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V. ExamplesIn order to visualize the relationships between Partch’s “Monophonic” resources, the ratios inputted are then plotted onto a chart Partch developed called the “One-Footed Bride.” The chart shows all forty-three tones, as well as the twelve tones of equal temperament. The ratios are presented in two columns opposite their “compliment” (inversion). The tones ascend from the 1/1 beginning at the bottom of the left column until the first compliment is reached, and then continue to ascend by coming back “down” the right column to the 2/1. The chart also shows “primary” and “secondary” ratios, which indicates whether they were produced by successive integers (and their multiples), or by adding ratios together. Finally, the One-Footed Bride also reveals categories of intervals, and their relative intensities, as perceived by Partch. There are four interval categories:
Intervals of Suspense=
27/20, 11/8, 7/5; 10/7, 16/11, 40/17 (tritone intervals) Intervals of Emotion=
32/21, 6/5, 11/9, 5/4, 14/11, 9/7, 21/16; 32/21, 14/9, 11/7, 8/5, 18/11,
5/3,12/7 (thirds and sixths) Intervals of Approach= 81/80, 33/32, 21/20, 16/15, 12/11, 11/10, 10/9, 9/8, 8/7; 7/4, 16/9, 9/5,20/11, 11/6, 15/8, 40/21, 64/33, 160/81 (seconds and sevenths)
Ex. 1 **Monophonic Resources* 43 Tones within the
Octave 1/1 81/80 33/32 21/20 16/15 12/11 11/10 10/9 9/8 8/7 7/6 32/27 6/5 11/9 5/4 14/11 9/7 21/16 4/3 27/20 11/8 7/5 10/7 16/11 40/27 3/2 32/21 14/9 11/7 8/5 18/11 5/3 27/16 12/7 7/4 16/9 9/5 20/11 11/6 15/8 40/21 64/33 160/81 2/1 ##
[NOTE: In examples 2-4,
1/1is added at the beginning of the file in order to provide a reference point.
1/1 is an interval of Power.]
*Intervals of Emotion 1/1 1/1 7/6 32/27 6/5 11/9 5/4 14/11 9/7 21/16 32/21 14/9 11/7 8/5 18/11 5/3 27/16 12/7 ##
*Intervals of Suspense 1/1 1/1 27/20 11/8 7/5 10/7 16/11 40/27 ##
Ex. 4 *Intervals of Approach 1/1 1/1 81/80 33/32 21/20 16/15 12/11 11/10 10/9 9/8 8/7 7/4 16/9 9/5 20/11 11/6 15/8 40/21 64/33 160/81 ##
Ex. 5 *Intervals of Power 1/1 1/1 4/3 3/2 2/1 ##
Examples 6-9 present
primary tonalities as they are revealed in Partch’s “Tonality Diamond.”
Otonalities (“major” tonalities) are read from 1/1-7/4, 16/9-14/9, 8/5-7/5,etc.;
Utonalities (minor tonalities) are read from 8/7-1/1, 9/7-9/8, 10/7-5/4,etc.
Tonalities are determined by the “numerary nexus” of ratios. For example, all
intervals with an under number of 7 (8/7, 9/7, 10/7, etc.) create a distinct
tonality.
Ex. 6 **Tonality Diamond
Hexachords *Otonality of Unity 1/1 9/8 5/4 11/8 3/2 7/4 ##
Ex. 7 *Utonality of Unity 8/7 4/3 11/16 8/5 16/9 1/1 ##
Ex. 8 *Primary Otonalities 1/1 9/8 5/4 11/8 3/2 7/4 16/9 9/9 10/9 11/9 4/3 14/9 8/5 9/5 5/5 11/10 6/5 7/5 16/11 18/11 20/11 11/11 12/11 14/11 4/3 3/2 5/3 11/6 3/3 7/6 8/7 9/7 10/7 11/7 12/7 7/7 ##
Ex. 9 *Primary Utonalities 8/7 4/3 11/16 8/5 16/9 1/1 9/7 3/2 18/11 9/5 9/9 9/8 10/7 5/3 20/11 5/5 10/9 5/4 11/7 11/6 11/11 11/10 11/9 11/8 12/7 3/3 12/11 6/5 4/3 3/2 7/7 7/6 14/11 7/5 14/9 7/4 ##
Ex. 10 **Ptolemaic Scales [from
Harmonics] *Enharmonic Tetrachord
1/1 28/27 16/15 4/3 ##
Ex. 11 *Chromatic Tetrachord 1/1 28/27 10/9 4/3 ##
Ex. 12 *Diatonic Tetrachord 1/1 16/15 6/5 4/3 ##
Ex. 13 *Enharmonic Scale 3/2 14/9 8/5 1/1 9/8 7/6 6/5 3/2 ##
Ex. 14 *Chromatic Scale 1/1 16/15 10/9 4/3 3/2 8/5 5/3 2/1 ##
Ex. 15 *Diatonic Scale 1/1 12/11 6/5 4/3 3/2 18/11 9/5 2/1 ##
Ex. 16 *Olympos Pentatonic 9/8 6/5 3/2 8/5 2/1 9/8 ##
Ex. 17 **Non-Western Scales *Chinese sequence
[circa300BC] 1/1 8/7 6/5 5/4 4/3 3/2 5/3 2/1 ##
Ex. 18 **Score Excerpt *”By the Rivers of
Babylon” [mes. 21-29]
a0503+a0403 a0503+a0403 a0503+a0403 a0503+a0403 a0503 a0503
a0503 a0503 a0503+a0706 a0503+a0706 a0503+a0706 a0503+a0101 a0503+a2011 a0503
a0503+a0504 a0503 a0504+a0503 a0504+a0503 a0504+a0503 a0504+a0503
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