Homometry

HOMOMETRY TOM JOHNSON BEFORE THE MATHEMATICS T WAS ALL RIGHT THERE on page 1 of Allen Forte’s seminal 1973 book, The Structure of Atonal Music. Taking an example of the (0,1,4,6) tetrachord as found in Schoenberg’s George Lieder and in Webern’s Six Pieces for Orchestra, Forte wrote: This pitch combination, which is reducible to one form of the allinterval tetrachord, has a very special place in atonal music. It could occur in a tonal composition only under extraordinary conditions, and even then its meaning would be determined by harmonic-contrapuntal constraints. Here, where such constraints are not operative, one is obliged to seek other explanations. Accordingly, in the sections that follow, terminology and notation will be introduced which will facilitate the discussion of certain properties of such combinations. The rst task is to formulate a more general notion to replace that of pitch combination. So the two all-interval tetrachords become the key example of pitchclass sets that go beyond the tonal language. Why? Because they are I 72 Perspectives of New Music homometric. As shown in Example 1, the elements of (0,1,3,7) and (0,1,4,6) have the same six differences, one of each. Twenty pages later, Forte is more speci c: One might conclude that each prime form has a distinct interval vector. That is, however, not the case. For example, the two forms of the all-interval tetrachord, 4-Z15 and 4-Z29, are so constructed that they are not reducible by transposition or by inversion followed by transposition to a single prime form, but they have identical vectors (111111). Thus, he singles out the all-interval tetrachord rather than all the other dissonant chords and clusters of atonal music because it is homometric. He goes on to speak generally of this category, which he calls Z-related pairs: There are 19 such pairs in all. In addition to the one of cardinal number 4, there are three pairs of cardinal number 5 and 15 pairs of cardinal number 6. Always concerned to base his ideas on what he found in actual scores, he goes on to show two Z-related pairs as they occur in three measures of Webern’s Three Short Pieces, Op. 11 No. 1. EXAMPLE 1: ALL-INTERVAL TETRACHORDS Homometry 73 EXAMPLE 2 74 Perspectives of New Music EXAMPLE 3 Homometry 75 Today Forte’s ”Z-related pairs” are generally known as homometric pairs, though the word “homometric” is not yet in the English dictionary. Forte acknowledges in a footnote that these sets were rst described in the literature by David Lewin in 1960, and he might also have gone back to the earlier appearance of this idea in an article by Lindo Patterson in 1944, showing how certain crystal formations are also similar when they have the same sets of differences. Despite the unique and attractive qualities of homometric pairs, very few composers have used them in constructing their harmonies. I have read remarks about how Elliott Carter used the all-interval tetrachords, but really only as details. I am probably the only composer who has produced entire compositions with all-interval tetrachords. The two all-interval tetrachords (0,1,4,6) and (0,1,3,7), with their inversions and transpositions make a list of 48 chords and I wanted to use all 48 in a piano piece called Intervals (2013). The piece consists of short sections that single out each of the six intervals. Example 2 is a diagram of the 48 chords with their tritones singled out, followed by the piano music allowing us to hear the sequence (Example 3). My second piece using only the all-interval tetrachords is a twentyminute opera composed for a production of Grand Magasin in the Festival d’Automne in Paris in 2016. The lyrics, taken from the Essai sur l’entendement of Gottfried Wilhelm Leibnitz, deals with the tiny differences between leaves of the same tree, and the music demonstrates this with two utes and two sopranos. Example 4 shows one section using one of the (0,1,4,6) tetrachords. EXAMPLE 4 76 Perspectives of New Music THE MATHEMATICS The thinking...