against the bow string, as he shakes the rattle."2 How does this show the separation between noise and music? Let's see: When the bowman became intrigued with the "murmur of the bow" he began to make it murmur even when he wasn't hunting. Why did he do it at all? Why didn't he spend his spare time rearing more children? Or finding new gods to worship? Well, he did, but why did he strike the bow string too? And if he had to play with his bow, why did he choose the string instead of the bow, or the arrow? Why didn't he become intrigued with the sound of the bow when hit with an arrow (aside from rhythm possibilities)? Why pick on that little string? Certainly the bow, 24

the string and the arrow each had importance in his hunter's way of life, but nevertheless, the string became the object of his musical aspirations. Is it because of the climate in which he lived? Is it because wood is really less important than string in his society? Maybe it was an accident? If so, how can it be explained that this accident happened both in ancient Greece and "in Omaha," North America, where no one had ever heard of the Greeks? I think all of us have an idea why, and that is that there is a difference between the sound of the string when struck with the arrow, and the sound of the bow when struck with an arrow. That difference, recognized all over the world in all ages, is the difference between noise and music. But - according to cultural theories - all sounds, tones, etc., are equally capable of being considered musical or unmusical, and it depends only upon social forces, habit and custom to determine which sounds would be "noise" and which "music." If this is so, then how can the unique and universal separation of the same sounds for music, and all other sounds for noise, be explained? For if the cultural theory is the correct theory, then the selection made in history to define music from noise would be based upon social forces, which by differing in different societies, would cause different, not similar, concepts of what is noise and what is musical. But everywhere in history we see man making selections of some sounds as noise, certain other sounds as music, and in the overall development of all cultures, this distinction is made around the same sounds. Of course, within any particular culture, man has temporarily called some things noise which he later called music, and vice-versa. But in the general development of any musical culture from one society to the next, when the smoke had cleared, and the fires of old discarded temples had gone out, we find, for example, that such things as bow-string playing tended to remain as musical elements in any new society, while such things as "bow-banging" (if ever such things arose as musical concepts in the first place) died out. The separation of sounds into "music" and "noise" continued according to lawful means; according to means which had to be outside the influence of any particular or local social environ ment because the development followed the same way in other, different social environments. The only effect had by the internal forces of any social environment was to encourage or discourage the development, but not prevent it. 25

The Octave, 4th and 5th not explained No less an authority than the Harvard Dictionary of Music has, in its entry under Octave, this sentence: "The octave is the most perfect consonance, so perfect indeed that it gives the impression of a mere duplication of the original tone, a phenomenon for which no convincing explanation has ever been found . . . ."3 The second thing unexplained by cultural theories is, then, the wide existence of the octave, as well as the 4th and 5th, also "perfect" intervals. First, let me explain some terms. In this book, there will often be occasion to refer to notes which exist in the major diatonic scale. This scale is commonly known as the Do, Re, Mi, Fa, Sol, La, Ti, Do' scale. There are 8 notes in the scale (7 if you exclude the last Do'). The fifth note in the scale is called, simply, a "5th." The first plus the 5th note is also called a 5th, and the distance between the first and fifth note is called the interval of the 5th. The same is true of the other notes: The fourth note in the scale is called a 4th; the eighth note, an octave, and so on. 26

Therefore, the octave may be written (Do-Do'). The apostro phe means that the second Do is one octave higher than the first. Two apostrophes mean two octaves higher (Do"). The 5th may be written (Sol) or (Do-Sol). The 4th, (Fa) or (Do-Fa). If you can sing the Do, Re, Mi scale, you will be able to sing a 5th (Do-SoD, as well as other intervals. If you can't sing the scale, examples in this book will often be furnished with a picture of a piano keyboard. All of the several methods of calling the notes, by letter, by their names (Do, Re, etc.) and by the intervals they form with the first note of the scale (5th, etc.) will be used in this book. Although most keyboard examples herein will be in the key which starts on "C" what is important is the relationships that exist between the notes of the above scale, no matter what note is started upon to make that scale. The same relationships obtain for the scale in all keys, that is, no matter which note is Do. For the sake of simplicity, the major diatonic scale will be mostly referred to in the key of "C." Below is an illustration of the keyboard and the names of the notes in this scale.

DoReMi etc "X" • Middle C

It was the octave which led me to the study of music, and about which I have been engaged in much debate. Therefore, I will try to present my reasoning on it here very carefully and in detail so that it will be absolutely clear. If the purely cultural theory is true, then we must assume that "consonance" and "dissonnance" (or "harmony" and "discord") are terms which take on different musical content in different societies. Therefore, we must assume that no note has an "inherent" or "natural" capacity to be consonant with any other 27

note, and that any two or more notes which are considered consonant are only considered so because cultural history and practice dictated it. To put it slightly differently, all notes must be, in relation to each other, considered equal, but are subject to becoming called consonant or dissonant by subsequent human practice and cultural evolution. Two exponents of this theory have been Hermann Smith and A. J. Ellis. (There are more such viewpoints as can be seen by a trip to the library. I am not here preparing "straw men" to knock down.) After assuming some of the basic concepts of cultural theories, let's examine them in the light of the development of the octave, 4th and 5th. In the past, men and women,* who wanted to sing together, didn't sing the same note. The men may not have gone high enough, nor the women low enough, to sing the same note. Their voice ranges made it difficult. How then, were they going to sing the same song? To do that, the men and women singers had to sing notes which were different from each other. They had to sing notes which were apart from each other. But how far apart? Does a particular culture determine that? Is it arbitrary? One might say, "Of course, they used the 'same' note - only higher." The culturalists will then cry: "Why must we assume there is such a universal thing as the 'same note - only higher?'" They'll ask, "Why aren't the notes above and below any given note all really different from that note? We know that some of them sound different. If others sound alike, why shouldn't we attrib ute the cause of this to habit and training rather than some innate characteristic of the notes themselves?" This is a correct question. But it is incorrectly answered by the culturalists. The point is that history sides with those who are prompted to think there is such a thing as "the same note - only higher." In history, the distance between the notes which were sung by men and women were in intervals of the octave, 4th and 5th. Even the names of the notes belie cultural theories. We and others start on "A" (or some such symbol) for example, and several notes later consider another, different note to be "A" * In many times and places women were not allowed such functions as being singers; boys took their roles, from ancient Greece till lately in the Church. (Hence, "Castrati.") However, in the earliest societies women were not always so proscribed. 28

again, and this is done almost universally regarding the same two notes. Why is this, if as cultural theories indicate, all notes are really equally distinct by nature; why, if no two notes have any "natural" similarities? If there is nothing similar about the octave, 4th and 5th "inherently", then we should expect to find some cultures in which these intervals were considered dissonant and not used as the intervals by which men and women could sing the same songs or chants. For if any two notes are consonant or dissonant only as social conditions dictate - then some place, in the entire world, past or present, it seems there would have been cultural conditions such as to dictate that some other intervals.instead of the octave, 4th and 5th, were considered consonant, and that by comparison, the octave, 4th and 5th were considered dissonant. But there is not a single example of that in the whole known cosmos. Baltzell says: "The performer while singing a melody accompanied himself on the lute, playing the same melody a fourth or fifth above."4 Jeans: "The early Greeks seem to have employed no other concord in their music" (the octave) "although they were certainly acquainted with others. Aristotle tells us that the voices of men and boys formed an octave in singing, and asks 'Why is only the consonance of the octave sung, for this alone is played on the lyre?'"5 Howard and Lyons: "In the earliest attempts at harmony, the only intervals considered consonant were the octave, the fourth, and the fifth."6 Censorinus, who lived in the third century, "makes mention" says Baltzell, "of a practice of using a melody in octaves accompanied by the fifth to the lower note of the octave, which is also the fourth to the upper."7 Max Weber: "Fourth and fifth parallels are predominant (among the Indonesians, bantu tribes and others)."8 Elsewhere he says, "As far as our knowledge extends today wherever the octave has been distinguished the fifth and fourth seem to have appeared as the first unique and harmonically perfect intervals. This seems to be the case in the overwhelming majority of all musical systems known to us."9 Helmholtz: "Now some of these intervals, the Octave, Fifth and Fourth, are found in all the musical scales known."10 Meyers: "It would undoubtedly simplify matters if one could 29

adopt ... a completely cultural theory of consonance without reservations. Yet, in spite of recognizing the social and cultural nature of musical experience . . . viewpoints such as this ... go too far. Such an exclusively cultural position is not only faced with the remarkable fact that the octave is the focal point in the music of all cultures but with the tendency for the fifth or fourth to become substantive tones and restful, consonant intervals."11 Bauer and Peyser: "The Greeks seem to have had no harmony outside the natural result of men's voices and women's singing together" (in octaves).12 The New Oxford History of Music: ". . . in the vocal music of primitive peoples these intervals" (octave, 4th and 5th) "not only frequently determine the tonal framework of the individual motives" (musical themes) "but are everywhere found to have the greatest influence in the formation of scales."13 Elsewhere, about the American Indian: "Nevertheless, even in these freely intoned songs there is a recognizable tendency to use a clearly defined fourth or fifth as the basic melodic formula and to group the other tonally indefinite notes more or less freely around it."*4 Raymond, about the octave as sung by a soprano and a tenor: "One unacquainted with music might not suppose that the two" (notes) "differed at all . . ,"15 We will have recourse to some of these statements later on. I could add more, but it is not my wish to fill half this book with a tedious list of repetitions and half with source references. Therefore, to return to the premises we assumed about cultural theories, we must revise them. We might still assume that consonance and dissonance are concepts which differ in different societies, but for now, we must make an exception of the octave, 4th and 5th. That notes do not have any inherent or natural capacities to be consonant or harmonious with another, we can no longer assume. Apparently, the octave, 4th and 5th do have that capacity. It can be claimed that perhaps the octave 4th and 5th had a common origin, and simply spread all over the world. Although I believe they had independent origins in different places, let's assume they didn't for the sake of the argument. It is true that between societies in contact with each other, some things in one society can spread to and influence another. Other things may not because social differences are too great. So, if one aspect or two of a society's musical culture spreads to a 3°

second society, and if these aspects which are accepted in this second society do not spread to or influence a third society then the rejection by the third society and acceptance by the second society of these musical aspects can be assumed to have a cultural basis or a local, singular cause. But if some things, like the octave, 4th and 5th, spread all over the earth, and are retained in all and different societies - even as these societies change; while other things, like certain songs, musical rituals and sounds, do not spread all over the earth and are not retained by different societies - because these societies change - and all this when many of these societies are still in contact with each other - then how is this special ability of the octave, 4th and 5th to spread in the face of any cultural conditions, favorable or not, going to be explained "culturally?" If the octave, 4th and 5th are culturally inspired and used, why don't they disappear with the disappearance of the society which inspired them - even in one instance? These intervals are always retained even though every new society discards old musical forms and styles and takes on some new ones different from their past and from neighboring societies. Culturalists will have to break down and admit there might be something natural about the octave, 4th and 5th which explains the phenomenon of its wide use - or deny that it is so widespread. Scales Not Explained The phenomenonof similar types of scales throughout the world is a third thing unexplained by cultural theories. The pentatonic (5-note) scale, loosely called the "Chinese" scale (the 5 black notes on the piano) is found in the music of most peoples in all continents and has been written on extensively. Willi Apel writes that this scale "occurs in nearly all the early musical cultures, in China (as far back as 2000 B.C.), Polynesia, Africa, as well as the American Indians," (including Mexican, Peruvian) "the Celts, and the Scots. It must be considered the prototype of all scales."16 Bruno Nettl: "The most widely used scale ... is the pentatonic . . . There are . . . as many varieties of the five tonescale as there are possible combinations of intervals, and the pentatonic in its varied forms is'* the scale most frequently encountered in primitive, folk, and Oriental music throughout the world. It dominates every major musical style except Western cultivated music, where it is subordinate to the far more preval ent diatonic scale." Continuing, Nettl describes the most 3*

common of the various pentatonic scales: "The most common pentatonic form is composed of major seconds and minor thirds, for example, c d e g a, with the tonic" (most central or important note in a melody) "occurring on any one of the tones. Scales that lack half steps, like this one, are called anhemitonic."17 (The avoidance of half-tones that Nettl mentions is also a common phenomenon in the history of musical scales, although by no means universally true of all scales.) Finally, Carl Engel writes: "The existence of the pentatonic scale at a very early period throughout so large a portion of the world as Asia, and also in Egypt, is a curious fact, not without importance in the history of music, and, moreover, leading the inquirer to various speculations. To conjecture that all the different nations have derived it from one source must appear bold indeed."18 And, proceeding boldly, Engel states his case for a common source, claiming that if the scale were suggested somehow by nature, then it would have developed in Africa, but he attests to no trace of it there. Since Engel wrote (1864), however, field research has uncovered enough data that there is little doubt of the wide use and influence of the pentatonic in Africa. As a result, the theory of a common source appears now even less satisfactory than before. Let us examine the properties of this scale and one man's explanation of it. The pentatonic scale is unequal, that is, unable to be formed by a simple, equidistant division of flute holes, nor is it explain able by a similar simple division of strings (Musicologist Curt Sach's "divisive principle"). In other words, if cutting 5 neat, equally spaced holes in a bamboo or cane flute (to nicely fit the 5 fingers) would have produced the pentatonic scale, then it would not be very difficult to imagine this scale having been independently generated by different peoples. Cutting equi distant holes is not an act which requires any complicated social or mental processes to achieve. But a flute, contructed in this manner, does not produce the common (anhemitonic) pentatonic scale, rather, it produces a scale with intervals quite different from any with which Western ears are familiar. Neither a common source theory, as shown above, or the ease in cutting equidistant holes, then, can explain the widespread pentatonic. To make a flute that will play the pentatonic, the holes would have to be arranged according to more complex principles, and would be askew. Such instruments are rarely found, even among 3a

the less primitive flutes, yet the pentatonic scale exists, vocally, in these places. Dividing strings in halves, thirds, quarters and fifths is a simple division and would produce $0M notes of the pentatonic, but as shown in the illustration, the arrangements

I Division of string ofair column into segments:

4 y5 7 -Aircolurnnot* ^ string J CMute length. =c)

,

segments)

I i j in

\\Flute holes: I A| (y/Koie = c)

Si

FfoEl (PFKITATOMlC)

fring- lengths (on fret):

(pemtatonic) A| G

t^wfer pitch, 33

which result from simple division are not simple when applied to instruments, rather, they are almost geometric: I: Note that the lengths of strings (or columns of air, in the flute) are simple divisions of the whole: divisions by 2, 3, 4 and 5. However, the selections of lengths, necessary to make the pentatonic, stand in a complex, not simple, relation to each other: II & HI: Notice the spacings on the flute and on the frets are uneven, or complex. Now, the pentatonic has been found in places and times which preceed the development of such instru ments or sophisticated numerical procedures. (It would not be surprising to learn that, in fact, the unevenness or inequality of the scale helped give rise to sophisticated numerical systems or procedures, which then found their way into use in other fields, such as mathematics and engineering.) Curt Sachs, while denying any explanation of the pentatonic scale in "natural law," will agree: "In any case, deriving scales from systems is putting the cart before the horse: all over the world, scales have been abstracted from living melodies and integrated" (later) "in systems."19 Sachs is unaware of the contradiction between his statement and the following: The term "systems" would also include his culturally inspired "divisive principle." Yet Sachs would have us believe this principle is the only possible explanation for the widespread similarity of scales and the universal use of octaves and other perfect intervals.20 But scales cannot be caused by, and yet precede, systems; the same being true of perfect intervals. Surely, too, a complex scale cannot arise independently of both, "systems" on one hand (as Sachs agrees), and also free from any natural pressures on the other hand (which Sachs also claims). If that were so, then agree ment of the scales in so many different areas and periods in the world would have to be a miracle of coincidence. One cannot exclude all the explanations of the scale. The origin of the scale, therefore, in the face of so many other possible (equal) scales and among peoples who have no "system" of tuning or arriving at scales; its persistence in so many primitive societies on the one hand and more technologically developed civilizations on the other; its popularity against the use of other scales or systems of music; the widespread use of its tones in melodies even among groups who do not conceive of "scales" as such - all these strongly indicate that formation of the penta tonic, far from being either culturally arbitrary, or widely distributed merely by habit, is due to the operation of more 34

universal and natural impulses. This is especially true considering that many cultures indulge in painstaking efforts to achieve symmetry and equality, in design, in mathematics, and in measurement tools involved in other human pursuits. Yet, at the same time, some of these societies uncomplainingly maintain and enjoy the asymmetrical series of pitch that is the pentatonic scale. We have seen so far how cultural explanations of the penta tonic scale are not satisfactory. And this is true also regarding the widespread use of our familiar 7-note Do, Re, Mi scale. In Africa, although the pentatonic is used there, one finds wider use of this 7-note kind of scale (heptatonic, diatonic), similar to our own. Ward, quoted by Merriam, wrote, "I am not prepared to accept the theory of a peculiarly African scale." (That is, a scale which is peculiar to Africa and different from all other scales.) "I think that African music is perfectly intelligible on a diatonic basis." (Meaning that the notes are similar to those in the West.) Merriam adds later, "Ward further points out that in his own experience African musicians accepted African music played on the piano save for the beginning and ending portamento ... 'In other words, I see no reason to suppose the existence of an "African" scale, but rather I think that African melodies are essentially diatonic in structure, modified by a liberal, and unregulated, use of portamento . . ."'21 (Portamento is the sliding or slurring of the voice in singing from one note to another; sometimes called glissando.) Writing of the 7-note heptatonic in general, Bruno Nettl notes: The hexatonic and heptatonic scales in primitive music are almost always parallel to the diatonic scale: the former resemble the hexachords of Guido d'Arezzo; the latter, the modes of Western music." (Guido d'Arezzo was a medieval musician and teacher who worked up a set of hexachords, or 6-note scales.) Nettl continues, saying that neither the hexatonic nor diatonic "are in much use, but a few hexatonic and heptatonic melodies are found in most areas. Only in Negro Africa do they occur frequently, and even there they do not predominate. The inter vals employed in heptatonic scales are standardized to an exceptional degree: major seconds are almost always included; minor and augmented seconds are rare, as are any segments of a chromatic scale."22 This scale, then, is also widely found, though not as much as the pentatonic, but enough to be a problem for any cultural 35

