Harry Partch's 43-tone scale

The Quadrangularis Reversum, one of Partch's instruments featuring the 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 and refined by Harry Partch.

See Partch 43 for the scale as a scala file.

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
41edo	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

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
41edo	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

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. 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.

Comparison with 41edo

The 43-note scale is almost epimorphic under the 41edo patent val. The only exceptions are the pair {11/10, 10/9} and its octave complement {9/5, 20/11}, which are tempered together in 41edo. Other than those, 41edo does a decent job of representing everything, for an EDO (although of course Partch himself would scoff at such a claim).

41edo steps	Partch ratio(s)	Partch cents	EDO cents	Error (cents)
0	1/1	0.00	0.00	0.00
1	81/80	21.51	29.27	+7.76
2	33/32	53.27	58.54	+5.26
3	21/20	84.47	87.80	+3.34
4	16/15	111.73	117.07	+5.34
5	12/11	150.64	146.34	-4.30
6	11/10, 10/9	165.00, 182.40	175.61	10.61, -6.79
7	9/8	203.91	204.88	+0.97
8	8/7	231.17	234.15	+2.97
9	7/6	266.87	263.41	-3.46
10	32/27	294.13	292.68	-1.45
11	6/5	315.64	321.95	+6.31
12	11/9	347.41	351.22	+3.81
13	5/4	386.31	380.49	-5.83
14	14/11	417.51	409.76	-7.75
15	9/7	435.08	439.02	+3.94
16	21/16	470.78	468.29	-2.49
17	4/3	498.04	497.56	-0.48
18	27/20	519.55	526.83	+7.28
19	11/8	551.32	556.10	+4.78
20	7/5	582.51	585.37	+2.85
21	10/7	617.49	614.63	-2.85
22	16/11	648.68	643.90	-4.78
23	40/27	680.45	673.17	-7.28
24	3/2	701.96	702.44	+0.48
25	32/21	729.22	731.71	+2.49
26	14/9	764.92	760.98	-3.94
27	11/7	782.49	790.24	+7.75
28	8/5	813.69	819.51	+5.83
29	18/11	852.59	848.78	-3.81
30	5/3	884.36	878.05	-6.31
31	27/16	905.87	907.32	+1.45
32	12/7	933.13	936.59	+3.46
33	7/4	968.83	965.85	-2.97
34	16/9	996.09	995.12	-0.97
35	9/5, 20/11	1017.60, 1035.00	1024.39	+6.79, -10.61
36	11/6	1049.36	1053.66	+4.30
37	15/8	1088.27	1082.93	-5.34
38	40/21	1115.53	1112.20	-3.34
39	64/33	1146.73	1141.46	-5.26
40	160/81	1178.49	1170.73	-7.76
41	2/1	1200.00	1200.00	0.00

Further reading

• "Musical Mathematics: Meyer's Diamond" at Chrysalis-Foundation.org