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Part 1: Analysis of Iannis Xenakis’ Herma!

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In the 1961 solo piano piece Herma (a Greek word, translating to ‘bond’ in English with

alternate translations of ‘foundation’ and ‘embryo’) Iannis Xenakis presents and develops classes of pitches through the use of different logical structures, creating what he titles musique symbolique. Over the course of the piece, the composer develops his original material to a logical conclusion through the use of mathematical operations and in doing so creates a unique musical work.! !

The ‘embryo’ of Xenakis’ piece consists of three stochastically chosen classes of pitches (A, B

and C) derived from one referent class of pitches (R, consisting of all the notes on the piano). These sets and the complements of these sets are presented one after the other at the beginning of the piece. The lineaire presentation of these pitch sets is differentiated through the use of clearer articulation, longer rhythmic values and a louder dynamic. The nuage material by contrast is presented at a softer dynamic, using smaller rhythmic subdivisions and is often accompanied by the use of pedal, highlighting its greater harmonic and temporal density. Despite being treated separately in many ways, it is only through the union of both nuage and linear presentations of pitch sets that one can derive Xenakis’ original three classes. The use of what the composer terms ‘arborescences’, branch-like figures of notes, expanding in contrary motion, adds another layer of interest, with the delineation between linear and nuage arborescences delineated once again through the use of the pedal.! !

After this original statement, Xenakis uses a range of Boolean algebra operations (an extension

of regular set theory) to create variations on his three classes. The pitch class AB, for example, is derived from the intersection of classes A and B, resulting in a new set of 8 pitches. The composer uses other techniques such as combining A and B and taking the negation of a particular class or intersection of classes (in other words, taking every pitch that falls in the referent class and that is not included in the one chosen to be negated). These variations build from simple to complex concluding in set F (see figures 1 and 2), evident in the piece in the climactic final thirty seconds.!

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Images from Sward, Rosalie La Grow. An examination of mathematical systems used in the work of Iannis Xenakis and Milton Babbit. University Microfilms International, 1981! ! Figure 1. Venn diagram and equation for set F

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Figure 2. Pitch class derived from equation!

! ! ! This results in a piece that while sounding quite random has an inherent mathematical logic to it. In a similar way to a Classical composer deriving an entire piece from motifs developed through standard compositional techniques and structures, Xenakis develops his pitch sets through the use of a mix of musical and mathematical techniques and structures. Indeed, in the introduction to the score he suggests, “One might consider comparing this formal structure to a sonata.” Xenakis’ treatment of pitches as individual and non-octavating, as well as a solid grasp of mathematics results in a more complex fusion of these two distinct disciplines than that of serialism and a unique statement of musique symbolique representative of and aided by an intense numerological process.!

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Fenn Idle!

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Words: 520!

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! Bibliography! Sward, Rosalie La Grow.!

An examination of mathematical systems used in selected !!

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compositions of Iannis Xenakis and Milton Babbit. Ann Arbor, Michigan:!

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U. M. I., 1981, pp. 68-77; pp. 373-400!

Xenakis, Iannis!

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Herma: Musique symbolique pour piano. Boosey and Hawkes, Paris,

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1961!

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