theories to explain. It too, like the pentatonic, is unequal or uneven. Notice that in the pentatonic scale, there are large "gaps" shown by dashes: Do, Re, -, Fa, Sol, La, -, Do'. If one fills in these gaps, between Re and Fa, and between La and Do', then a number of 7-note scales may be formed (depending what notes are used to "fill in the gaps"), among them, our familiar major and minor scales. These scales do not now become equally spaced by filling in the gaps in the pentatonic. Now there are two "squeezed" areas - the opposite of gaps - between Mi and Fa, and between Ti and Do' in the major scale, called semitones. (The other notes of the scale are separated by a whole tone.) As is it still unequal, then, like the pentatonic, it also could not be expected to be found in so many nations merely by accultura tion, habit, or, on the other hand, by accident. (A look at the piano keyboard will show that both diatonic and pentatonic scales are uneven, as there are different numbers of notes between some of the notes of both scales.) The difference between the two scales (pentatonic and the 7-note diatonic) are the 3rd and 7th notes in the latter scale. These two (Mi andTi) are recurring as our discussion progresses and are worth remembering. For example, another similarity among the music of different peoples becomes apparent in the following points, and is centered around these two notes. In Scottish, Irish, some aspects of Eastern music, and in other places, the 7-note scale appears to grow out of the pentatonic and this happens in accord with a relatively singular pattern of development: The added notes to the gaps (the 3rd and the 7th) in the pentatonic are often tuned with either unsureness, or variety, of pitch.* Carl Engel, quoting an example of a Scottish Air, writes of it, "The words are more modern than the melody, which is strictly pentatonic, with the exception of the fifth (measure), where the minor seventh . . . occurs," and a little *Various ethnomusicologists and others often parade this^leeway" in tuning as an example of the "arbitrary" nature of tones and intervals among the musics of various people and claim it as evidence against any kind of natural or non-cultural theories on the causes of scales. A discussion of this properly belongs to a later part of the book. However, the existence in general of similar scales and of widespread justly-tuned ("perfect") intervals is well enough established to provide reason to search further than cultural relativism for an explanation of these phenomena. 36

later, he writes, "Traces of the pentatonic scale are perceptible also in Irish national tunes, although to a less extent than in Scottish," due to the occasional introduction of "notes-in-thegaps." In the Irish melody Speic Seoach, which was transmitted to a writer in c. 1786, Engel notes the "major seventh . . . occurs therein twice, but it does not constitute an essential note of the melody; in fact, it rather gives the impression of having accidentally crept in from carelessly drawing the voice over from the sixth to the octave."23 (My em ph.) Above, we see the seventh (Ti) of two types (major and minor) being added to basic pentatonic melodies. Regarding Africa, Jones writes: "I have lived in Central Africa for over twenty years but to my knowledge / have never heard an African sing the 3rd and 7th degrees of a major scale in tune. "24 Merriam notes: "There has been some discussion of an African scale in which the third and seventh degrees are flatted or, more specifically, neutral between a major and minor interval. This concept has been advanced especially by those concerned with analysis of jazz music, since in jazz usage these two degrees of the scale - called 'blue' notes - are commonly flatted and since the third degree, especially, is frequently given a variety of pitches in any single jazz performance."25 Sachs reports this phenomenon of the East Asians: "The ryo scale . . ." (an anhemitonic pentatonic scale) "was heptatonized . . by the insertion of a sharpened fourth and a major seventh: F GAbCDeF." (However, in a slightly different species of the same scale, beginning instead on C, this would be a 3rd and sharpened 7th: C D e F G A b C.) Sachs continues, "The addi tional notes kept a transitional, auxiliary character and had not even the privilege of individual names: the Chinese called them by the name of the note directly above with the epithet pien, which means 'on the way to,' 'becoming.'" Notice the similarity here, of concepts surrounding these notes, with that of the Scot tish and Irish view of these same notes (as in the earlier quote by Engel), in which the notes are formed by "drawing the voice over from the sixth to the octave" and are considered inessential. In later Western music these notes become termed "leading notes" or "passing notes." Later, Sachs concludes: "The evolution of East Asiatic scales now begins to stand out. It starts from strictly pentatonic scales with thirds of any size. In a second stage, heptatonics appear in the form of seven loci for strictly penta tonic scales. In a third, the two 'skipped' loci are admitted to the 37

scale, though only as passing notes. Finally, they are fully incorporated."26 Similar processes, similar scales, similar intervals, even similar concepts as language usage shows: "passing notes" in the West, "on the way to," or "becoming" in the East - all indicate a lawful, rather than a culturally relative or arbitrary, development. There is still another reason to believe that these scales have developed independently. The Apache Indians appear to have developed a fiddle by themselves. (The peculiarity of its con-

Apache Indian fiddle

struction denies Asian influence.) Such a spontaneous development in one case indicates the possibility in other cases, such as scales. Edgerly points out that the "musical bow was found among the California Indians, and at wide intervals elsewhere. Even fiddles, of a kind, appeared; the Apache had a peculiar one string fiddle, sometimes made from the stalk of a yucca flower."27 Mrs. Jean Jenkins, member of the staff of the Horniman Museum says that the "Chukchansi Indians of California use" (the musical bow) "for mourning the dead."28 (Sachs says that stringed instruments are entirely lacking in the Western Hemisphere. It is true they are certainly scarce. But their development, however rare, is an example of the very process (Fegun all over again by American Indians) which occurred in 38

Europe and Africa, where stringed instruments were brought to a

When all of the above is reviewed, we see a grand array of phenomena which cultural theories cannot and have never 39

explained. But the acoustical nature of sound does explain them (we will see next), in the most exact way and with the most amazing parallels. This is true in spite of the fact that music was evolved by people who until this and the last century had virtually no knowledge of acoustics. Of course, it is true that no matter how well a theory explains facts, it may not be a true theory. But the theory which is expounded in this book cannot be rejected just because a better theory might be evolved. So far, the theory of acoustics, together with an accounting of real cultural and social influences, best explains the development of music, and I believe the theory in this book on the development of the scale and harmony best explains the change from non-chord music (homophonic music) to harmonic music (polyphonic music). No other theories can explain the development without serious inadequacies. A dialectical* approach is needed in order to account for the often contradictory phenomena. Some writers can't think of or accept two opposing realities at once and so they reject one reality to "explain" the other. Other authors have touched on the answers accidentally or unsurely. Meyer says: "Thus once we leave the octave, and perhaps the fifth and fourth . . . cultural factors play an increasingly important role in the development . . ."29 (My emphasis.) As for the octave, 4th and 5th, Meyer proposes further "study." This is Meyer's conclusion. Starting from this, we can begin the study, with some simple fundamentals of acoustics, explaining what has already been presented as unsolved problems in this section, as well as going on to other aspects of music, the scale, harmony, etc.

*Dialectics, among other things, deals with the simultaneous existence of opposites and attempts to explain the resulting contradictions. 40

The Theory

Development of the Scale and Melody

Overtones and Intervals The existence of overtones has been one of the bases on which scales have evolved. To begin with, let's examine overtones and the characteristics they have. Overtones are often known by other terms, such as "partials," "harmonic series," and "harmonics." The illustration below gives an example of how overtones are produced. What is shown is

^

^ 2^

'

^

f three of the many ways in which a single violin string vibrates, when one note is produced. However, "one" note is really a 43

complex of many notes: Each of the three examples a, b, and c produces a different note. Other notes are produced by vibra tions of the string in even smaller segments producing successively higher notes. The string vibrates in all these ways simultaneously. All of the notes thus produced make up what is called the overtone series. If the string in example is a C-string, then the overtones it produces are C\ G, C", E, G', Bb and C"\

1111 11 llil 111 till 111 I 1 HI II 1111

'overtone C Series

1

G

C E

G'

2

3

5 6 7

4

C"

All together they sound to us like one note, but this one note we hear is really a compound thing, made up of all these overtones. Overtones also have relative strengths of audibility. In the example the key-board also illustrates the overtone series in the order of strength of audibility. Each smaller segment in which the string vibrates produces a weaker overtone until,after the 6th or 7th overtone, they are beyond human perception. But those shown above are the ones which are audible to the ear. "Theo retically," says Jeans, the overtone series "ascends to infinity; often in practice harmonics beyond the sixth or seventh" (overtone) "are too weak to affect the ear . . ."30 The above overtones are the ones which we hear although we are not conscious of it. But proof that we do hear them is that, if we didn't, we wouldn't know the difference between a note on the violin from the same one on a trumpet, or between a plucked string and one which is bowed. Overtones are usually referred to by the number which is written beneath them in the above example. However, they are often called by the name of the interval which they form with their original note. For example, the 2nd overtone of C is G, but G forms a 5th with the original C when set within the range of 44

one octave. In the example below, the overtones are shown all within the range of an octave to illustrate the intervals which are formed.

Minor ,— 7th— Bb

I

Octave -1

The more audible overtones, when played in a chord, form the C major chord, which is the basic type of chord in Western harmonic music. Also, it can be readily heard, by playing the two keyboard examples above as chords, that the first example sounds more richly harmonious than the second. This is because the first example is the one which represents the actual arrangement which overtones form in relation to the parent note. The second example, within the range of an octave, is a distortion of the overtone series. It is already possible to explain the prevalence of the octave in the musical systems of all cultures, because the octave is the first and most audible overtone of any note. If the note is C, the first overtone is C - an octave higher. By existing in nature, it is discovered by men in all nations whatever their culture may be in any nation. The same is true of the interval of the 5th. The prevalence of the 5 th can be explained because the 2nd overtone of any note, when played an octave lower, forms a 5th with that note. The 4th, which we said in Part One was also present in every musical scale known, is not formed by being an overtone of any note. The explanation of the 4th is a little different. To begin a partial explanation here, it should first be pointed out that there are mathematical ratios which exist between sets of any two 45

notes. These ratios represent the vibrations of strings or columns of air, for different notes, in relation to each other. For example, Middle C has 264 vibrations per second. C an octave higher has double that amount. If any starting note is assigned the number 1, then the octave to that note would be, accordingly, 2. In this way any octave has a ratio of 2*.l. In the 5th, the upper note of that interval vibrates 3 times to the lower note's 2 vibrations, and so has a ratio of 2-3; the 4th has a ratio of 3:4 and the 3rd has a ratio of 4:5. The remainder of sets have increasingly complex ratios. Those notes which have complex ratios have wave lengths which do not match: This causes the crests and valleys of these waves to interrupt each other, causing rapid intermittent weak ness and strength, and these are registered by the ear in what are called "beats."

SOLID LINE: — One note DOTTED LINE — Another, die sonant note

Pythagoras' question, "Why is consonance determined by the ratios of small whole numbers?" is answered by Helmholtz who says that the ear "regards as harmonious only such excitements of the nerves as continue without disturbance."31 (My em phasis.) In other words, the more complex ratios reflect combinations of tones which cause "beats" and these are heard as "disturbing sensations in the ear. Only those notes whose number of vibrations can be reduced down to simple ratios, such as those of the octave, 4th and 5th, have the least beats. That these interruptions are unpleasant is understood if likened to the 4e

eye, which finds rapidly flickering light unpleasant. In the same way, the ear, and the nerves within, are strained to have to so rapidly adjust to the sudden changes in the wave pattern origina ting from tones which have complex ratios. (The question here is simplified. See Appendix II, the latter half of article: "Two Arguments Against Overtone Theories.") We can now account for the wide existence of the 4th, even though it is not among the overtones in the overtone series. It is widespread in all musical systems because it has a low, simple number ratio: 3:4. This is the third simplest ratio, only those of the 5th and octave being more simple. The history of what has been considered consonant has been the history of those tones with such simple ratios, in general. The reason why the 4th has a simple ratio and is consonant is because it is a 5th "in reverse" so to speak. What is meant is this: If we start with C, we can produce its 2nd overtone, G, and as stated, this G can be placed within the range of an octave and form the interval of a 5th with the original C. However, what note do we start from which will produce C as its 2nd overtone (and thus, the interval of a 5 th to that unknown note)? That note is F. That is, just as the 2nd overtone of C is its 5th (G), the C itself is the 2nd overtone of an F below it. This F, if placed above the C, will make the interval of the 4th to our C. The illustration below shows this: The F produces the C as an over-

*4ths" tone (and 5th). But when we place the F above the C, it produces a 4th. Thus the elusive 4th is a 5th by inversion or "in reverse." Not an overtone itself of C, the F produces the C as its overtone, and thus, with an inverted overtone relationship, has, like the 5th, a simple ratio. (One may reasonably ask when do we call the interval of the two notes F and C a fourth, and when a fifth? The current usage 47

of numbers and terms in musicology is unfortunate, and leads to this confusion. Intervals should be reckoned upward from the note which we start from because the overtone series runs upward. From the starting note C, F should be the "4th" of it, no matter where the F is played after the C, above or below it. Similarly, if we start on F and then go to C, this, then.should be called a 5th. The numbers originally come from the number of keys between the intervals on the piano, as we can see above. But whether you go up or down will affect the number of keys between C and F. If C-F was no longer called by a number, but, for example, was called by a name, such as cadence, the con fusion would end, and C to any F would be a cadence, and that would be that. The importance of which note one starts from in forming an interval of two successive notes cannot here be taken up, but will be at the end of the chapter, in the section on cadence.) However, at this point, before we have gotten into any real depth of analysis, we have a powerful indication of why the octave, 4th and 5th are so widely used in all musical scales: No other notes have the close overtone relations as do the octave, 4th and 5th. Here is how Helmholtz describes the octave: "When, then, a higher voice afterwards executes the same . . an octave higher, we hear again a part of what we heard before, namely the even numbered partial tones of the former . . and at the same time we hear nothing that we had not previously heard."32 (His emphasis.) Helmholtz goes on to explain that the same is true of the 5th and its "reverse," the 4th, although we hear this to a lesser degree because the 5th is a less audible overtone than the octave. Howard and Lyons say of this: "The art of music and the practice of harmony have been developed according to what has pleased human ears; they have been evolved by musicians, not by scientists. Nevertheless, as one compares the growth of the art of music and the extension of its basic principles with the laws of acoustics, he finds an interesting parallel between the two. In other words, men have found most pleasing to their ears the combinations of those tones that bear certain mathematical relationships" (ratios) "of vibrations to one another, even though they may not have been aware that those relationships existed." A little later they say: "Authorities may differ as to the actual connection between 4s

the so-called harmonic series of overtones and the development of tonal combinations in music. It is impossible, however, to ignore the parallel between the two, one a science and the other an art, and to fail to observe that the tones which have been accepted by Western ears as producing agreeable, or consonant, sounds in combination with other given tones have corresponded roughly with the natural overtones of those given tones. More over, the historic order in which these tones have come into the musical vocabulary forms an almost identical pattern with the harmonic series."33 In other words, it's likely that it is because of these mathe matical ratios that man found certain notes consonant, and the major chord agreeable, and not likely that the whole thing is a coincidence, especially when the historic entrance of notes into musical scales occurred in the same order as the harmonic series, that is, in the order of the degree of audibility of the overtones. The only exception to this is the 4th, which is not an overtone of the tonic, but whose 2nd overtone is that tonic. Relativity of Consonances The consonances we have already mentioned, when they are played in different ways, change their physical degree of consonance. When they are played in the bass, they are much less con sonant, because the overtones of each note of the consonance become more audible. The reason overtones of notes are more audible in the bass is because they are brought, by the lower tonic (or starting note), more closely to the center of the range of human hearing. If you go low enough, you will not even hear the original note, but will hear its overtones. (The same is true in reverse with high notes. Their overtones are weaker because they sound out of the range of human hearing, or at the edge of that range) As the overtones get louder, some of them form beats with the others. For example, the 5th: C-G: The overtones of C are C, G, E, Bb. The overtones of G are G, D, B and F. Now three overtones of G (G, D, B) are in a relation of 5ths to three over tones of C (C, G, E). (The others are virtually inaudible.) But these six also form a 2nd (C-D), a minor third (E-G) and a semi tone (B-C). These are not so consonant with each other (they cause beats), and are more easily heard in the bass, when the interval C-G is played there. Another way of affecting the degree of consonance of the 49

intervals is to spread them apart from each other. This can be done by raising the upper note of the interval an octave, or lowering the lower note an octave. This, like the range in which consonances are played, will also perceptibly change their degree of consonance. For example, the C below middle C, played with the G above middle C, produces a more consonant 5th (now it is really a 12th, 12 notes apart) than the 5th played within the range of only one octave.

This is because the larger 5th, or 12th, is simpler than the smaller. The larger has a ratio of 1 :3 while the smaller has a ratio of 2:3. 1 :3 is simpler than 2:3. The 4th, made into an 11th, becomes notably dissonant, and its ratio changes from 3:4 to a more complex 3:8.

The only notes whose interval ratios become simpler, and whose consonances improve, when spread over a wider range, are those notes which are part of the overtone series of any given note (of C in the example above). This is because by being spread apart, the notes of the consonance approach, or match, the 50

actual distances apart which they have as overtones of their tonic. When they are played within the range of only one octave, their arrangement is a distortion of the overtone series, although the general relations of consonance still remain between the two notes. The differences in richness and sonority can be heard easily when the examples are compared on the piano. The 3rd, which has a ratio of 4:5, when played in its form of a 10th, has a ratio of 2:5, and in its even larger form of a 17th has a ratio of 1:5, which makes it more consonant than either the ordinary 5th (2:3) or the 4th (3:4).

