Harry Partch's 43-tone scale

The 43-tone scale is a just intonation scale with 43 pitches in each octave. It is based on an eleven-limit tonality diamond, similar to the seven-limit diamond previously devised by Max Friedrich Meyer[1] and refined by Harry Partch.[2]

The Quadrangularis Reversum, one of Partch's instruments featuring the 43-tone scale

The first of Partch's "four concepts" is "The scale of musical intervals begins with absolute consonance (1 to 1) and gradually progresses into an infinitude of dissonance, the consonance of the intervals decreasing as the odd numbers of their ratios increase."[3][4] Almost all of Partch's music is written in the 43-tone scale, and although most of his instruments can play only subsets of the full scale, he used it as an all-encompassing framework.

Construction

Partch chose the 11 limit (i.e. all rational numbers with odd factors of numerator and denominator not exceeding 11) as the basis of his music, because the 11th harmonic is the first that is utterly foreign to Western ears. The seventh harmonic is poorly approximated by 12-tone equal temperament, but it appears in ancient Greek scales, is well-approximated by meantone temperament, and it is familiar from the barbershop quartet;[5][6] the ninth harmonic is comparatively well approximated by equal temperament and it exists in Pythagorean tuning (because 3 × 3 = 9); but the 11th harmonic falls right in the middle between two pitches of 12-tone equal temperament (551.3 cents). Although theorists like Hindemith and Schoenberg have suggested that the 11th harmonic is implied by, e.g. F♯ in the key of C, Partch's opinion is that it is simply too far out of tune, and "if the ear does not realize an implication, it does not exist."[7]^:126

Ratios of the 11 Limit

Here are all the ratios within the octave with odd factors up to and including 11, known as the 11-limit tonality diamond. Note that the inversion of every interval is also present, so the set is symmetric about the octave.

Cents	0	150.6	165.0	182.4	203.9	231.2	266.9	315.6	347.4	386.3	417.5	435.1	498.0	551.3	582.5
Ratio	1/1	12/11	11/10	10/9	9/8	8/7	7/6	6/5	11/9	5/4	14/11	9/7	4/3	11/8	7/5
41-ET	0.0	5.1	5.6	6.2	7.0	7.9	9.1	10.8	11.9	13.2	14.3	14.9	17.0	18.8	19.9
Audio	Play	Play	Play	Play	Play	Play	Play	Play	Play	Play	Play	Play	Play	Play	Play

Cents	617.5	648.7	702.0	764.9	782.5	813.7	852.6	884.4	933.1	968.8	996.1	1017.6	1035.0	1049.4	1200
Ratio	10/7	16/11	3/2	14/9	11/7	8/5	18/11	5/3	12/7	7/4	16/9	9/5	20/11	11/6	2/1
41-ET	21.1	22.2	24.0	26.1	26.7	27.8	29.1	30.2	31.9	33.1	34.0	34.8	35.4	35.9	41.0
Audio	Play	Play	Play	Play	Play	Play	Play	Play	Play	Play	Play	Play	Play	Play	Play

Filling in the gaps

There are two reasons why the 11-limit ratios by themselves would not make a good scale. First, the scale only contains a complete set of chords (otonalities and utonalities) based on one tonic pitch. Second, it contains large gaps, between the tonic and the two pitches to either side, as well as several other places. Both problems can be solved by filling in the gaps with "multiple-number ratios", or intervals obtained from the product or quotient of other intervals within the 11 limit.

Cents	0	21.5	53.2	84.5	111.7	150.6
Ratio	1/1	81/80	33/32	21/20	16/15	12/11

Cents	266.9	294.1	315.6
Ratio	7/6	32/27	6/5

Cents	435.1	470.8	498.0	519.5	551.3
Ratio	9/7	21/16	4/3	27/20	11/8

Cents	648.7	680.5	702.0	729.2	764.9
Ratio	16/11	40/27	3/2	32/21	14/9

Cents	884.4	905.9	933.1
Ratio	5/3	27/16	12/7

Cents	1049.4	1088.3	1115.5	1146.8	1178.5	1200
Ratio	11/6	15/8	40/21	64/33	160/81	2/1

Together with the 29 ratios of the 11 limit, these 14 multiple-number ratios make up the full 43-tone scale.

Erv Wilson who worked with Partch has pointed out that these added tones form a constant structure of 41 tones with two variables.[8] A constant structure giving one the property of anytime a ratio appears it will be subtended by the same number of steps. In this way Partch resolved his harmonic and melodic symmetry in one of the best ways possible.[8]

Other Partch scales

The 43-tone scale was published in Genesis of a Music, and is sometimes known as the Genesis scale, or Partch's pure scale. Other scales he used or considered include a 29-tone scale for adapted viola from 1928, 29-, 37-, and 55-tone scales from an unpublished manuscript titled "Exposition of Monophony" from 1928–33,[9] a 39-tone scale proposed for a keyboard, and a 41-tone scale and an alternative 43-tone scale from "Exposition of Monophony".

Besides the 11-limit diamond, he also published 5- and 13-limit diamonds, and in an unpublished manuscript worked out a 17-limit diamond.[10]

Erv Wilson who did the original drawings in Partch's Genesis of a Music has made a series of diagrams of Partch's diamond as well as others like Diamonds. [11]

References

"Musical Mathematics: Meyer's Diamond", Chrysalis-Foundation.org.
Kassel, R. (2001, January 20). Partch, Harry. Grove Music Online.
Gilmore, Bob (1992). Harry Partch: "the early vocal works 1930-33". British Harry Partch Society. p. 57. ISBN 978-0-9529504-0-0.
Partch 1974, 87.
Abbott, Lynn (1992): Play That Barber Shop Chord: A Case for the African-American Origin of Barbershop Harmony. American Music 10, no. 3 (1992), 289–325.
Döhl, Frédéric (2014): "From Harmonic Style to Genre. The Early History (1890s–1940s) of the Uniquely American Musical Term Barbershop." American Music 32, no. 2, p.123–171.
Partch, Harry (1974) [First published 1947]. Genesis of a Music (2nd ed.). Da Capo Press. ISBN 978-0-306-80106-8. Lay summary.
"Letter to John from ERV Wilson, 19 October 1964 - SH 5 Chalmers" (PDF). Anaphoria.com. Retrieved 2016-10-28.page 11
Bob Gilmore, "Changing the Metaphor: Ratio Models of Musical Pitch in the Work of Harry Partch, Ben Johnston, and James Tenney", Perspectives of New Music 33, nos. 1 & 2 (Winter–Summer 1995): 458–503. Citation on 462.
Bob Gilmore, "Changing the Metaphor: Ratio Models of Musical Pitch in the Work of Harry Partch, Ben Johnston, and James Tenney", Perspectives of New Music 33, nos. 1 & 2 (Winter–Summer 1995): 458–503. Citation on 467.
"Wilson Archives The Diamond and other Lambdoma". Anaphoria.com. Retrieved 2016-10-28.