This harmonisation uses Harry Partch's 43-tone-per-octave "Genesis Scale" which is an extension of just intonation. The first note of the scale is G, or 1/1, and all other notes are named as ratios with relation to this "unity." For example, the note 3/2 is a perfect fifth above 1/1 and is roughly equivalent to the note D.
The scale construction of the scale begins with the first eleven overtones of the harmonic series, which results in six different tones after removing duplicates. The tones are 1/1, 3/1, 5/1, 7/1, 9/1, 11/1, (every odd overtone) but are conventionally expressed transposed into the same octave: 1/1, 3/2, 5/4, 7/4, 9/4, 11/8. Partch calls this collection of tones an "Otonality" and this particular one would be labelled 1/1-O as it begins on 1/1. Partch considers Otonalities somewhat analogous to major tonalities, and the closest approximation in 12-tone equal temperament would be dominant 9 #11 chord.
A complementary minor tonality, or "Utonality" 1/1-U is then created by reciprocating these ratios. In other words, the previous intervals are taken as descending intervals from 1/1 rather than ascending. (The term Utonality is derived from the synthetic undertone series, which is the upside-down overtone series). The tones of 1/1-U are therefore 1/1, 1/3, 1/5, 1/7, 1/9, 1/11, and are conventionally expressed transposed into the same octave as 1/1, 4/3, 8/5, 8/7, 16/9, 16/11.
From here, things begin to get complicated and I would recommend consulting Harry Partch's book "Genesis of a Music" for a detailed explanation with diagrams. I'll summarise to give a basic idea of how the rest of the scale is generated.
After generating the 1/1 Utonality, five more Utonalities are generated - one for each of the remaining tones of the 1/1 Otonality. This results in a total of 29 different tones. An alternative method to get these same tones is to instead generate five more Otonalities from the remaining tones of the 1/1 Utonality. The symmetry of these relationships is easily seen in tonality diamond diagram that Partch details in his book.
An important property of the six Otonalities and six Utonalities that have been generated is that each contains the unity 1/1 at some point. Thinking back to 12-tet to try to make sense of this, if 1/1-O is like a G9#11 chord (containing the notes G, B, D, F, A, C#) then six Otonalities generated thus far are all of the dominant 9 #11 chords that contain G (1/1). These would be G9#11, Eb9#11, C9#11, A9#11, F9#11, Db9#11. Since just intonation is used rather than 12-tet, there are a total of 29 tones present in this collection of chords - six per chord minus the five duplicates of G and also a duplicate 4/3 (C) and 3/2 (D). The same process works with the Utonalities, which would resemble major 9 #5 #11 chords.
From here, the remaining 14 tones are added by generating five additional symmetrical pairs of Otonalities and Utonalities to fill in the gaps (even with the initial 29 tones there are gaps larger than 150 cents). These are added somewhat arbitrarily based on what Partch considered important, and some of these secondary tonalities are incomplete.
The result is a scale with 43 unequal intervals per octave. There are all kinds of microtonal effects and inflections available in this scale that Partch makes use of, but the scale is primarily a gamut of tones that includes a range of just intonation chords and untempered scales such the ancient Greek scales formulated by Ptolemy and Pythagoras. The Greek influence on Partch's music is prominent also in some of his instrument design (kithara), his use of dramatic speech-based music (monody), and also in his taste for Greek cuisine like rose petal jam.
22 апр 2020