Most primitive instruments never had such a wide range, and only when the range increased were musicians able to produce the 3rd in the form of the 10th or 17th. The development of stringed instruments provided this range and allowed primitives to make clearer and more defined low notes than on the drum. Around the time of the development of stringed instruments, 3rds began to be included in the scale as a melodic step, becoming slowly, by the increased range in which they could be played, a consonance of higher merit. But this is only part of the reason for the 3rd ultimately becoming a member of the scale. The most important is the fact that it is an overtone of the tonic, the original note of the scale, although a weaker one than the 5i

5th, the 4th, or the octave. As the relative values of the consonances can change by having them spread apart in multiples of octaves, various societies and cultures were bound to slowly change their appreciation of the different consonances when they developed the ability to produce these more widely spread intervals. Nevertheless, it was slow, and the ancient Greeks, who were able to produce the larger intervals, and who firmly included the 3rd in the scale, kept their music mostly within the range of an octave, because it was always to be sung or used to accompany poetry. This had the effect of keeping the 3rd in its least consonant form and holding back its being included among the more important consonances for a long time. (To be in the scale, the 3rd did not have to be also considered a consonance of the highest order.) The interval of the 6th, like the 4th, is less consonant as it is spread apart. However, the interval of the minor 7th (the 6th overtone of a note) is very much more consonant when spread over a wider range:

What we see here, then, is not only that there is a general historical parallel between changes in concepts as different intervals were able to be made, but concretely, some intervals, such as the octave, 5th, and3rd, improve with the increased range capacities of early music, while others either become less important in relation to the others, or retain their original appeal, such as the 4th and 6 th, which are most consonant in their narrower ranges. (In late medieval times, the 4th eventually came less and less to be favored; the third, more and more.) We can partly understand why the third today is considered as a high consonance even though the octave and 5th, accousti 52

cally, in all these intervals' narrow range spans, are really more consonant. However, the thoughtful reader may have realized that the word consonance has become one which now encom passes, in the discussion, more than one idea: that of acoustic consonance and that of appeal, or beauty. Such a use of terms becomes unfortunate if continued so vaguely defined. A quote from Helmholtz reflects our modern taste, and also begins to try to make a distinction between consonance and esthetics. He also throws a "fly" into the ointment regarding the 6th: "Esthetically it should be remarked, that of all ... the major Sixth and major Third have . . . the highest degree of thorough beauty. This possibly depends upon their position at the limit of clearly intelligible intervals. The steps of a Fifth or Fourth are too clear, and hence are, as it were, drily intelligible .."34 It seems natural for one to wonder, if the 6th's most "beauti ful" range span was already at the disposal of primitives in their music, why this interval did not find a more important place in the scale sooner. To answer that it is necessary to further define the differences between consonance and beauty, a most im portant concept to our pursuits from here on. Consonance and Beauty There is a distinction between consonance and beauty. The octave, when played as two notes simultaneously, is a conson ance; it is indeed hard to tell, on good instruments, that two notes and not merely one are actually being played. It is easy to understand that the octave to any tonic, by being so similar to the tonic, does not offer the musician much beauty. It is, rather, the almost total lack ofdissonance which interests the musician in the octave. The octave is used when two melodies cannot be played or sung together by beginning on the same note. The same is true of the 4th and 5th, which have been used as intervals expressing sameness, although to a lesser degree than the octave. On the other hand, the 3rd is less similar to its tonic than different from it, although it is an overtone and part of the tonic. Except for this similarity as an overtone of the tonic, all the overtones of the 3rd are different from those of the tonic, and do not form very consonant intervals with them. (Compare the over tones of C (C, G, E, Bb) with those of its 3rd, E (E, B, G#, D): The only matching, necessary to consider it a consonance at all, is of the 3rd itself with the E in the overtones of C.) As a result, 53

the 3rd, E, lies on the border of consonance and dissonance, is both the r^.me and different in relation to the tonic, and is to us, especially in its wider range, more interesting and beautiful than the other consonances. In its smaller range, such as at the command of primitives, it is not as noticeable. Despite its potential beauty, it had to wait until Greek times to get into the scale. Another reason for its exclusion from the scale is given below regarding the 6th, but also applies to the 3rd. The 6th, which for the same reasons as the 3rd is interesting and beautiful to us, but whose most consonant range is within that at the command of primitives, also was excluded from the scale until after the octave, 4th and 5th were firmly established. The above distinction, or contradiction, between consonance and beauty, provides the ability to explain this. Although the 6th is beautiful to us, it is so only because of our harmonic context in which the 6th is always placed, and this context is remembered by us at all times. The primitive, who was without possession of this harmony context, found the 6th more dissonant than con sonant. To put it another way, that side of the above contra diction which shows the relationship or sameness between the notes of the 3rd (or 6th), is only clear, or made more apparent, when certain other notes come before and after these intervals. Without such notes, then only that side of the contradiction which expresses the separateness or distinctiveness of the notes forming the intervals can be strongly felt. In the case of the 6th, this separateness is clear because the 6th note in the scale has no overtone relationship to its tonic. Regarding the third, as noted earlier, only one of its overtones match that of its tonic. (More about the above harmonic context and an example of it will be given in the chapter on harmony. Why primitives were not able to provide themselves with such a context will also be under stood in the chapter on harmony.) The lack of means to integrate the 3rd and 6th in primitive music is why Helmholtz reports that the 3rd and 6th were inter vals which "all antiquity . . . refused to accept as consonances'.'35 Nevertheless, for other reasons, the 6th finally came into the scale and was followed later by the 3rd. These other reasons will be seen to ultimately relate to the same causes as the origin of the octave, 4th and 5th. So far, the whole pattern of how the notes entered the scale is almost exactly explained by the first laws of acoustics. It is as if the very dry laws themselves were a history of musical develop 54

ment. The further we examine this, the better it gets (or worse) depending what one would rather believe about the cause of the development of the scale. Up to now, I have tried to show how overtones strongly influenced the discovery of the octave, 4th and 5th and other, lesser consonances. Before we go into the theory behind the formation of the whole scale, let's examine a cultural viewpoint on how the musical scales of man were formed. A Cultural Theory of the Scale We must remember that we grow up in a culture in which the diatonic scale is the only scale. As far as we are concerned, the notes on the piano are the "only" notes. This is not so of the music of primitive people, or of people with primitive scales. In the West, we have 12 notes between an octave and in addition, a 7-note scale. Primitives usually had a 5-note scale and other scales consisting of various numbers of notes besides. The Arabic and Persian system of music had 1 7 notes dividing an octave and also several 7 and 8 note scales. Many of these scales included notes which cannot be reproduced on the piano. Herrman Smith, author of The World's Earliest Music, explains his version of the origin of these scales: "Those who, seeing the holes that are cut upon a common flute, or oboe, consider that in the origin of the instrument they were done in order fitly to comport with a musical scale, are wrong in their supposition . . . the distances of the holes, spaced for the convenience of the fingers, ordained the musical scales."36 This is undoubtedly true. However, Smith also writes: "The holes were cut to suit the spread of the fingers, and the scales which followed . . . were accepted by primitive man; the ear got to like the sequence of sounds, and so it worked into the brain of the race, that ages after, it became an intellectually accepted musical scale or relation of notes and was varied by evolution . . . Our so-called divine music is to the Chinese miser able . . . and the sounds which please Asiatics . . are to us distracting din, positively painful to listen to. The liking of the ear in music is a liking by inheritance, transmitted as a facial type is."37 (My emphasis.) To conclude, Smith quotes his friend, Dr. A. J. Ellis (translator of Helmholtz's work on acoustics and founder of modern ethomusicology): 55

"The final conclusion is, that the musical scale is not one, not 'natural,' nor even founded necessarily on the laws of the consti tution of sound, but very diverse, very artificial, very captious." Smith adds, "He has actually caught the scale in the act of changing by a caprice at the bidding of a finger. On the lute, in the very early Persian and Arabic scales, the middle finger had nothing to do, and to find employment for the lazy finger, a ligature was . . tied halfway between two existing notes . . . and so added two notes to the scale."38 The flaws in elementary logic by these men are based in their incapacity to generalize. They cannot generalize that an overall process exists despite momentary and singular vagaries within that process. On the other hand, when they do generalize, it is from an example which is atypical. If, as Smith says, the scale was so liked by the ear, and had worked into the "brain of the race," then what caused its evolu tion? That is, what caused men to like something new something tmworked into the brain of the race - in place of the former, established and liked scale? Something caused it. Was it the same "caprice" which gave rise to the first scale? Is caprice the cause of all development? And what caused this evolution to take place, not just any old way, but in the long run - parallel to the laws of acoustics, after all? Smith and Ellis offer no explanation for this. They seize upon a peculiarity in a nation, caused by that nation's cultural rituals, institutions, concepts or practices, and try to prove from it that nothing, therefore, is natural about the scale at all. It is true that the uses of certain sounds in Chinese music may have had their origin in the chance thickness of the flute-maker's fingers. The notes so formed often came to be used in connection with Chinese religion or ritual, and they then became codified by an institution or decree. All this is not a natural development, but it doesn't mean that the scale is wholly artificial. The notes produced by the ligature Ellis describes are not a development which is found in every musical system. (Although it has parallels in several.) Neither is chow mein found in every society's cuisine. But does this logically mean that eating is, therefore, a cultural peculiarity? Of course not. Similarly, the act of music making: of making scales which include the octave, 4th and 5th, is not proven to be artificial in origin because of a possible artificial origin of the above Arabic additionsto the scale. (Even at that, the notes produced in this case are not even by 56

"caprice" as we will see in this chapter in the section on comple tion of the scale.) Flowing also from ideas like those of Ellis and Smith is one explanation of the octave, 4th and 5th, which goes like this: As one note is actually heard as a combination of that note with its overtones, we got used to this. Then, when we began to play these overtones as separate notes, instead of merely hearing them subliminally, we consciously recognized them as related to the original note. Moreover, as any note with its overtones were in the "brain of the race" anyway, we not only recognized rela tions, but liked them because we had always been used to them. Although there is something true in such an explanation, it is not simply a question of having gotten used to notes. Man could have liked many other sounds if it were only a question of having gotten used to them, such as thunder, banging, howling wind, and so on. But man has always made a distinction between noise and music, both of which he was and is "used to." Noise and Music It is true that man has been able to find noise "pleasant", but only because of what he associates with it, not because he gets used to it. Let's look at what Smith says about the Chinese: "One of the most curious traits in the character of the human animal is an unfeigned delight in super-exaggerated noise . . . everywhere we find that this sheer delight in noise, called music, is manifest and on record. Not merely called so, but dignified and accepted as music." Smith is here referring to ear-splitting orgies of natives engaged in trying to scare away "dragons threatening to devour the moon."39 But Smith is not correct when he says that the Chinese considered this "music." (They may have called it so only euphemistically, in the same way we might call other people "sweet." We don't go round tasting each other, but our descriptions a hundred years from now might make a foolish historian think so.) Smith's own description of the purpose of the noise-making by the Chinese demonstrates this: Notice that the more immusical they became, the more they believed this would scare dragons. This is, on their part, an admission of the distinction between noise and music and is a similar distinction to ours. We, also, believe that noise has "repelling" qualities. The more success they had in this social undertaking in defense of the moon, the more esthetic pleasure the Chinese got. Not their noise, but their success, pleased them, whatever they may have 57

called it. However, the Chinese no longer view the moon as threatened by dragons. What happens to the above "music" when the super stition associated with it is no longer believed? — It also is given up. (Today, in China, the state has endorsed both Western style harmony and Western instruments (including the piano) as "revolutionary." In Japan, My Fair Lady was a relative popular success.) Smith says, "The love of noise belongs to us ... I confess to thoroughly enjoying a thunderstorm, my nature is absorbed in an energy greater than the individual, and I revel in it."40 I confess the same thing too. On the other hand, some are not so happily affected, and the thunder, which identifies the storm, as it does wind, rain and darkness, is an unpleasant sound to them. To primitive farmers, thunder was alternately "music to their ears" or unpleasant, depending whether they needed rain or had had too much of it for the sake of their crops. There are many sounds which are enjoyable in so far as they herald a total event, which we have reason to view as desirable or fascinating. But the various possible attitudes to such events do not help us to understand if the sounds in question are pleasant "in themselves." Can we ever separate such sounds from association with the event? Well, Smith, perhaps, never heard of the hi-fi phonograph. But if we could play the sound of thunder on a high-fidelity phonograph for him (assuming he'd not be a mad-faddist about super-frequency-modulation sets) would he have enjoyed it then? The owner of the set will enjoy anything his set reproduces faith fully. Realism is his watchword, not just beauty, if that. Would Smith, however, enjoy the sound of thunder without the accompanying grand spectacle of the clouds, colors and motion, without the whistling wind blowing across his face, without the cool burst of rain? To the extent he might find the remarkable reproduction of the sounds on the set interesting is perhaps the only way he would enjoy it. But other than that, the sound is not the same thing. The sound of thunder is noise, which is only beautiful (to Smith and me) in association with the total event of a storm. At any rate, the sound of thunder and such are not best-sellers on records, hi -fi bugs being the main purchasers. The Chinese (and Smith, unconsciously), make, then, the distinction between noise and music. Why do the Chinese choose 5»

noise and not music to scare their dragons?* That is, what quali ties does noise have that makes the Chinese think that it will be repulsive to their dragons? If culturalists' theories cannot answer

this, we must look elsewhere. The answer lies in the acoustical difference between noise and music. This will also help us to understand the completion of the scale. First, here is a brief summary and conclusion about some of the things already taken up. In the first part of the book we said that two things are responsible for the overall development of music. One is the state of economic development of society with all the ideas, institu tions and practices which flow from this economic base. The other is the nature of the senses and of sound. How do these two motive forces operate in relation to each other? They are contradictory. They combine with each other, but like two fighters in a clinch. This contradiction causes pheno mena that confuse many musicologists. They find it difficult to *In Chinese temples where emphasis was on music to inspire reverence, and not on noise to scare dragons, there are found lovely melodies which are, if strange, still sensible to Western ears. There is no harmony, of course, because primitives rarely combined sounds deliberately. 59

explain one thing when its opposite exists with it at the same time. They want history to be one way or another, but not more than one way at a time. The use to which music is put varies in different societies, at different times. Its use is part and parcel of some productive process, or part of, and subservient to, some institution which arises in a social system, like the church, for example. Standards of beauty, as has been shown, often originated from the produc tive process or its institutions. As long as the ritual, religion, institution or ceremony with which music (whether consonant or dissonant) is associated is historically justified and is viable, then its success and the respect it engenders are transferred to the music, and the music is considered beautiful (or reverent, sacred, soulful, etc.). This is the same as the hunter who finds certain animals "beautiful" on sight, as he looks down the shaft of his arrow pointed at the animal's heart. It means the hunter eats. The beauty of that thought is transferred to the animal he sights. The earlier food-gatherer hated the sight of animals because they competed with him for food and got in his way. He thought they were ugly. In general, as soon as one social system decays, most of the trappings developed by that system, including certain standards of beauty, also decay, and are replaced by other standards, and sometimes by their opposites. It would not be unreasonable to conclude that whatever artistic or esthetic elements do not decay with the dying society, but grow and prosper even more, must still have reason for their continued existence, must have some other source of viability besides association with any institution which may have gone to ashes with the old society. It must have a source which is not culturally inspired. One of the things not decaying in the music of any people were separations made of musical tones from certain natural and man-made noises, which included ungraded tones, such as the whistling of wind, splashing of water, etc., and also from those which may have been earlier enjoyed by the same peoples because of associations made with them. In the illustration below, we see the reason for the separation. The first line represents the wave of a simple musical tone. The air carries this motion to the ear. It is musical because it is simple. The second line depicts the wave motion of noise. The ear picks up that complex motion and registers it, but in a relatively confused way, because it is irregular. 60

At what point does musical tone become noise? By looking at the illustrations it can be seen that the pattern of the first one is easy to decipher with the eye, the other is not. This is true of the ear, too. When the pattern or regularity of the sound becomes imperceptible to the ear, when it cannot be understood (and when it produces beats), then we have crossed over into noise. Jeans says of wavelengths, "Indeed, it is this regularity which distinguishes music from mere noise. "41 The siren, or sliding sound, although its pitch at any given instant is a musical one, is nevertheless avoided historically as noise or dissonance. It is true there is the use in some musical cultures of "speech-song," crying sounds, howling and the rest of such sounds, and these are examples of the sliding sound. But societies have in the long run shown a certain development from that, and on the other hand, such things represent more a form of mimicry than music. It is imitation and drama that count, and not the abstract value of the sound. (Today in Jazz this has become slightly different, but more of this later.) "When the wind howls and its pitch rises or falls in insensible gradations without any break," says Helmholtz, "we have nothing to measure the variations of pitch, nothing by which we can compare the later with the earlier sounds, and comprehend the extent of the change. The whole phenomenon produces a confused, unpleasant impression. The musical scale is as it were the dividing rod, by which we measure progressions in pitch, as 61

rhythm measures progression in time . . ." (This is easier to see by comparing to change in people. We know from our own experience that over a period of time, the change in a person's appearance, due to age, or gain or loss of weight, is difficult to fully appreciate by one close to the changed person, especially if the change has been slight, because the association of the persons is continuous. Only the use of old photographs (or of reports from casual acquaintances), which by being representative of steps in the change, by being suspensions of the continuous gradations of the change, capturing it at its different stages, can make the actual amount of change apparent and measurable. In the formation of scales, composed of differ ent, but definite, pitches of notes, we see the desire expressed to gage change in musical pitch.*) Helmholtz continues: "We consequently find the most complete agreement among all nations that use music at all, from the earliest to the latest times, as to the separation of certain determinate degrees of tone from the possible mass of continuous gradations of sound, all of which are audible, and these degrees form the scale . . "M2, You can try an experiment relating to this, if you have a long cylindrical tube, about 30 or 40 inches long, and about \Yi inches in diameter, open at both ends. Put your mouth at one end and begin whistling upward, like a siren, without making any distinct steps to mark change of pitch. The column of air, forced to vibrate in the tube, will make the steps for you. It will sympa thetically vibrate only to those frequencies which are part of the *In every other field in which change must be measured, steps are selected from the possible continuous lines of change. In temperature, degrees are used, and some of these are placed at natural points of heat and cold, such as the boiling and freezing points of water, or at the total cessation of molecular motion (absolute zero), etc. Those "scales" of temperature not hung upon such points come to be less and less useful. Distance is measured more and more by steps which are simple divisions of the size of the earth (meters) and are more valued (except for the weight of tradition, which keeps the use of meters from being universal) than those measurements based on a dead English King's hand, foot, or other length of body. We have been showing, and will show further, that in music, not only are scales formed from the continuous stretch of pitch, but the notes of the scales historically come more and more to be pegged at pitches which have natural physical relationships to each other. 62

basic tone represented by the length of the tube. Is it possible that in the spaces of air in the cavities of the skull (culturalists' skulls have bigger cavities than any others), and in the areas of the inner ear, that here too is a physical lack of "sympathy" for ungraduated changes of pitch? I believe so, and that the forma tions of scales of any kind is impelled by this. Now let's see just what scale emerged, and what forces molded the particular form of it which we know today. Completion of the Scale Music slowly developed as an art separate from the other arts, and broke away from being used only in connection with one or another social institution, to which it was subservient. If we go to a concert today, it has nothing to do with making a car, or building machines, factories or houses. Each one is a relatively separate activity, with its own techniques and methods. This sense of "separate " is how separate music came to be at one point in history. When the direct influence of social institutions was removed from music by this development, the influence of the laws of acoustics took fuller sway and music developed more rapidly and more consistently along lawful lines. The fact that some religious sects still sing chants, discants, organum and non-harmonic music is an example of the effect of an institution in keeping music from changing there as rapidly as it did outside these sects and religions. The Chinese scale and music did not develop into such a separate activity at the same time it did in the West because Chinese society and custom did not change as rapidly as in the West. The result of this constitutes the gulf which developed between Eastern and Western music. Let's see how the comple tion of the scale illustrates this point: If we take the old trio, the tonic, 4th and 5th, and write out their overtones, in the order of their audibility, we can add up the different overtones of the tonic, 4th and 5th, and get the modern diatonic (Do, Re, Mi) scale: Tonic: C - Overtones are: C, G, E, Bb Fifth: G - Overtones are: G, D, B, F Fourth: F - Overtones are: F, C, A, Eb When you add up the three most audible overtones of each note, you arrive at the C major diatonic scale: C, D, E, F, G, A, B,C. When you add up all of them, you get the same scale with a 63

couple of notes left over, Eb and Bb. The hardly audible Bb is cancelled out by the more audible B. Why both notes, Bb and B, were not included together is because "Many nations avoided the use of intervals of less than a tone . . ."43 says Helmholtz. The B and Bb form a half-tone, or what is called a semitone, and are therefore avoided by including only the most audible of the two. The same is true of the Eb which is cancelled out by the more audible overtone of E. This is why we have a 7-note scale. Of course, the E to F is a half-tone, but the presence of the E in the scale is due to its being an audible overtone of the important tonic note, C. The same is true of the B which forms a semitone with the C. It is in the scale because it is an audible overtone of the 5th, G, which is an important member of the scale. No other semitones are admitted which are not justified by this overtone relationship. We will see later, in more detail, why the E and B are included despite the semitones they form in the scale. Nevertheless, the E and B are weak members of the scale. The C, which appears twice in the table of overtones above, is the strongest member of the scale, and is the tonic. The G appears twice. The F appears twice, although the second appear ance of the F is as a weak overtone of the G. The D appears once, but as a strong overtone of the G. The E, A and B are the weakest members of the scale, but the A is the strongest of these weak three, because it forms no half-tones with any other note in the scale. (Or, in other terms, the 3rd (Mi), the 7th (Ti), and the 6th (La), are weakest, but the 6th is strongest of these weak three.) Being weak members of the scale, we might expect there would have been historic deviation or uncertainty around these three notes. And there was. If we replace the E with the Eb, the B with the Bb and A with an Ab, we have the minor scale:

Eb

AbBb

64

Finally, when two of these three weak notes are removed from the scale entirely, namely the E and the B (leaving in the A which I showed was the strongest of these three) then we are left with the pentatonic (5-note) scale - which is the Chinese scale.

PENTATONIC

1 2 DbEb

3 4 5

GbAbBb

1

Db

(same Scale on Black Keys)

The differences between Chinese music and taste with that of the West has been cited as "proof or, at least, as an example, that there is nothing natural about our diatonic scale or about Eastern scales. We have been told that each people uses and likes their own scale and views the other's music as noise. But the above explanation of the scale begins to show the contrary. Namely, the two scales, diatonic and pentatonic, are not opposite scales nor opposed, but rather, the pentatonic is part of the diatonic. It is less than the diatonic by those notes which are acoustically weak and which have been the subject of vacillation on being settled in the scale. To digress momentarily, with these weak notes in mind,we can now follow up a point raised by Smith and Ellis earlier. Smith told us that Ellis caught the scale in the act of changing by 65

"caprice," which ordained two new notes into the Arabic scale. To be accurate, it should be said that these notes were not exactly added to the scale. They were added to those notes which are the Arabic division of the octave. In the West, the octave is divided by 1 2 semitones. From these, we select our 7-note scales of minor and major. Similarly, the two notes in question, introduced by an Arabic musician and theorist named Zalzal 1100 years ago, were among the 17 which divide the octave in the Arabic and Persian system. From them, 12 scales were formed, most of them 7-note scales, some 8-note. Zalzal's two notes found their way into 6 of these 12 scales. (The two Arabic scales which are equivalent to our major and minor scale do not have these notes. In addition, several of these scales perfectly match those made by the Greeks, by beginning the diatonic scale on each of its notes, thus forming a variety of scales, ^or example, if you begin the C scale with A, it makes the A-minor scaled)

The actual notes in question are not described by Smith? how ever, the two notes which Zalzal introduced were a 3rd and a 6th. But they are called "neutral" 3rds and 6ths because they are neither major nor minor; but between these notes. In the C scale a neutral 3rd would be between the E and Eb; a neutral 6th, between the A and Ab. The difference between them and their minor or major counterpart is a quartertone. These notes reflect the vacillation we have mentioned. They are "compromises" based on 2 of the 3 weak areas in the diatonic scale. In addition, one of them, the neutral third, is a deviation of the most important area distinguishing the pentatonic from the diatonic scales. So, in all, these two notes of Zalzal's are hardly "caprice!" The impulse for this "caprice" is based on the nature of 66

acoustics and is universal wherever scales developed beyond the tonic, 4th and 5th. The particular response to this impulse in the Arabic system is not altogether uncommon or unjustified in relation to the laws of acoustics and their effects on music. In the West, such a "compromise" is represented by the "blue" notes in Jazz, based on the 3rd, the 6th and the 7th. The continued use of them as an artistic deviation does not appear to me to be contradictory to the view of the scale as natural and basic. This point will be better appreciated when we take up the question of the role of dissonance in harmony and of tonality before that. But here it will suffice to say that the use of these notes is really a confirmation of this view, because by their existence and use, they show (in a negative way) an unconscious recognition of the influence of the laws of acoustics: This influence is weak only concerning the area of these notes, and so gives rise to the deviations around them. The Jazz quartertone cannot be made on the piano but can only be approximated by playing both the major and minor notes together. Jazz singers, however, hit these notes directly. But, however it is done in the melody, what is interesting is that the harmonic accompaniment, especially in earlier Jazz, does not generally include these "blue" notes. It remains based on the chords of the tonic, 4th and 5th most of the time. The harmonies are strictly major or minor, and the dissonant effect of the blue notes against this harmonic background is of great tonal value; is justified, because it "plays with," and thus enhances, the strong consonances of the diatonic system. In such a system, melody is freer to flirt with the notes of the scale, even to the extent of quartertones and "gliding" pitches, without losing its musical "sense," because the harmonies underneath show the relation ships and retain the integrity of the diatonic system. (Such deviations are more difficult in a period of music without harmony and the diatonic scale developed in such a period. The importance of the scale is considerably lessened today because of harmony.) At any rate, this "flirtation" is a consequence of accepting the diatonic scale and harmony as, in fact, standard and basic. This acceptance is true not only in Jazz, but, regarding the scale, true of the Arabic and Persian system. The thing to explain is not the artistic deviations from the standard, but what determines the standard? Is that "caprice," or is it lawful? In general, regarding the weak notes we have been discussing, all these vacillations and uncertainties in the past were overcome 67

only in the West, and allowed the scale, as a standard, to be definitively settled with the present two major and minor forma tions. The deviations which remain are now deliberate, artistic; and not a reflection of unsureness regarding what the standard scale should be. The reason why this happened only in the West was because social change was more rapid than in the East. This meant that associations, formed by connecting various "unacoustical" musical scales and practices with Western social institutions, were more often changed as these institutions were abandoned or overthrown. In the more frequent interims, or "intermissions" between social changes, the music was able to undergo only those changes which were inspired by the internal impulses given it by the effects of acoustical laws. As later, music reached the threshhold of a "separate" art, an institution of its own, social effects and associations with them became fewer and weaker, to the time when musical changes were almost wholly consistent with acoustical impulses. If, as culturalists claim, the laws of acoustics had nothing to do with all this: with the avoidance of most half-tones, which are the worst dissonances; with the acceptance into the diatonic scale of only those half-tones which had overtone relationships with the tonic, 4th and 5th, etc., - then it is beyond understanding what did cause the scale. The diatonic scale is the basis of our Western music and whole melodies are made out of it. Handel's Joy To the World Christmas Carol is the C-major scale, played from the top down (except in a different rhythm than that in which the scale is usually played). A whole song is explained and accounted for, note by note, by the laws of acoustics. The words, of course, are not, nor is its use as a Christmas Carol. Later we'll see that the chords which harmonize this carol are also explained by these same laws. There is still a lot to be said about the development of the scale, but for now, a general conclusion can be made. In the long evolution necessary for the development of the completed scale from the tonic, 4th and 5th, some nations did not go as far as others. The tie-up of their social institutions with their musical scales and forms retarded the development. Every stage in the development, instead of being seen as part of a process by these primitives, was each looked upon as sacred and not subject to change, or further change. The West, therefore, did not develop "differently" from the Chinese, but only further. And up until the recent Chinese revolution, the West developed 68

further in every other way too. The difference in scales of the West and East, seemingly slight, is the difference that causes the great gulf between the musics built upon the two scales. (There is not so vast a gulf in the music of the Scottish, who also used the pentatonic scale, because Scottish music occasionally included the 3rd, which is missing in the Chinese scale. We see in Scottish music the historic juncture between the two scales.) We cannot appreciate the Old Chinese religions and ideas as our own; we don't have the associations they had between their music and their culture. As a result of not being able to con ceptualize in the same way as the Chinese, then, when we hear their music, it is another step removed from our understanding and appreciation. This can be illustrated by a comparison to opera music. When opera music is played without the words and acting accompanying it, then often the music is harder to understand and follow. The motion and direction of the music, the stops and starts, the jumps and glides, are many of them determined out side the music itself, by the nature of the opera's plot ( or in Chinese music, by the nature of the historical ritual or institu tion). Unless you know what is going on in the music, unless you know the plot or ritual, the music loses its "sense," at least as a totality. But can the Chinese appreciate our music? Will they adapt the diatonic scale to their music now that the old institutions in China are dead or dying, and, as I have claimed, because the diatonic scale is capable of being understood by any ear, Chinese or Western? Will they take our music, which is "free" of all but a minimum of cultural effects in its physical makeup (cultural effects which are understood only by Westerners), and use it from now on? I will tell you what Jeans says in his Science and Music: "If we visited another planet, we might expect to find them" (possible inhabitants of human type) "employing the same diatonic scale as ourselves."44 Up to now, we have seen the parallels between acoustics and the diatonic scale. Still, it remains to be understood how the effects of acoustics made themselves felt upon a human race unconscious of their action. How does the effect of acoustics filter with such precision into the scales which men have formed that we can now remark how exactly parallel are the two? What is this process which can work without the necessity of human 69

awareness? On the other hand, if acoustics was exerting such a strong influence, why did the scale based on it take so long to form? Let me try to illustrate the process graphically. First, about why it took so long: If you found the "illustration" (shown below) on an artist's scratch pad in his studio, you might assume he was either doodling, or trying to blunt the point of a pencil which was too sharp for his work. But now that the illustration is in this book,

70

you are immediately aware it must be of more purpose than that. In this, you have an advantage over the primitive, who wasn't even aware that he was going to discover anything about music. Nevertheless, your awareness will not dull the point I'm trying to make. If I merely included the illustration in the book without comment, even though you would have the advantage of believing that there is some reason for it, it would take you a great deal of time to realize what it is. Perhaps now you are thinking there is no meaning to the illustration, and that I'm playing a trick. However, as soon as you look at the next illustration below, which reveals the relationships that are hidden in the first illustra tion, you can then look back at the first illustration and quickly see all the pictures which were otherwise obscure. In the same way, mankind, not even suspecting that there was anything to discover about the relationships of sounds, took thousands of years to discover them. But when we (or primitives) are presented with the finished music based on these relations, music which is designed to show these relations clearly and openly, then it is more quickly recog nized that these relationships are really there. Therefore, Western music, so "obvious" to us now in its musical "rightness," had to take eons to be developed. In this way I hope I have illustrated this part of the process. The idea that men are often unable to see, in advance, results of the process in which they are engaged and a part of, is an important idea. It helps to correctly under stand not only the slow, peculiar and difficult development of the scale, harmony (the subject of the next chapter), and things in general, but also why discoveries of things, after they are

made, are so rapidly understood. Anybody can be a Monday morning Quarterback. Let's remember this point as we try to follow the vagaries within the development of harmony later on. Another aspect of the analogy illustrates that even though the effects of acoustical relationships takes time to discover, especially without human awareness, nevertheless, the relation ships will be discovered. If in the course of your life, you were to see my first illustration (in which the horse, hand, etc., are hidden by other lines) on trees, on the ground, in the sky, in the corner of a page in the newspaper; over and over again, all without explanation; although it would take time, you would, through such constant contact with the apparently meaningless pattern, realize the relations there. You would see the horse's head, etc., becausethey are there, and it would all be without prior awarness that this pattern really had those relationships. This constant contact is the meaning of the principle I outline in the next section, which I call "Use," and is the method by which men were able to discover the relations of sounds in music. Here is what Helmholtz says about the same process: "A feeling for the melodic relationship of consecutive tones, was first developed, commencing with the Octave and Fifth and advancing to the Third. We have taken pains to prove that this feeling of relationship was founded on the perception of identical partial tones in the corresponding compound tones. Now these partial tones are of course present in the sensations excited in our auditory apparatus, and yet they are not generally the subject of conscious perception as independent sensations. The conscious perception of everyday life is limited to the apprehension of the tone compounded of these partials, as a whole, just as we apprehend the taste of a very compound dish as a whole, without clearly feeling how much of it is due to the salt, or the pepper, or other spices and condiments . . . Hence the real reason . . . the melodic relationship of two tones remained so long undiscovered . . ,"45 (My emphasis.) Continues Helmholtz a little later: "We recognize the resemblance between the faces of two near relations, without being at all able to say in what the resemblance consists, especially when age and sex are different, and the coarser outlines of the features . . . present striking differences. And yet notwithstanding these differences . . . the resemblance is often so extraordinarily striking and convincing, that we have not a 72

moment's doubt about it. Precisely the same thing occurs in recognizing the relationship between two . . . tones." 4^ (My emphasis.) It has been my experience, just as often, that I was surprised to discover that two men were brothers, because the differences between them were so strong; but once I was told they were related, it was then easy to see the resemblances, and they became so obvious that I wondered why I hadn't seen them earlier. However, either way, this analogy reflects the un conscious and vague aspects in the history of the discovery of relationships of sounds. To conclude this section, Jeans outlines the history of the development, saying that ". . . vast numbers of tribes and peoples . . . developed music independently, and in the most varied surroundings . . . and the principles which guided them - to choose pleasant noises rather than unpleasant, consonances rather than dissonances . . . led" (them) "to much the same result . . . and this with a unanimity which is remarkable. They exhibit enormous differences in their language, customs, clothes, modes of life and so forth, but all who have advanced beyond homophonic" (one note at a time) "music have, if not precisely the same musical scale, at least scales which are all built on the same principle." (Jeans is referring to the "cycle of fifths" principle which is next discussed.) He explains: "The main differences are found in the numbers of notes which form the scale. By stopping at different places in the sequence F-C-G-D-A ... we obtain the various scales which have figured in the musics of practically all those races which have advanced beyond the one-part music of primitive man. "47 The principle of the scales mentioned by Jeans at the end of the paragraph quoted above, the "cycle of 5ths," is the stumbling block which both he and Helmholtz did not overcome. Because of it they could not fully explain the development of the scale, as we will see next. Jeans also seems to think that "pleasant" and "unpleasant" are the guiding principles in music. It's true that in the development of music and the scale they are basic principles, and at that, the most important too, but they are not the only ones. It is apparent that I use the tonic, 4th and 5th to explain the development of the scale, and not the cycle of 5ths. This latter principle plays a role, but a smaller one than supposed. As a result of this different approach, things which were before 73

unclear will become clear, and things which could not before be explained will be explained. The two approaches will be examined and compared together. The Cycle of Fifths What is the cycle of 5ths? If, for example, you start on note F, the 5th of that note is C. Then, if you play C and after it, its 5th, which is G, and its 5th next, D, and so on, playing the 5ths of each preceding note, you will wind up with all the notes, passing first through all the notes of the pentatonic scale, then through all the notes of the diatonic scale, and finally through all the notes in the 12-tone system. The fact that each sounded 5th is Cysts of fifth*

r F#

t-5th4-5th«4-Etc-H

simply the repeating of the 2nd overtone of each note before it (which, next to the octave, is the most audible overtone), is among the reasons why many musicologists have assumed the scales to have been formed by this succession of 5ths. But there are problems about this: First, it seems unlikely that the cycle of 5ths could have played more than an incomplete role in the development of scales until stringed instruments had been invented. Wind instruments earlier than this didn't have the range necessary to make the whole cycle. Nevertheless, we know the pentatonic and other similar scales were formed in this period. Secondly, as Helmholtz complains: "Finally there is no perceptible reason in the series of Fifths why they should not be carried further, after the gaps in the diatonic scale have been supplied. Why do we not go on till we reach the chromatic scale of Semitones? To what purpose do we conclude our diatonic scale with the following singularly unequal arrangement of intervals I, I, Vz, I, I, I, Vi" (What he means by his illustration "I, I, Vz, I, I, I, Vz" is that in the diatonic scale, the distances between notes are in an arrange 74

ment of both whole-tone intervals and half-tone intervals. On the other hand, in the pentatonic scale, (T>elow) a different arrange ment of intervals is formed by the notes of that scale and is shown beneath those notes:) Helmholtz continues: "The new Intervals of Pentatonic

Intervals of Diatonic

1■l ■ JL I M 1 1'» JL Of 't O tones introduced by continuing the series of Fifths would lead to no closer intervals than those which already exist. The old scale of five tones appears to have avoided Semitones as being too close. But when two such intervals already appear in the scale, why not introduce more?" 48 Helmholtz sees no answer. (As a result of the fact that no note of the pentatonic scale forms an interval closer than a whole tone to any other note, it has often been called the "whole-tone scale.") Now, as Helmholtz asked, if there are 2 half-tones in the diatonic scale, why stop with only two? Why wasn't the cycle of 5ths continued further (making the following equal arrangement of 12 half-tones) in the same way the same people equally divided their measurements of time, distance, temperature, etc.? CTDbTDTEbTETFTGbTGTAbTATBbTBTC Vx Vi Vi Vi Vi xh Vi Vi Vi Vi Vi Vi 75

(The above is the 12-tone series or system. It is all the notes in the Western system of music, between an octave. The above begins on C, but by its nature as a set of equal divisions of the octave, it can be played beginning on any note and will sound the same. The diatonic scale, on the other hand, cannot be started on any of its notes except C and still produce the Do, Re, Mi scale. A start on any other of its notes will make a scale which sounds different from all the others (the start on A making the A-minor scale, as was shown earlieT). The Greeks as well as the Arabics and Persians, made a set of scales in this way, but most have been abandoned in Western music and sound distorted or incomplete to us.) There is a reason for the cycle of 5ths to be stopped at different stages. The reason is due to tonality. If the avoidance of half-tones was responsible for the stop made at the pentatonic scale, that alone cannot explain why a stop was made at the diatonic scale. Tonality and the Cycle of 5ths Tonality is a musical form. It is analogous to nature in many ways: For example, the seasons appear, they change, and they return. Things thrown up, come down. The sun rises, it sets. People are born weak and helpless, grow strong, and become weak and helpless again. They are born, live and return to non-existence. The rain starts, stops, and the cycle is repeated. All around us we see a starting point, departure or change, and then a form of return to the starting point. This is not universal, but it is all which men can grasp so far. It has always been difficult for people to imagine infinity, although it is recognized. There is for us always a beginning, middle and end, and the end often appears similar to the beginning. In music, to let Helmholtz describe it, tonality is "that the whole mass of tones and the connection of harmonies must stand in a close and always distinctly perceptible relation to some arbitrarily selected tonic, and that the mass of tone which forms the whole composition, must be developed from this tonic, and must finally return to it." Helmholtz adds after this, "The ancient world developed this principle in homophonic music, the modern world in harmonic music. But it is evident that this is merely an esthetic principle, not a natural law . . . The origin of such esthetic principles should not be ascribed to a natural 76

necessity. They are the invention of genius . . ."49 It would have been better to say that the ancient world discovered, not "developed," this principle, because it does appear to be the result of natural necessity, as we will see in a moment. But not thinking so, Helmholtz doesn't find its presence in the earliest development of the scale, but only in ancient forms of art-music based on the already developed scale, beginning around the time of Aristotle. Helmholtz writes: "It is indeed remarkable that though the musical writings of the Greeks often treat subtle points at great length . . . they say nothing intelligible about a relation which in our modern system stands first of all . . . The only hints to be found concerning the existence of the tonic are ... in the works of Aristotle . . . 'Why do the other tones sound badly when the tone of the middle string is altered? But if the tone of the middle string remains, and one of the others is altered, the altered one alone is spoiled? Is it because all are tuned and have a certain relation to the tone of the middle string, and the order of each is determined by that?' ... In these sentences" (quoted by Helmholtz) "the esthetic significance of the tonic, under the name 'the tone of the middle string,' is very accurately described." 50 Aristotle may have been the only one who mentions the idea and shows a level of consciousness of it, but that doesn't mean tonality wasn't already in operation for centuries before that. If tonality isn't the result of natural necessity, it must be culturally inspired. If so, what culture gave rise to it? To what culture does it "belong" (as the language of French belongs to France, etc.)? But if it belongs to all cultures, why? Back in chapter II, under the Octave 4th and 5th section, we read that the American Indians, too, had a tonal concept of notes (even if they had no Aristotle to ask why or even notice it). We know, of course, that modern cultures exhibit tonality in music as well. So did the Feudal musical system, and so do Chinese, Scottish, Arabic, African and Greek musical systems, ancient and modern. If Helmholtz hadn't passed it off as "the invention of genius" he could have explained everything, including the whole develop ment of harmony, which to him seems to have simply happened. He offers no explanation of the cause for its arrival at the particular time that it did arrive. He understands the function of harmony in music, and he often points it out in places, but, as we will see, he missed the connection which tonality has in the origin of harmony and why the diatonic scale marked the end of 77

the cycle of 5ths in the formation of scales. Not only will an understanding of tonality explain things hitherto unexplained, but the origin of tonality itself will be explained. We have read that two things determined the history and development of music, and other arts. The one we are now concerned with is that of the nature of sound, of man's senses, and their relation to each other. Within this category are two aspects which work in the shaping of this relation. They are USE - or PRACTICE - over eons of time, and TONALITY. If Helmholtz failed to see tonality in antiquity it was because the effects of it were sometimes hidden by the greater effects of Practice, or Use. By Use is not meant the various uses to which music is put, but simply the actual using of sounds, however they are used. In general, by using a thing, one becomes familiar with that thing, and slowly this thing reveals its traits, properties or qualities, and its inner relationships, to the user. They are discovered simply from enough use, just as facts are eventually uncovered by pursuing an investigation of any kind. Until the inner relation ships are clearly brought out, the forms of music (such as tonality) do not always reflect these relationships and the music appears haphazard. From this one could incorrectly conclude, as did Helmholtz, that tonality never existed at all in early periods. In antiquity, the influence of tonality in the shaping of scales was weaker than that of Use. (Use can be taken to include the cycle of 5ths, although it includes more than this.) Use of a note revealed its 5th (its 2nd overtone), and use of that 5th revealed the next 5th and so on. Tonality originates as a result of the discovery that these kinds of relations exist. The existence of these relations causes tonality to develop as a necessary method by which primitives could record and present to themselves and to each other these relations of tones. By originating as an effect of the natural relations of sound, it is proper to view tonality not as the product of culture, or "genius," etc., but as the product of natural necessity. // an art of music is a necessary result of the existence of men, then tonality is a necessary result of the existence of an art of music. In other words, use engenders tonality; but tonality, once developed, begins to overcome Use as an effective operating principle. Tonality puts an end to the cycle of 5ths at certain points by demanding that all notes relate to the tonic; that is, 78

that no new 5ths can be added to the scale unless they have a relationship of some kind to the original tonic. What kind of relationship is determined by the level of sophistication which the concept of tonality achieves at various points in history as it is developed by Use. We can see this process more concretely below step by step in the formation of the scale. The reader should note that the following process is proposed by the author as an abstraction of the real events; a generalization culled from an average of the real events. The reality may not, in one or another specific case of the music of any nation or people, exactly follow this process in all its particulars. The process is proposed as an underlying tendency which helped to determine the development of the scale. It is proposed as a main impulse, originating in nature, or acoustics; but in specific cases, other social and cultural peculiarities were also at work, however minor, and these altered the process to varying degrees. (Although it may have been distorted or retarded, the basic process, however, did take place.) "In the beginning" . . . was the octave. Also in the beginning there was no applied notion of tonality, although a similar principle may have existed in other art-forms. Each note was taken separately, as something in its own right, so to speak. Melody, if it can be called that, showed no disposition to have been formed out of recognition of definite relations of notes. Notes were as haphazardly placed together almost as if they had each been separately taken out of a hat. Gongs in the East were an example of this. We know this must have been partly true even as late as Aristotle, with whom tonality was becoming a conscious element, because the fruit of tonality - melody - was but little known. Helmholtz says of Aristotle's time, ". . . homophonic music . . . played an utterly subordinate part" (to poetry and epics). "The musical turns" (the way the melody "went") "must have entirely depended on the changing sense of the words, and could have had no independent artistic value or connection" (between notes) "without them. A peculiar melody for singing . . . throughout an epic, or . . . throughout a tragedy, would have been unsupportable," if played apart from the epic or tragedy. 51 (My emphasis.) Also, the tuning of the lyre, down to the time of Orpheus, was of an octave, 4th and 5th, "which," says Helmholtz, "certainly it was scarcely possible to construct a melody ... a lyre of this kind might possibly have served to accompany declamation." 52 79

(Declamation is the ending of a sentence or verse.) As the octave was used throughout antiquity, the ever-present louder overtones, such as the 5th (and the 4th, which is seen as a "backward" 5th) were discovered, and seen as consonant rela tions to the original note or tonic which was compounded of these overtones. As melody could not be constructed, and as other notes were still viewed as unrelated to the tonic (such as the 3rd, 6th, etc.), we had scales based on what we will call Early Tonality, or homophonic, non-melodic tonality, which means (one-note-at-a-time) (non-melodic) (notes-only-in-relation-to-thetonic) musical scales. Such a musical scale would have been exactly the notes of the above mentioned lyre. In the key of C, it would be C, F, G, and the octave C\ As a result of the emergence of the 4th, 5th and early tonality from the Use of the octave, the very first step in the cycle of 5ths was also the last one for a long time. No other notes seemed to be related to the tonic note. More use of the tonic, 4th and 5th was needed to discover the less audible, or subtle relations to the tonic. (For example, the 5th of the existing 5th was probably audible and known, but it could not be seen as related to the tonic until the first 5th was so firmly rooted to the tonic by time and association that it seemed inseparable from the tonic.) Use of the octave, 4th and 5th in time allowed all three to be seen as basic notes around which other notes could be grouped if they related to any one of them. In this way, the 5th of the 5th was eventually seen as closely related to the tonic via the first 5th. Then it was added to the scale. Here is how the scale would stand at this point:

MOTES of SCALE oveeTONES

(The 5th of the 5th, note D above, is called a 2nd, because it forms an interval of a 2nd with the tonic note. C. It also became 8o

the second note in the arrangement above, although I listed it in the order in which it was often likely included.) An interval of the 2nd (whole tone) is not very consonant, but as the 4th with the 5th already produces such an interval, the addition of the D is not a radical one. It can be seen that not only does the principle of the cycle of 5ths indicate D is next (F-C-G - D), but if we were to use the principle of Use based only on the existing first three notes taken together, and whose overtones are above written in the order of their audibility, the D stands as the loudest overtone of these three, and this principle, too, without sole reliance on the cycle, can explain the D as next (circled above). Immediately upon adding the D, observe that the 2nd over tone, or the 5th of this D, is A, which is also an overtone of the 4th, F. Soon the A (circled below) became appreciated as a tonal relation, not directly to the tonic, but to F, which is directly related to the tonic. This A forms a 6th to the tonic, and both this 6th and the 2nd appear to have come into the scale at about the same time because they are the most audible tonal relations to the tonic, 4th and 5th taken together as a "trio." What the A lacked in strength as an overtone, it made up for by the relative consonance which it formed with the tonic. All of the notes now add up to the pentatonic scale, as follows:

NOTES of SCM£

Second? RsurttiWtK:

OVERTONES c> J>>

J> T* G

K K (AJ B* fry ®

A, 1 ZV G* (nowadded)

Although these notes, the 2nd and the 6th, had weak and indirect relations to the tonic, they were not dissonant with the tonic or the other existing notes, as they are at least a whole-tone away from all of them. So far, the cycle of 5ths can also explain all the added notes, 8i

and is an important influence up to this point, although not the only influence, for it alone cannot explain why the additions stop at certain points. For example, the "next" note, E, which is the 5th of the newly added A, and which also has a direct relation to the tonic as its 4th overtone, should come next in the scale. But Lo! It doesn't, because at this point, the addition of the E will form a half-tone with the 4th, F, which, by being a strong consonance, is a "big wheel" relation to the tonic. It has priority, so to speak, and it remains while the E is left out. Besides, even if "Use" revealed the E as next in the 5ths cycle (that is, the 5th of A), it probably didn't reveal it as also related to the tonic (that is, as the 3rd of the tonic, C), and so it isn't added. Early non-melodic tonality reflected the outlook that notes must relate to the tonic only, giving us the 5th, 4th and octave. But as Use further revealed tonal relations among notes, then notes were allowed which related not only to the tonic, but to the 4th and 5th of the tonic. But not all such notes were allowed, such as the 3rd, E, but only those which did not form disson ances with the other notes. This represented another halt in the process for a long time: The Pentatonic. Tonality was further developed by Use of the pentatonic scale and became what can be called Melodic Tonality. The formation of primitive melodies, possible with the notes at command in the pentatonic scale, created a need to fill the gaps in the pentatonic scale for the sake of further melodic formations. Use provided the notes for these gaps as it finally revealed relations between the existing notes. As we will see, these "missing" notes came to be looked upon not as individual dissonances with one or two notes in the scale, but as melodic steps of a tonal character grouped around all three, the tonic, 4th and 5th. This newer and later concept of melodic tonality demanded a note between the Re and Fa (between the D and F in the pentatonic C scale). The gap was too big for the increasingly melodic musician. (Such a note appears to be caught in the act of being added to the Scottish pentatonic, where it sometimes appears and is sometimes avoided.) What note will fill the gap? The Eb? Something between the E and Eb? We know that the discovery of the third note in our scale was hesitant: The "history of musical systems shows that there was much and long hesitation as to the tuning of the Thirds," wrote Helmholtz.53 82

The E stands high on the list of candidates for the third note in the scale, if we review our table. It is an overtone of the tonic, as well as of the 6th, A, and it forms various consonances with the other notes. But what about the half-tone it will form with the 4th, F? Well, that is overcome by the growing view of the note as a melodic step to the 4th. Any note in the gap would be viewed as such, but the E is finally settled upon because Use reveals over a period of time its above additional and subtle overtone relation to the tonic, as well as its more obvious over tone relation as the 5th of A, in the cycle. As an overtone, E appears twice in the pentatonic scale, and forms consonances with the other notes, except for the 4th. Melodicism thus over comes) the half-tone and makes it acceptable by considering it a note in a melodic sequence. The E is then added, and likewise, the B, which fills the gap at the other end of the scale. Finally we have, with the addition of the major 3rd and the major 7th, the full diatonic scale. If here, the only operating principle was the cycle of 5ths, these notes would not have been the last added. But a new halt is called to the cycle of 5ths, and the reason is tonality, or melo dicism. Even though other notes may be added as melodic steps to some of the other notes, such as an F# as a step to the 5th, no other notes which can be so added have any audible overtone relation to the tonic, 4th or 5th. (For example, the last note added, B, is an overtone of the 5th, G.) Helmholtz understands only part of the reason for the addition of only these notes in the scale when he explains the concept of the "leading note" which is the newly added B in the scale. ". . . the interval of a Semitone plays a peculiar part as the introduction ... to another note . . . "Hence the major Seventh" (B) "in its character of leading note to the tonic" (C) "acquires a new and closer relationship to it, unattainable by the minor Seventh" (Bb). "And in this way the note which is most distantly related to the tonic becomes peculiarly valuable in the scale. This circumstance has continually grown in importance in modern music, which aims at referring every tone to the tonic in the clearest possible manner; and hence, in ascending passages going to the tonic, a preference has been given to the major Seventh in all modern keys, even in those to which it did not properly belong. This transformation appears to have begun in Europe during the period of polyphonic music, »3

but not in part songs only, for we find it also in the homophonic" (music without harmonies) "Cantus Firmus of the Roman Catholic Church."54 (My emphasis.) The "preference" Helmholtz wrote of meant that singers made a habit of writing and singing a sharpened major 7th, all the more to show its relation as "herald" of the tonic. Even when Pope John XXII in 1322 made an edict against this sharpening, and music writers could no longer indicate it in the written music, singers persisted in supplying the sharpness. Today it is still a practice. The keys in which this leading note found itself and where "it did not properly belong" is, for one, the minor key. When, in that key, the minor 7th (Bb) is replaced with the major 7th (B) as leading note, it forms what is called the "harmonic" minor scale, and is very much more beautiful than the ordinary minor scale with the Bb as leading note:

Eb

Ab

To sum up: The overcoming of the existence of semitones in forming the diatonic scale from the pentatonic scale is rooted historically in the increasing concept of tonality and melodicism. This stemmed from ages of Use of the pentatonic scale. In the same manner as Use of the tonic revealed the existence of the 5th, Use of the pentatonic scale revealed all the notes related to the tonic, 4th and 5th, and gave rise to melodicism, which allowed the weaker relations of notes to find their place as strong melodic consonances, if not as strong individual consonances, of the tonic, 4th and 5th. The semitone was overcome and modern tonality, or Melodicism was born. A look at the cycle of 5ths when compared to a list of the overtones of the tonic, 4th and 5th below, shows the areas where the two principles, the cycle of s4

5ths and tonality, reinforce each other and where they conflict:

ONEBTONSS of TONIC, 4tK} Jth,:

F-C-G-D-A-B-B

Crete of piFms*. F-C-G-D-A-E-B "(H£ K°7

In the early period of the formation of scales, the principle of the cycle of 5ths played an important role, and represented the easiest discoveries led to by Use. These are the most audible overtones (5ths) of each preceding note (that is, most audible except for the octaves of those preceding notes). More Use of these discoveries led to further discoveries of additional, although more subtle, relations among these notes. This bred tonality as a means to integrate all the relations into one whole. Tonality eliminated all those notes not able to be unified with the others. If still another note was to be added to the diatonic scale, not stopping after 2 half-tones were in it, melodicism, the highest expression of tonality in non-harmonic music, would demand the addition of only a note which had both: a tonal relation to the tonic, 4th or 5th, and which had a melodic usefulness. Further Use would not reveal any more such relationships, with the exception of the very weakest overtones of our "trio," which would be Bb and Eb (circled above). But these had no melodic usefulness. The only other audible overtones which could be discovered by further Use would be those next in the cycle of 5ths, the F# and C# (circled above), but these had no tonal (overtone) relations to any strong notes in the scale; and are, in fact, dissonant: The F# is the audible 5th of the 7th (B), the leading note in the scale. But the F# is not related to any other note besides this weak B in the scale, and forms a dissonance with most of them, especially with all the strong and most tonally related notes, the tonic, 4th and 5th. The lone quality of the F#, that of possibly serving as a "leading note" to the 5th, was not enough to justify its addition. Not consistent with tonality, it is therefore left out, as is the C# (a "downward" leading note), and tonality stops the 85

cycle of 5ths. As a result, Use wears itself out, so to speak, after being earlier, through its vehicle of the cycle of 5ths, the strong influence. The strong influence remaining to develop music (toward harmony) is the highly developed tonality. The reason why the Eb and E, or the Bb and B, alternate with each other, forming two scales, major and minor, instead of both being included together all in one grand scale, is because the Eb and Bb, if included with the E and B, would form dissonant intervals with each other in the scale, but the E and B alone (or the Eb and Bb alone) form dissonant intervals with few notes. Also, the addition of all four notes would make an unmelodic scale preponderant with half-tones: CT-Dr-EbT-ET-F-r-G-pAT-Bb-rB-rC 1, V2, V2, Vi, 1, 1, V2, V2, Vi The very fact minor scales are formed by the alteration of the weakest overtones (E and B) of the tonic, 4th and 5th, and not around the next two notes (F# and C#) of the cycle of 5ths, is further proof that the scale is based primarily on the tonic, 4th and 5th and not only on the cycle of 5ths. Regarding the cycle of 5ths, as each note is presumed by Helmholtz to have been added to the scale by this same method each time, then each should have equal value in the scale, as indeed, have all the notes of the 12-tone series, which is the full outcome of the cycle (See illustration of this just prior to this section). If all the notes were equally derived, then the diatonic scale would not have been historically seen as having "strong" and "weak" notes in it. But the historic fact that the scale does exhibit strong and weak notes (as shown by the vacillations in the tuning of the 3rd mentioned earlier) must be explained. Therefore, to view the scale basically as the addition of the over tones of the tonic, 4th and 5th, does explain the relative values of strength that are and have been attached to notes in the scale as they were added, and explains why these additions stopped at certain points. In addition, notice that the overtones of just the tonic, C, which are, in the order of their audibility, C, G, E, Bb, came into the scale in this same order. (The Bb never entered the major scale, but it came into the minor scale.) Again, if all the notes in the scale were really equally formed, then the chords necessary to harmonize the notes of the scale would be major chords, each one derived from the note it is supposed to harmonize with. 86

For Example:

The Scale: c (C Maj) C £ G

(D Maj) (B Maj) D E F# G# A B

(G Maj) (A Maj) G A B D £

(B Maj) B D# F#

(P Maj) F A C

(C Maj) C £ G

(The chords named in brackets are those derived from the notes of the scale (shown above the chords). Below the brackets are the notes which make up each chord.) ~ This harmonization can be heard to be unsatisfactory when compared to the harmonization below, where the scale is harmonized only by chords of the tonic, 4th and 5th. This second harmonization reflects the fact that all the notes in the scale are derived from the tonic, 4th and 5th. C D B (C Maj) (G Maj) (C Maj) C C G £ B B G G D

F CP Maj> F A C

CG Maj) G B D

CC Maj) C 8 G

(G Maj) G B D

CP Maj) F A C

The method of harmonization in the first example above has

also been rejected historically as we will see in the chapter on harmony. Let's remember that the latter, modern system was developed not by people who wished to have all the aspects of this system neatly fit a preconceived theory of music, but by people without, for the most part, conscious knowledge of what they were doing. By not seeing tonality and melodicism in operation from the beginning, Helmholtz took the notes of the scale as all equally formed by the cycle of 5ths, and consequently could not under stand why the cycle stopped at the diatonic scale, after half-tones had once been included. Seeing the scale as an unequal arrange ment of whole-tones and half-tones which have "no reason" to be that way, explains why the scale appeared to Helmholtz as arbitrary, as a product of Western culture, or "genius," with no foundation in any natural laws or necessity. Finally, let me summarize the three stages of the development of the scale and modern tonality, or melodicism: Early tonality: Notes must relate to tonic only. Melodic tonality: Notes must relate to tonic, 4th and 5th, except those which form half-tones with other notes in the scale. Melodicism: Notes must relate to tonic, 4th and 5th, but half tone dissonances are allowed when they contrib ute to the overall melodic unity of the scale. The Fourth and Cadence If a man who lives in a room upstairs from us decides to go to bed, the first thing he will do is take off his shoes. When one of them is off, he drops it, and we hear it downstairs in our room. If he never drops the second one, some of us might not sleep for the want of the second shoe to fall. In the same way, should a capricious pianist, just before the final notes of an exciting performance, decide to forsake these final notes, stand up and take his bow early, and then retire to his dressing room, he might find a few dozen people from his audience waiting for him in the alley of his stage exit, prepared to realize the unfinished notes upon his nose. Both examples above are of frustration resulting from anticipa tion and expectation having been betrayed. In the case of the missing shoe, no one can deny that expectation of two of them falling is what causes frustration when only one of them does 88

fall. But suppose the person downstairs is not ourselves, but an Australian Aborigine. Obviously, not a wearer of shoes himself, he won't expect two of them to fall, nor even one. It is because we know that shoes come in sets of two that we expect two. In other words, our anticipation and expectation is culturally conditioned. It is a learned response for us to expect to hear the second shoe fall. How about the case of the capricious pianist? Is the source of disappointment due to the fact that certain standard cadences (final notes) are prevalent in Western music, and by this, we are taught to expect them, and irked when they don't occur? It might seem so. But if we examine the cadence, the 4th, we will see it is endowed by nature with physical qualities which make it very suitable for being a cadence, and all the other questions raised before about the 4th will be answered too. First of all, the idea of "cadence" is "a coming to rest" in music. Many intervals, such as the 3rd or 5th, etc., are examples of motion away, or separation, from the tonic. So is the 4th, but it will be seen that the 4th represents the least motion away from the tonic, and because of tonality, is able to represent "close ness" to the tonic; a "return" to the tonic. Let's compare two intervals, the 5th and the 4th: 1 . ) If we play C and then G, we make the interval of the 5th. We are also going from C to one of its overtones, G. C: Overtones are C, G, B, Bb When, after playing the C, we play G, then the overtones of G become audible too, and are different (except for G itself) from all the overtones of C: C: Overtones are C, G, E, Bb G: Overtones are G, D, B, F Because G's overtones are all different from those of C, a certain sense of dissonance, or motion, or conflict, sets in. (Of course G forms with C a powerful consonance, but the only perfect consonance, without any dissonance at all, is silence.) This conflict, however weak, is nevertheless the beginning of movement away from the first note, C. 2. ) If we reverse the interval in example 1., and go from G down to C (or G to C above - it's no different for our illustra tion), it follows that the overtone relationships are also the reverse of those in example 1 : G: Overtones are G, D, B, F C: Overtones are C, G, E, Bb 89

By reversing the interval we are also making a 4th, and not a 5th. It would appear that the order in which the notes are played should make no difference. But there is a difference, and it is this: In example 1., starting on C, we see that in the aftersounded G, no overtone C is present to help us remember the prior C. All effects of the C are blotted out when we hit G. (Of course, we heard the G in the overtones of the first-played C, and this indicates the relationship between that C and the G follow ing.) But once the G is played, we feel we have left the C behind.

tA STILL WITH YOU, FKi£NP C, YOUCKN'T 6ET VEKYjj fMS FROM ME.!' /s\

, WT KEEP FOKMER.

OVERTONE Of* C

4tF 5th (ex.*)

By comparing to example 2, we can hear the difference: In example 2, the reverse is true. By starting on G, although we hear no C in advance of playing it, as no overtone of G is a C, when we do hit the C next, we are still able to hear, in C's overtones, the first G. (See example 2 above.) The G is "carried along" with us as we go from G to C, so to speak, and we can feel that we have not left the G behind in example 2 to as great a degree as we left the C behind in example 1 . The interval of the 4th (G to C), therefore, has less separation from the starting note than does the interval of the 5th (C to G). The 4th then, can be said to be more restful than the 5th, because it has less conflict than the„5th. Further, the interval of the 4th can be said to have a smaller separation to the second note of its interval than does the 5 th, because, by its overtones, we have seen that it "carries" the starting note with it. Whatever interval represents less separation from its starting note would, accordingly, represent less separa tion to its second note. Which side of the above relation is more important to view in 9°

any given instance depends upon whether the first or second note of an interval is the starting note, or tonic. (Usually the first note in an isolated interval is taken to be the tonic, but within a musical context, either note of the interval may, by that context, be the tonic.) Looking at the intervals below, let's assume the first note of both intervals is the tonic. C to G (5th) G to C (4th) If the first note of both is the tonic, then the C to G (5th) represents a greater separation from the tonic (as we have shown by the overtone structure of this interval) than the 4th., If the second note of both intervals is the tonic, then the C to G (5th) represents a greater separation, or distance, to the tonic than the 4th, because in the 4th, G to C, the first-played G is found again in the following C and in this sense only moves - to itself. (Of course, in C to G (5th), the G in the overtones of the first-played C is also moving to itself in the next played G, but this is less audible, less apparent, than in the 4th, where the first G is a played note and not just an overtone.) Therefore, the interval of the 4th, regardless of which note is tonic, is by its physical nature, absolutely suited as a restful, or cadence-like interval. Whether going to its second note, or from its first note, the interval expresses least separation, and so both its notes appear close to each other. A number of questions may have entered the reader's mind and anticipating some of them, I'll take them up here. First question: Why, when in all the preceding pages of the book where I use C to F as an example of the 4th, do I now use G to C as an example? The answer is that what is true of one 4th is true of them all, whether C-F, G-C, E-A, etc. But for the purpose of illustrating the relationship between example 1 and 2, it was better to keep the same two notes because this shows that there is a difference made by the order in which 2 notes are played. This also answers the question raised at the beginning of the chapter why intervals should be reckoned upward. The interval of C to the F below it (or above it), in that order, has all the qualities and relationships which sound like a 4th; and even though C to F below is a distance of 5 notes, it is nevertheless properly called (in isolation) a 4th, or "cadence." (Likewise, the reverse interval of F to the C below it, should be called a 5th, even though it is 4 notes wide.) The order in which notes are 91

played is more important than the actual number of notes between them. There is a little more to say about the question of reckoning music upward and downward, but of that later. We can now transpose the 4th used in the above examples (G to C) and say that the 4th note and interval in the scale, C to F, is a more restful, cadence-like interval than the 5th in the scale, C to G. (Play these two and the sound difference between them is even more apparent, in relation to the tonic, C.) In addition, the above approach of using the same two notes to illustrate the difference between the 4th and 5th also illus trates what probably happened in history concerning the origin of the 4th. I believe that the 4th was discovered through use of the sequence below:

p-jtti-jj— 4^ —1 Cupte G,do*mtoC (ovuptoC*) This sequence is a combination of a 5th and then 4th. After this first historical step came the 4th which entered the scale, C-F. This took place in a second historical step which may have happened as follows: The C-G-C' sequence gave the primitive a 4th in the last two notes of this sequence. As said, this was not the 4th (C-F) which we know in the scale. But when he heard this latter 4th, among the general mass of tones and sounds which he and others made, he soon recognized in it a similar relationship of notes which he knew in G-C, and he began to take special notice of it too, because the F also had a C as an overtone. This 4th, C-F, has the characteristic of having the same first note as the already discovered 5th, C to G. In the first case, G-C, this 4th is a return to the tonic in the above sequence. In the second case, C-F, the 4th, by beginning on the tonic, provides a new, related note, (F), one which "leaves" the tonic, but less than does the 5th, C-G. As both 4ths are consonant, both came to be used, and finally the 4th came into the scale. All this, of course, probably took a zillion years. Second question: Why do I, in example 1 and 2, earlier, allow myself to switch tonics around? How is a tonic determined? Isn't it always the starting note? If not, how does a musical context determine it? 92

Often the tonic is a starting note, but more accurately, it should be said that it is primary in a piece, which means that it can be other than a starting note, without losing its basic importance. How the tonic is formed is clearly evident in an analogy to rhythm. Let's take a given rhythm: (7, 2, 3, 4, _/, 2, 3, 4,) etc., with the heavy accent on the first beat, as underlined above. If we were to tap out this rhythm a few times, we would find our attention drawn to the first beat. Then, if we begin to omit this beat, that is, (- 2, 3, 4, - 2, 3, 4,) the silence in place of this heavy beat in no way detracts from our attention to it. The other beats, 2, 3, and 4, by the way in which they define the empty silence, keep us aware of the missing beat. The first, heavy, or "missing" beat is analogous to the tonic in a piece of music. Whether it is actually played, or started upon, or whether its existence is only known through the behavior of other notes, it is, in tonal music, always defined and an ever-present point of reference. In a piece of music, the tonic is either being empha sized, as at the end of a piece, or obscured, as in the middle. The cadence, which here interests us, comes at the end. What matters in our comparison of intervals is not which note of the interval may be the tonic, but which, of all the intervals, has the greatest effect in "coming home" to the tonic at the end of a piece. The 4th is better used to define the tonic than any other interval, historically, and for the physical reasons I have attempted to make clear. The tonic, or any note, is not simple, but compound. It is made up of overtones, etc. If a piece of music is in motion away from its tonic, any of the notes, which as overtones comprise the tonic, can well serve as graduated links via which to leave that tonic "behind." As these "exit" notes are played, they in turn provide their own overtones and more links by which to vary from the tonic. (Including the use of rhythm, harmony, etc., the possible ways to vary from the tonic are virtually infinite.) But at some point, according to the historically developed esthetic of tonality, you must return to the tonic. If the over tones of a tonic are part of those notes which can serve as related links by which to leave the tonic, it seems reasonable that the return to the tonic would be the same process in reverse. And it is just that way, in general. When returning to the tonic, in fact, the tonic note is rebuilt by the use of notes representing some of the overtones of which the tonic is compounded. In the return, 93

the closest one can get to the tonic without actually playing it is the note of which the tonic forms a 4th. If the tonic is C, then that note is G (G to C is a 4th), which represents the most audible overtone of the tonic besides the octave of the tonic itself. In such a musical "return," the G, when heard, makes the tonic push its way into our consciousness. We are always aware of the tonic as a reference point throughout the music, but the G represents the closest of a series of notes which are aiming to bring that tonic most to the fore. Finally it is played, and the sequence comes to an end. Again, the ways in which this can be, and has been, done in music are limitless. The above is a super-simplified version of what is happening in the process of forming a cadence. Harmony, which has made the physical relations within the 4th much more apparent, is responsible for the almost universal use of the 4th as cadence in Western music. It is difficult for us to take any other interval for a cadence. In harmony, the 4th as cadence is done as follows: G to C is harmonized respectively by the chords of G and then of C. To give the expression greater power and clarity, add an F to the chord of G (G, D, B, F) and the whole chord leads more urgently to the following C chord (C, E, G). This is because the F produces a loud C as an overtone and forecasts the ensuing C. Without the P

With the P

Not only is the G "carried along" when the C is finally played, as has been shown earlier, but adding the F makes a C overtone (of that F) which also forecasts the tonic and the C, too, is like the G, "carried along," as if from a vague, dark corner in the Q chord, into the light, when the C chord is played. In primitive music, in which no harmony existed, the relations 94

of the 4th are less apparent and it is less directly used as a cadence. But it nevertheless has been used immensely as a cadence. Below is an example of a pentatonic folk-melody which, when we examine it, will show the 4th as cadence, and which also illustrates the variety and capacity for creativity which can be used to form a cadence. First, the melody, without any harmony, as it was meant:

f2_ f—| 1

^

Now the ending again, and in the bass I have added a harmony: (The three notes I have added would, together, make a D-minor chord, which, by harmonizing well with the melody, shows how the notes of the melody maintain reference to its tonic, D.) ^+ ■

The last four notes are the cadence. The first two of these, 95

added to by the harmony of A beneath, are two notes which are part of an A-minor chord, which provides the start of the 4th. The D following, the tonic, (D-minor, actually), makes a 4th to the A-minor before it. And, even though indirect, the 4th is nevertheless present here as a cadence. This type of ending cannot be harmonized any other way without destroying the mood and integrity of the melody. It is a typical ending in many of the early folk-songs of Scottish music. One last point on the upward or downward effects of music. In musical terminology, intervals are calculated upwards, because the first known intervals came from the overtone series, which goes upward from the original note. Tonality expresses this too. Motion away from the tonic tends in musical works to lead upwards, and the ending, or return to the tonic, tends to be expressed downward, and intervals are therefore reckoned in "reverse," in this return. Aristotle asked, in his Problems, "Why is it more convenient to sing from high to low than from low to high? ... is it because the low note after the high one is nobler and more harmonious?"55 Bauer and Peyser report: "When the" (American) "Indian sings, he starts on the highest tone he can reach and gradually drops to the lowest."56 And Smith, discussing the cycle of 5ths, says, "Moreover, our modern method of counting from the low note upwards seems to be an inversion of the more primitive method, which proceeded from above downward."57 Of course, in the cycle of 5ths, we count upward, but in most Western musical compositions, we are like our predecessors. Our music ends most often on notes lower than the general range of the piece of music. Examples of this are legion. Why is landing on low notes from high notes an expression of tonality and cadence (or "return") in American Indian music or Greek music, if they don't start on low notes first? It is true, in the above examples, the start is not on the tonic, or low notes, but already on the high notes.One reason for this is that the tonic came to be "understood" before starting on the high notes and returning down.* This is much the same as words in language are *The use of this technique by Beethoven is done in the first part of his 9th Symphony, only here a whole theme is worked toward, instead of just a tonic, and is done before the theme has even been played once. He puts together bits and pieces of the 96

"understood" without being spoken. For example, the title of this book is "The Universality of Music." But the words: "The title of this book is" are all understood when you look at the cover. Years of tonal practice can bring the same thing about in music. Another reason you can "return" to a tonic which has not been started upon first is because the ear, upon the playing of combinations of high notes, will produce, by the very inner construction of the ear itself, the lower tones; will produce the tonic in instances, and "call for it," so to speak, after high notes. The tones so produced by the ear are called by Helmholtz, "Summation tones" and "Difference tones." But these are produced only by combinations of higher notes, and so, strengthens the cadence feature of lower notes only in Western harmonic music.

theme to come, as well as hinting at its key, so that one feels like a land-hungry sailor aboard ship who sees bits of wood and debris, flotsam and jetsam, floating in the water, which signifies the whole land mass of which these bits are a part. Below is the piano score for the introduction of the first movement of Beethoven's ninth symphony. The ominous entrance of the low D in the 5th measure tells you that this is the key of the impending theme, which, if one hears the piece, will be seen to be accurately described by Meyer: ". . . the music moves on without pause, without pity, to its stark and awful declaration."58

97

Harmony

Jeans says, about the evolution of harmony from homophonic music, "ridiculous though it seems," (it) "remains one of the unsolved problems of music. "59 Since 1937, when Jeans wrote the above lines, there have been no notable advances on Helmholtz's theory of harmony, which he admitted was incomplete. It is the purpose of the following theory to completely explain the development of harmony with out sacrificing the ability to explain other facts of music. Old and new cultural and psychological theories, whatever else they can explain, are unable to explain even the Octave, 4th and 5th, as I have shown. The reason why harmony has not been explained even by the best of theoreticians is due to a number of errors. One error is the belief that the scale was formed only by the cycle of 5ths. This error has been avoided in this theory, replaced by the concept of the "trio" — tonic, 5th and 4th - as the main impulse. Another error was the failure to appreciate the change in musicians' concepts which were due to each new musical discovery and the ensuing "Use" of each discovery. The main change necessary to harmonic development was from the 98

position of "melodic tonality" (which allowed melodic steps between the Octave, 4th and 5th, but allowed only those steps which formed no dissonances with any other notes) to "melodicism" (which allows all tonally related steps between the Octave, 4th and 5th even though some are dissonant to each other). The concept necessary to melodicism is willingness to view the whole as more important than any of its parts. Finally, harmony remained unexplained because of lack of understanding of the reason why early attempts at harmony failed. This is a judgement based on admissions by early writers thruout the centuries that their own music was less than satis factory. Concrete evidence of this will be seen as we review the development. We wili examine each of the main items more or less separ ately. "More or less," because these currents overlap and are interdependent.

Changing Concepts of Tonality The first of the preconditions for harmony to develop is the change in the concept of music. Curt Sachs gives an example of the earlier concept which we have called early tonality: "The single note actually counted for more than melody; chimes, numerous in all kinds of" (primitive) "orchestras, were merely sets of single stones, metal slabs, or bells, united in one frame, it is true, but not in any scale arrangement . . . Cosmological connotations were given individual notes, not, as in the West, to melodic patterns. And notation consisted in separate pitch symbols." 60 This did not mean that interest in scales was altogether lacking. Sachs assures us that "the opposite is true." However, the single note, up until Greek times, counted for more than melodic sets of notes. The change to when melody began to count for more than single notes was due to years of "Use" of existing music, which revealed that many actual relationships existed between different notes. Each new awareness of these relationships in turn increased the tonalism of the musician. What this change in concepts has to do with the development of harmony is this: A chord, however consonant, is always more dissonant than a single note. The reason why a chord is more dissonant than a single note is 99

this: A single note produces overtones which are audibly consonant with one another, although we don't hear them consciously. When all these overtones are played out loud in a chord, even though the result is bound to be consonant, there is an added element of dissonance because each of these additional tones in turn produces its own overtones, many of which are not consonant with each other. These elements can be easily heard upon comparing a chord to a single tone. Only the highly melodic musician can overlook the dissonant elements in a major chord. He does this because his demand for melody and tonality needs the tensions, dissonance or unrest that a chord has, and which it has more powerfully than a single note. These tensions are like hooks, which catch hold of notes before and after them, helping to relate these notes to each other melodically. This satisfies the melodic musician and at the same time makes the chord acceptable in the light of this use to which it is put. This is the function of harmony, as it historically developed. The primitive musician didn't find any chords consonant or useful to him compared to his single notes. (This is assuming he may even have stumbled across these chords.) Neither did he have the knowledge of, nor the ability to make, all other chords, whose relationships to even the most consonant of chords, in a tonal or melodic pattern, could make any of these chords acceptable and meaningful to him. An analogy might help here. I hope all my readers have seen motion pictures. We have no doubt experienced, at one time or another in a movie show, a part when the projectionist has to bring the picture into focus. Imagine the picture-in-focus as the consonant major chord. Imagine the picture-coming-into-focus-but-not-yet-in-focus as another chord, more dissonant than the major chord, but related to it. Both the second, more dissonant chord and the picture out of focus, have qualities in common. One is that they reveal what they are an imperfect example of: The basic elements of what you are trying to see on the movie screen are still on the screen even though distorted and out of focus. You see what is "almost" there, and you want it to be there. Similarly, our second chord, even though, on one hand, it is a distorted, dis sonant form of our first chord, on the other hand, it has the capacity to reveal and keep us aware of that first chord. This two-sidedness of the second chord gives it the appearance of 100

being in "motion." This creates the desire for the completion or resolution of that motion, just as we desire a rapid and full arrival of focus of the elements on the screen. It seems clear too, that if the chord is too dissonant, or the movie too much out of focus, we lose our perception of just what the thing is that is out of focus, or out of "harmony." Without that perception, we don't know what we want of the blur on the screen nor of the dissonant chord. We don't know, because "motion" is lost for lack of apparent destination by which to measure the motion. We are "repelled" instead of "impelled." (Of course, the analogy should not be taken as perfect. We go to the movies to see the story and characters in it, not to enjoy the abstract pleasures of watching the film get into focus. How ever, it is just this abstract element of coming-into-harmony that we seek out and enjoy in music for itself. The reason why the same is not possible in movies is twofold. If the movie were abstract (in order not to distract us with some story or character) we would't know if it were out of focus or not, only our knowledgeof real things being the measure of that. Secondly, both,the physiological capacities of shapes and colors,andthe number of ways to "get into focus'/ are more limited than the number of ways to resolve harmonies in music. That's why several such attempts at "abstract" movies have failed even to approach the popularity of musical concerts.) But to return to the analogy : The pleasure we get when a chord resolves into harmony, similar to the relief we feel when the movie image "resolves" into focus, indicates that neither the unfocus of the film nor the dissonant elements of the chord can be meaningful in themselves. They can take on meaning only because they are ultimately resolved. Vice versa, the thing they resolve to becomes more meaningful because of the prior dissonance or unfocus. These serve as a created frustration, which by being resolved, provide relief. This seems to be the principle about which the esthetic of harmony and melody developed, historically. As shown above, however, this principle cannot be arbitrarily applied to just any art. Both the artistically created need and the fulfillment of that need are possible in our diatonic harmony system. The consonant major chord, formerly rejected by primitives (because the method by which to resolve to that chord was not known, and because it therefore stood in unfavorable dissonant contrast to 101

single tones), later takes on value for the more developed melodic musician: All of the various types of chords can be put into a sort of "spectrum" of chords, growing one out of another, and varying by degrees of consonance and dissonance. In this spectrum some types of chords are the most consonant. These although in relation to single notes are dissonant - in relation to the spectrum are acceptable as a new, although more complex, form of single "note" or "pure" consonance. Below are examples of the process of harmony. First, a musical example:

$ 9

1 rw—

n

The chord shown above is an example of a very dissonant chord. So dissonant, it may seem that it has no quality or side to it that shows any relationship of notes. But the notes do have a relationship to each other, or would, if a proper context brought them out. Supplying the context below, we see that our chord, marked (*), shows that the relations within it were ones in

suspended motion, which are here continued and completed: (The example also shows that sound and the art of music based on it may have qualitative differences from arts based on other senses. No disagreeable smell or taste can be made more 102

acceptable by any added context of smells or tastes. Color, in its way, doesn't seem to have any shade which can be considered disagreeable to begin with. But I am getting ahead of myself. This question can only be touched upon here.) Secondly^- a graphic example of the process of harmony is shown below, illustrating that what is taken for granted (focus), becomes a downright pleasure when it is challenged.

^It^Nor

music

Without the spectrum of chords necessary to justify the simplest of chords, there can be no role for the consonant chord. What is this spectrum of chords? The most noticeable lead in the search for an answer is what music has become today: All the popular music, folk songs, and literally millions of other types of music, from classical to jazz, illustrate one glaring phenomenon common to almost all of them today: The music is basically (not entirely) harmonized by chords of the tonic, 4th and 5th. In the last chapter we learned that these same three notes produced a combined list of overtones which produce the notes of the scale. As the scale is a reflection of these overtones, we can say that the scale is tonally organized, that is, organized in reference to a tonic. As most melodies are formed from scales, especially major or minor, we can say that the scale itself repre sents the first melody (speaking only in formal terms. The scale was really developed from melodies and later codified as an abstraction which represented all melody.) Therefore, to harmonize this "scale—melody," if we were to invent a harmony system for it, we would assign to each note a chord which reflects that note's role in the scale (relative to its tonic). The tonic, then, would get its own chord. The next note (which would be D, in the scale of C) derived from the 5th of the tonic, G, and it would get the chord based on the 5th. The 3rd, which derives from an overtone of the tonic again, would get the tonic chord again, etc. (See last chapter, under "Completion of the Scale" for the diagram illustrating the derivation of the notes of the scale.) When we look at our invention, we see that we have perfectly matched the existing phenomenon of harmony in so much of today's music. In fact, we have matched that of music for several hundreds of years. (It is true that most art-music is more 103 /

f

harmonically complex than this, but later on in this chapter, as well as in the "Summary" in Appendix II, the reader can learn that our basic scheme is not contradicted by the historic use of a variety of other chords as well as those of the tonic, 4th and 5th.) The condition which impelled the scale, then (tonic, 4th and 5th), is seen to also be reflected in harmony. We can see that it does this, but the problem is to understand how this process took place, considering that harmony was evolved, not invented, and evolved by musicians with no "theory," as presented here, to guide them. Understanding the change in musical thought, historically, from non-melodicism to melodicism, may explain why the melodic musician can appreciate harmony once he has it, for harmony serves to strengthen the tonal relations of notes in melody. But changes in concepts alone do not tell us how it was discovered. Such a harmony system would not be "obvious" to any ancient musician, as we will see next. (With the wrong theories, a search into history to trace the evolution has proved frustrating for many researchers. But, now that we see the tonic, 4th and 5th is everywhere to be the core of the matter, like red permeates the colors of the sunset evening, our search again into history promises to be more rewarding. We know what we're looking for.) Max Weber tells us that "no evolution toward harmonic music could have begun ... if other conditions, especially pure diatonicism as a basis of the tone system of art music, had not . . . existed." 6* In other words, only after the diatonic scale had been formed and used for a long time could harmony have developed. Why is this true? Why was the development necessarily prevented from occurring earlier? Why couldn't the primitive musician have stumbled across chords just by the laws of chance, without the big process outlined above, and have recognized or learned, from these chords themselves, the role that chords could play, and also have learned from them to earlier have become more tonally or melodically-minded? The chords necessary to justify the use of even the most consonant major chord, against the use of more consonant single tones, which were readily available, needed all 8 diatonic tones to form them. All these notes may have. existed in vaguely defined form at a very early period. Couldn't a thousand years of music have accidentally and occasionally produced some of these notes in just the right combinations to form the needed chords? 104

The answer is that not only these 8 tones, but many others just as vaguely defined were in use too. From such a world of notes the possible combinations are limitless. The permutations are so great that even the laws of chance would be hard put to have provided by accident even a couple of consecutive chords in just the right transpositions necessary to reveal the possibility of a workable harmony system. (Most other progressions have proved, historically, to be relatively unsatisfactory as we will see later.) Secondly, to find such chords, or pay attention to them, one must first be disposed to think they can be found, or that study of them, if they are found, would reveal something. Even if this was thought, a deliberate process of elimination, of trial and error, to find such a system of chords (even from among only 8 well defined notes, not to mention the dozens which existed then in a poorly defined state) would have been more than two lifetimes of work. Again, if the primitive had all the above lucky happenstances, including two lifetimes in which to work at it, such events would have had to happen many times before the significance of them could be felt in the development of music. No "genius" can be responsible for a thing, unless that genius is a reflection of a commonly-inspired development. (And in general, no commonly inspired developments can have more than temporal and local meaning without being backed up by natural, or at least techno logical forces, which continue to exist even as social conditions change.) Finally, the lone major chord itself cannot be too easily assumed to have been known of by primitives. The only major chord that can be formed from the tonic of the pentatonic scale would have been (in the key of C) C, F, and A - which is really an F chord - but in an upside-down, more dissonant form: the tonic of this chord itself, F, is not the bottom, most audible note. As the rise of the diatonic scale from the pentatonic scale took place at the same time as the change from melodic tonality to melodicism (which is our first precondition for harmony), then the musical difference between these two scales may provide us with a clue about how harmony was discovered by the more melodically-oriented musicians. The difference between the 5-note pentatonic scale and the 7-note diatonic scale is the 3rd and 7th (Mi and Ti in the Do, Re, Mi scale). Therefore, let's look at these, especially the 3rd. 105

The 3rd - How It Spurred the Discovery of Harmony The 3rd was allowed into the scale because its relation to the tonic was realized by the principle of Use, as I mentioned earlier. (Long after it was in the scale it was later officially declared by the Church to be a strong consonance as well as a melodic step.) The initial dissonance of the 3rd with the 4th, the note after it, and with the 2nd, the note before it in the scale, was overlooked by the growing melodicism of the time. However, it became, first, only a "consonance" as part of a successive melodic phrase - C, D, E, F (Do, Re, Mi, Fa) or Tonic, 2nd, 3rd and 4th - before it became a consonance in harmony with the tonic. (The 7th was also viewed as a melodic step. But it never became a consonance to the tonic, because it is only a half-tone away from the octave of the tonic. This fact made it a special melodic step however, as "leading note," shown in the preceding chapter) When stringed instruments were developed, it was possible to produce lower notes than before, and to do it with a higher degree of quality than those which could be made on a primitive drum. It was then possible for the 3rd, accepted as a melodic step in the scale, to eventually become more of an isolated harmonic consonance of importance too, for, as is shown in the section on "Relativity of Consonances" in Chapter I, when the notes of the 3rd are played as a 10th (by using a tonic an octave lower) they are more consonant - more so than the 4th. (If any musician had ever heard this 10th earlier, he was never able to hold onto it, its production probably occurring occasionally when a drum beat hit the low tonic, and a singer or flute, by chance, the 10th. It seems to be a law of development that things must be discovered over and over again in primitive times for them to be considered discovered.) At any rate, only hundreds of years after the 3rd was included in the Greek scale, was man able to get a good enough look at it, and he may have then cried (in Medieval times) "Just what I've always wanted!" For here, indeed, did man have a peculiar consonance: Other harmonic consonances to date were consonances of "similars" as explained in Chapter IB. Nothing of special attractiveness exists by playing the octave to a note. It is only noted as "the same note, only higher." The 4th and 5th too, are more similar to the tonic than different from it. But the 3rd, we can say, is a consonance of "difference" and implied the possibility of a workable harmony. "The most attractive of the intervals, melodically and harmonically, are . . . the Thirds and Sixths, - the intervals which 106

lie at the very boundary of those that the ear can grasp" as consonances, says Helmholtz.^2 The history of music illustrates that the first clearly recorded forms of harmony began to develop in the Church at least as early as the 9th Century. But this was long before the 3rd was admitted into the family of the "official" harmonic consonances of the Church. The 3rd was in the scale from early Greek times, but because it was not an official consonance until very late in Medieval times, the new attempts at harmony (called "organum" - a word to remember) could not at once use it. The harmony that was developed was based on 4ths and 5ths; on consonances of "similars." One may naturally wonder how the 3rd spurred the develop ment of harmony when for hundreds of years it was not used as an harmonic interval. The 3rd, by making the major diatonic scale (the first melody as we know "melody" today) contributed, thereby, to the development indirectly. For, as harmony can be used to strengthen the relations of notes in a melody, then with out the advent of melody in this form (familiar to our own concepts of melody) harmony would have had little function. But this is only half an answer. It is necessary now to learn that there are really two histories of music (and of the 3rd) in this period: that of Church music and that of Secular music. The Church kept records of much of their music. Secular music was vocal and spontaneous, and little of it was noted, even for the sake of history, by Church historians. It's important to be careful not to take written church music or its own history of music as too accurate in these matters, as the anti-paganism of the clerics surely affected their objectivity. From St. Augustine on through the various meetings of the Council of Trent to the present day, we read of sporadic diatribes by the church condemning "wantonness," "vulgarity," etc., picking at certain rhythms and harmonies which have insidiously crept into the music. And in many cases, until these fits of pique occurred and were recorded, historians have had no other way of discovering exactly what had been going on in the music up to then; for surely, while these evil practices were done by musicians on a daily basis, no one ever was allowed to write about them nor justify them in any theoretical writings. The written music most often did not reflect what the real sound was like. 107

As secular music had no recourse to written music or written history, therefore, we may, in most cases, only infer, too, what it was like, but later we'll see that we can infer a great deal without fear of missing the mark by too much. The history of the major diatonic scale, whose distinguishing characteristics are the 3rd and 7th, is divided into these "two" histories. Let's examine this history in more detail. The scales most tonally organized (previous chapter) and which imply the ingredients of harmony are the major and minor scales. It has been claimed by some historians that the major diatonic scale, as part of the "major-minor system," was well on its way to predominance long before recorded history would lead us to believe. We know that the medieval church basically relied upon a system of modes, that is, scales formed by beginning on different notes of the diatonic scale. For example, beginning on D, we have DE*FGAB*CD, a scale known as the "Dorian" scale or mode. (Others, beginning on other notes, have names like "Phrygian," "Mixolydian," "Lydian," etc.) The difference between these scales is the arrangement of tones and semitones. In the Dorian mode, the second and third notes (marked above by the asterisk), and the sixth and seventh notes, form a half tone. The other notes form whole-tones. In the Phrygian mode, beginning on E, the first and second, and the fifth and sixth, notes form the semitones. Our major mode or scale has half-tones between the third and fourth, and seventh and eighth, notes. Melodies were written in these modes, and each had its own characteristic sound or emotional quality. The Greeks had a similar set of modes. The names were the same, but the names were applied differently, and several ways, as their music evolved into these modes, than the way they were applied during the middle ages. Although we don't know for sure which series of tones ("harmonies" — prior to Greek modes) was Lydian or Phrygian in ancient Greek times, we do know that they, too, had an emotional quality and were written about by Plato and Aristotle. We know the later Greek modes, but little is written about their effect. In most cases, the effect of the music was mainly due to different arrangements of whole and half-tones. Among these modes in the middle ages, we know that accord ing to Church-recorded history our minor and major modes were absent until a medieval theorist, Glarean, added them in the 1 6th century, calling them Aeolian and Ionian. But were they really "absent" until then? No one knows if the late Greeks or early 108

Romans likewise omitted these modes or relegated them to a status of lesser importance to any other of their modes. There is no reason why they would have, but the church may have had a reason to slight them if they had been in vogue: As they opposed all pagan devices, among these may have been the major and minor modes. Grout says about the "tracts," which are the longest chants in the liturgy, and among the oldest in the Christian church: "There are certain recurring melodic formulas which are found in many different tracts, and regularly in the same place . . . These features - the construction of so many melodies on only two basic patterns ... all suggest that in the tracts we have a survival, probably in elaborated form, of some of the most ancient music of the Church. If this is true, it would suggest that originally there were only two modes . . . and that these two original modes were what we should now call 'minor' and 'major' in character."63 In the name of anti-paganism perhaps these modes were eventually and successfully organized out of the official music. However, we know that accidentals (Bb) were allowed in church music. "The only accidental properly used in notating Gregorian Chants is B-flat. Under certain conditions the B was flatted in modes I and II, and also occasionally in modes V and VI; if this was consistently done, these modes became exact facsimiles of the modern . . . minor and major scales respectively." 64 Prior to church organum there were "troubadours" and "trouveres," traveling musicians, whose music Grout describes: "The modes used were chiefly the first and seventh . . . certain notes in these modes were probably altered chromatically by the singers in such a way as to make them almost equivalent to the modern minor and major."65 (That is, tending away from modality.) In other words, the claims of an early popularity (or at least persistence) of minor and major scales isn't unfounded. The history of the 3rd itself, like that of the scale, is also involved in the dual accounts about early harmony. The desire for combinations of tones had already been partly developed before Christianity: I believe that very early, despite the lack of success or even of developed melodicism, man often wanted to make harmony. This is evidenced by the very early occasional uses of simultaneous 4ths and 5ths, from the uses of various rhythm patterns in combination (as we will see in the chapter on 109

Rhythm), and from such things as the very ancient "drone" accompaniment of bagpipes in Scottish music. (See Appendix II, "Harmony in Ancient and Primitive Music" for a fuller discussion of this question.) In these earliest attempts at harmony, the Secular history of the 3rd figures much more prominently than it does in Church music. "The practice of singing a given melody in thirds was called gymel or cantus gemellus (i.e., 'twin song'). This practice seems to have had no connection with ecclesiastical developments in organum and it may have existed prior to organum. It was probably of Welsh or English origin."66 No one knows whether any such use of thirds existed in the Roman-early Christian areas. Also, no one knows that it didn't exist then in some form or other either. But the widespread use of 3rds melodically; and, in almost all places untouched or not under the strongest influence of the church, the non-modal character of secular music indicates that one of the "pagan" aspects banned by the church easily could have been thirds in harmonic form. Sachs wrote of the relation between Church and secular music: "Contemporary sources, written by clerics and for clerics, did not condescend to mention it;" (secular music) "and folk music itself, depending on oral tradition, seldom availed itself of notation. All conclusions must be drawn from indirect evidences - from a few songs of the time preserved in later notation, from the popular music of jugglers and minstrels from ages after A.D. 1000, and from the folk-songs of today, with their amazing tenacity. "All these evidences bear witness to one fact: The nonMediterranean secular music of Europe did not follow the tetrachordal and modal style of the Church (except in cases of willful imitation). Instead, it organized its tunes in thirds: their framework consisted of one third, or of two, three, four and even, as in today's folksongs of Iceland and Scandinavia, five thirds heaped one above another. Such chains complement each two consecutive thirds to form a fifth. Therefore, the thirds are major and minor in regular alternation. "The resulting 'chains' were not scales, but loose organiza tions, pieced together out of single elements without any thought given to higher units beyond the fifth. The decisive change forward came from the organizing power of the octave . . . For no

we find tertial chains in which the third of the thirds, a seventh away from the groundtone (as C-E, E-G, G-B), was sung but was at once pulled up to make an octave. Wherever this happened, the third-fifth-octave skeleton of later Western music was established . . ."67 With these reservations about the two histories in mind, we can now trace the development of harmony in the Church, according to the Church's own history and records. And we will see, nevertheless, that things like the 3rd, the harmonic system (based on the tonic, 4th and 5th), and other such aspects of music, all fought their way to the surface through the Dark Ages, and created, despite the originally powerful social pressure and bans against them, an almost "acoustically perfect" music. As was pointed out, the first Church harmony was called "organum," begun around the ninth century. Initially, the introduction of the 3rd allowed this new attempt at harmony to take place by forming the first real melody: the diatonic scale. Also developed from the scale itself were the remainder of notes, the 12-tone series, which made possible the relating of whole scales to each other in keys* of the 4th and 5th. Let's examine this process in more detail. Because the diatonic scale and modes were the "first" real melodies (and new melodies could be arranged from them), they were wanted in transposed keys, that is, starting on other notes. Here the aim was to transpose completely, not just start on another note and get another and different mode with a different pattern of whole and half-tones. For example, in order to have the major scale (starting on C) transposed so that it will start on G, a new note is needed.

The scale in the key of G

"Key" is similar to tonic, in that it is the note around which in

The note needed is F-sharp. F# is not an already existing note in the scale from which we wish to transpose: CDE FGABC. To play the scale in the key of G or in any other key, F# and all such "new" notes are needed. These notes, together with the scale, formed a 1 2-note series, or a 1 2-note division of the octave, such as all the notes, black and white, which we find between C and C on our piano. (Another reason for the additional notes was the desire to alter existing modes and melodies to suit them to a minor or major sound, or for causa pulchritudinus (for the sake of beauty). These alterations are known as accidentals, and throughout the Middle Ages, were often not written in the music. The result was that the real sounds were called musica ficta (ficticious, or false music). Much of our knowledge of what music was like is speculative due to several centuries of unfortunate things like that.) Just a side point: Prior to the diatonic scale and the desire to relate the scale in keys, the 1 2-tone series, which had been earlier discovered, had no use until melodicism was highly enough developed to use them for the above reason. Forsythe points out: "Chinese theorists also invented a chromatic series ... in which two whole-tone scales were twisted together like the strands of a rope to form a complete whole." ^One scale D C#

P D#

G F#

A G#

C A#

~C#

"Another scale "But," he adds, "this remained a theory and no more. It was absolutely unused in both ritual and popular music."68 Smith wrote, "The Japanese have twelve semitones to the octave, as the Chinese have, the root of their civilization being the same . . . The scale, however, is not used to play music proceeding by semitones, but is used for the purpose of transall other notes are centered, but the word key also implies all of these other notes. Hence G can be considered a note in the key of C, because it is related to C. If all the notes of a chord are related to a tonic, then the chord is said to be in that key. 112

position of melody to high or low position . . ,"69 In other words, this series (which may have been arrived at in other places through the cycle of 5ths) was rarely used and nowhere was it used as a scale. Even in Greece, it was a set of auxiliary notes plus the diatonic scale, not a scale by itself. But to return: How did the 12 notes and the ability to play the scale in various keys contribute to the development of organum? That is, to the beginning of harmony? We know, as was mentioned, that Aristotle and Plato wrote much concerning the effect (ethos) of different modes in their music. In any poem which was sung, often one line may have required or been appropriate to one mode, and another line to another mode. Hence, (if our modern notation of the few Greek fragments of music is correct), a song would be in and out of different modes, successively. (It is well established that in medieval times, even in early chants, "multi-modalism" formed a part of the construction of these melodic lines.) This represents one reason for transposition of the modes and the extension of the scale to include the auxiliary notes we have already discussed. To give a melody to another voice, higher or lower, or to a particular instrument, modes and melodies were for this, too, transposed. (One may easily believe that, "Well, now we've got a melody or two. What does it take to have someone, experimentally or accidentally, find out that a few more notes might be added and sound good? Or try to play two different melodies at the same time?" (Ah, but that is asking too much. It would have taken a great deal. To do that would have been a great historic leap in the field of music, skipping over many steps. (In those days, music was still inseparably bound up with words. Instrumentalism was extremely rare. Often the words and their meanings had more to do with the shaping of the music than anything else. In any "short" stretch of history, especially when an art is not yet an independent art, social aspects strongly intervene.) The next step was a long time in coming, but resulted from what was brought into existence by all this transposing: The same melody began to exist in different keys. Once it so existed, in whatever crude notation may have existed, then it is not asking a great deal for someone to have thought (albeit 113

cautiously) about taking the same melodies and playing or sing ing them in the different keys together; simultaneously. The earliest known harmony consisted of occasional 4ths and 5ths played simultaneously with the tonic. Forsythe says that in China, "At most the guitar-player is occasionally allowed to touch a fourth, a fifth, or an octave when accompanying the voice."70 As a result of the use of the 1 2-tone system, the end of the Roman empire saw this old harmony give way to organum, the playing of whole scales and melodies, not just occasional notes, in 4ths and 5ths simultaneously. The melodicism of musicians allows them to do with whole melodies what they formerly did only with single notes: the playing of them in harmonies of 4ths and 5ths, using the 12 tones to form these melodies in the different keys and modes. This actually represents the first stage of the influence of the 3rd. By completing the diatonic scale and melody as we more or less understand it today, it led to organum, the earliest recorded form of harmony on any broad scale. Now, if the 3rd hadn't been introduced, forming the scale and the various modes; if among these modes the major and minor modes hadn't been of some importance (how important they were or would have had to be may have to remain speculative), then even the beginning process of harmony so far outlined above might never have been continued. Any forms of harmony attempted earlier would have been done without possession of all the notes necessary to form "consonant" chords, that is, chords which also have the potential of being part of a spectrum of chords. As has been said, without such a context or "spectrum," no chord could have been accepted as "consonant." However, the process had only begun. It was not finished with the advent of organum. Certainly, the first organum did not provide our "spectrum." While it was possible because of the diatonicism of the period, it was, I believe, another "failure" at harmony. (I use the word "failure" advisedly and only in a longrange historic sense.) Its duration was due to its techniques having been frozen by church regulation. The evidence for this belief is that music outside the church, to a greater degree than within, was developing the 3rd as a "consonance of difference," and may actually have been an unrecorded influence which spurred the church to its own "non-pagan" harmony in organum. A reason to suppose the possibility that organum was a "failure," is admission by churchmen themselves from time to

time concerning the course of their organum. Speaking of a later form of it, Franco of Cologne was not unaware of the cacophony of the sounds emanating from the faithful in praise of the Almighty. "He who shall wish to contruct a quadruplum or quintuplum" (4 or 5 voice writing) "ought to have in mind the melodies already written, so that if it be discordant with one, it will be in concord with the others." 71 There is less testimony regarding the early organum, but the parallel 5ths involved in it were soon discarded. "Strict organum was probably still used in the 1 1th century, although the interval of a fifth was no longer permitted in parallel motion." 72 (Secular music might have earlier developed a more successful harmony system except for lack of a method of notation, which was available only to the church, and which was another pre condition for the real development of harmony.) The point here is that if the music was in good measure admittedly discordant (saving us from imposing this judgment of it from our own prejudiced modern ears), for what reason did the process continue? One can infer that a good reason was the existence of the practice of a somewhat sweeter (although undoubtedly less complex or sophisticated) form of harmonic activity in the secular field. Although the church may have decried secular or pagan influences, it is well known that despite such attitudes throughout history, often the things disdained by some are never theless unconsciously adopted by the disdainers themselves. In this case, if such a secular influence existed, it was much modified. If it moved the Church toward harmony, the harmony initially used was modified to exclude the "secular" 3rd. However, organum itself, which began in the 9th century, by the 11th century began to finally develop the 3rd as an harmonic interval. The increase in harmonic activity, causing a growing cognizance of the 3rd as an harmonic consonance as well as a melodic one, gave rise to the next stage, which I shall call "polyvocality," and which happened around the 11th century. This stage, lasting several centuries, was to pave the way to the discovery of chords and chord connections (our spectrum). Polyvocality means the playing or singing of different melodies at the same time (compared to organum, which presents the same melody, but starts it simultaneously from notes a fourth or fifth apart). Other terms applicable to this in general are "counter point," "polyphony," etc. I believe polyvocality is a more "5

accurate and descriptive term. The first simultaneous 3rds, in any quantity that havebeen recorded, were produced by this polyvocality, and this is the initial success necessary to really inspire continued efforts at harmony. Although it was hardly clear to musicians then, a small part of the spectrum of chords, in their proper acoustical order, began accidentally to be formed. Social custom can keep many aberrations going for a while, but not forever. From this point forward, the history of music represents a thread toward finished tonality in both melody and harmony, and any socially inspired retrogressions from this thread, after having been introduced, were ultimately shunted aside later, or completely abandoned. I would like to introduce here an example of polyvocality. It isn't an historical example, but one by the author in a style illustrative of the idea. In the example below, voice 1 and voice 2 represent two different melodies. Notice how the accidental dissonances

FINK Voice 1 Slow S

zp si Voice f

116

formed by the two melodies can be viewed as if "in passing" and are not upsetting in that light. In fact, they add the necessary contrasts to the harmonies which make such pieces even more interesting than if there were only consonances in them. Around the time of the 11th century, it was noted that the playing of entirely different melodies could be done and could sound good too. The simultaneous 3rds created by this polyvocality was only one step forward from the playing of 3rds in the melodic scale. (Men — at least churchmen — have proceeded at a pace which consisted of the tiniest of steps.) Until the 3rd was long accepted as a melodic consonance in the scale, it could not be accepted as a harmonic consonance. Sachs has pointed out: "Whenever a musical style demands accompanying notes, it prefers them at intervals similar to those it favors as melodic steps." 73 Helmholtz writes as follows of the beginning and history of polyvocality, which he calls polyphonic music. "The oldest specimens of this kind of music which have been preserved are of the following description. Two entirely different melodies . . . were adapted to one another by slight changes in rhythm or pitch until they formed a tolerably consonant whole . . . The first of such examples could scarcely have been intended for more than musical tricks to amuse social meetings. It was a new and amusing discovery that two totally independent melodies might be sung together and yet sound well . . . Different voices, each proceeding independently and singing its own melody, had to be united in such a way as to produce either no dissonances, or merely transient ones which were readily resolved. Consonance was not the object in view, but its opposite, dissonance, was to be avoided. All interest was concentrated on the motion of the voices." 74 Here we see melodicism as the moving principle. And yet, although "7

Helmholtz could describe it, he failed to realize its importance in answering the question he posed on why chords were not discovered sooner: "It is scarcely possible for us, from our present point of view, to conceive the condition of an art which was able to build up the most complicated constructions of voice parts in chorus, and was yet incapable of adding a single accompaniment to the melody of a song ... for the purpose of filling up the harmony. And yet when we read how Giacomo Peri's invention of recita tive" (half-talk and half singing) "with a simple accompaniment of chords was applauded and admired and what contentions arose as to the renown of the invention; what attention Viadana* excited when he invented the addition of a Basso continuo for songs in one or two parts, as a dependent part serving only to fill up the harmony; it is impossible to doubt that this art of accompanying a melody by chords . . . was completely unknown to musicians up to the end of the sixteenth century. It was not until the sixteenth century that composers became aware of the meaning possessed by chords as forming an harmonic tissue independently of the progression of parts . . ."7^ "Since the involved progression of the parts gave rise to chords in extremely varied transpositions and sequences, the musicians of this period could not but hear these chords and become acquainted with their effects, however little skill they shewed in making use of them. At any rate, the experience of this period prepared the way for harmonic music proper, and made it possible for musicians to produce it . . ."76 So from the 11th century, when polyvocality began, to the 16th century, when chords were finally accepted in isolation, we see that the melodicism of musicians had to be whittled down a bit by the consonance of the individual chords which they accidentally formed in the progression of parts. It took this slight retreat for them to pay attention to and discover the relations that existed between various isolated combinations of notes that were formed by polyvocality; that is, for them to discover chords. If Helmholtz had been able to give due importance to the melodicism of the period, he could have understood part of the reason for the failure of musicians to quickly abstract chord combinations from music which was conceived and viewed only *Viadana is no longer credited with inventing the Basso Continuo. 118

as a progression of parts, or melodies. When Helmholtz could have talked about the cultural causes of things, of social effects in music, he doesn't. For here, the time span between polyvocality and the discovery of chords also has social causes. If abstract and isolated chords were wanted, the 3rd had to be used to form them. The 3rd was in the scale, but if the 3rd was not an officially approved consonance according to the Church, then musicians had to wait until it was before they could use it as one in harmony. To do so beforehand would have been as gross in taste then as to hav^ a church choir singing bawdy songs and doing the can-can at solemn Mass today. It took time for the 3rd to be viewed as a consonance in harmony as it already was as a melodic step. "It was not till towards the end of the twelfth century that Franco of Cologne included the Thirds among the consonances . . . "It was not till the thirteenth and fourteenth centuries that musicians began to include the Sixths among the consonances."77 (Here, of course, Helmholtz is referring to simultaneous combinations.) However, as has been mentioned, earlierusages of 3rds and 6ths are on record. "In the 11th century, a method of combining sounds, called the New Organum, was developed. This kind of organum admitted thirds and sixths."78 What seemed to take so long for Helmholtz, who I think relied too heavily on official church edicts for his estimate, might have been less of a question for him had he paid attention to the possible secular influences and to the "premature" activity of some church musicians who experimented with methods not yet officially approved. It should be stressed however, even had he understood this, that the process still was one which required more time than a first glance would expect, and this, again, was because having become highly melodically oriented, musicians did not concern themselves over ly with vertical harmonic structures. In addition, they also had no idea that they were "supposed" to discover and isolate chords, in order that they might "file" them away and pick them out to harmonize melodies as these melodies were composed and as they might require certain chords. At any rate, by the 16th century, the way was finally cleared for church musicians to use the 3rd "legally" in the writing of chords, and as an important consonance (at cadence points and on strong rhythmic points). The basic theory regarding harmonic structure (known as 119 \

harmonic rhythm) during these 400 years was often changing and nebulous. There seemed to be no definite notion of chord progression for the most part. As long as the movement of voices produces a tolerable whole, many unusual chord movements are found. Certain conventions arose concerning the cadences, and these tended quite directly, more and more toward the tradi tional V—I (that is, the 4th relationship) cadence. It seemed to have been more or less founded in the Burgundian school in the middle of the 15th century, and has lasted 500 years. And for several hundred years before that, the cadence was a bastardized incomplete form of the modern cadence, VII - I (Or, vii°- I, that is, f to 9. D £ In so far as any chords were thought of at all in an attempt to apply them deliberately at any given points, to bind two melodies together, or to alter them to suit a melody, one trait stands out in almost all the music in the 400 year span: Notes of any main mel ody were often harmonized by their own chord. More concretely, notes were frequently harmonized, one after another, with chordal structures which consisted of the 3rd and 5th of each note of the melody. (The illustration below is only an extension of the idea to the diatonic scale in order to illustrate the point.) In addition, this method was eventually rejected, although like everything else in history during these early periods, centuries went by as these things were modified. (It should be noted how ever, that the ultimate modifications conformed to natural laws, and followed them closely. Social events and influences of the time, which "should," according to cultural theories, have given rise (and did in the other arts) to modifications; in music, often found no reflection at all in any change in the music, and vice versa: the music made great historic junctures at a time when little else was changing.)* *Grout says: 'The period from 1450 to 1600 in the history of music is now generally known as 'the Renaissance . . .' Renais sance was first used by art historians'to designate fifteenth-and sixteenth-century styles of painting, sculpture, and architecture; its use was gradually extended to all cultural manifestations of these two hundred years, including music ... As the word gained in content, it lost in precision. Its literal meaning is 'rebirth,' and 120

The example below represented what was historically a "false harmonization technique.

(C maj) (D maj) (£ maj) (F C F#(«rtj) £ B G

maj) F A C

(G maj) (A maj) (B maj) (C B h A LX G D#(ortl) B F#(ort|) £ D

maj) C £ G

The top line is the scale, the next is the name of the chord which was used to harmonize each of its notes, and the notes forming each chord are listed below it. In this progression, mostly rejected now, none of the chord notes or overtones of, say, the D chord (2nd from the left above) "forecasts" any overtones or notes of the next scale note or its chord. In fact, the whole harmonic progression is one of disruption of relationships among the notes of the scale, even though the chords making up the progression are each major chords, the most consonant of chords. The more melodically-oriented musicians were, the more many writers and artists of the fifteenth and sixteenth centuries viewed the achievements of their own time as a revival of the glories of Greece and Rome, a revival stimulated in part by discoveries of many ancient works of art and literature. But no comparable discoveries of ancient music were made . . . Renaissance, then, in the sense of a rebirth of ancient art, is all but meaningless when applied to music of this period . . ."79 The reason for the mismatch is due to an earlier mismatch between society and effects on music: The "classic" age of architecture, sculpture, and later painting had already been (or were being) achieved long before the "classic" age in music was even on the distant horizon. There was nothing in music to be discovered from ancient times; nothing to be "reborn" and "revived." 121

unsatisfactory was this early attempt at harmony because of its disruptive character to melodic unity. One might ask, if forgetful of the previous chapter, why the D is in the scale at all, if all its overtones so poorly relate to those of the notes before and after it in the scale. Besides the role of melodicism, making the D a melodic step, there is this reason: is because D is an overtone of G, the 5th of the tonic, that it is in the scale, * and this, in spite of D's overtones. Therefore, if we use the chord of G, instead of the chord of D, to harmonize the D, we will have a better and smoother harmonization of the D — E — c —— [C maj) (G maj ) (C maj] CP maj) C P G C A fi B E C G (ibnic) tes) — g —— A - B —— CG maj) CP maj)