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]
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.[3] 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.[3]
Comparison with various equal tunings
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 |
Comparison with 46edo
Like 41edo, 46edo maps the 43 note scale to its patent val in a consistent way. However, this time, 12/11 and 11/10 (and their octave compliments) are tempered together rather than 11/10 and 10/9. In addition, several other intervals that are tempered closer together than JI in 41 edo are tempered further apart here, making the scale more uneven sounding in general.
The step sizes in this tuning are: 1 1 1 1 2 0 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 0 2 1 1 1 1
Comparison with 53edo
While 53edo is excellently tuned in the 5-limit and has more than enough steps to represent this scale without conflating any notes, it's poor representation of the 11th harmonic causes problems. Not only are 12/11 and 11/10 once again tempered together, but 14/11 is conflated with 9/7 in the patent val despite actually being closer to 81/64, which would mean only 39 of the scales notes can be cleanly defined under this system. Using the 53e val separates these notes, but now 11/10 and 10/9 are tempered together, which means it also does not cleanly map all 43 notes to different steps.
Comparison with 58edo
As the first edo that is distinctly consistent in the full 11-odd-limit, 58edo is also the first edo that can clearly represent every interval in this scale with no conflations or discrepancies between the patent val and the direct mapping. However, it is strongly sharp trending, which means compound harmonics stack their error, with 33/32 and its octave compliment the worst tuned intervals at 8.8 cents out.
The step sizes in this tuning are: 1 2 1 1 2 1 1 1 1 2 1 1 2 2 1 1 2 1 1 2 1 2 1 2 1 1 2 1 1 2 2 1 1 2 1 1 1 1 2 1 1 2 1
Comparison with 72edo
Since 72edo is distinctly consistent in the 11-limit and is a pepper ambiguity record in the 11-limit unsurpassed until 270, 72edo fits Harry Partch's 43-tone scale very well, with a maximum error of 5.9 cents, with the worst-tuned ratios the result of stacking the compound flatness of multiple 3/2's.
The step sizes in this tuning are: 1 2 2 2 2 1 1 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 1 1 2 2 2 2 1
Comparison with 270edo
In 270edo the step sizes are somewhat more uneven, but far more accurate, with a maximum error of approximately 0.8 cents from JI.
The step sizes in this tuning are: 5 7 7 6 9 3 4 5 6 8 6 5 7 9 7 4 8 6 5 7 7 8 7 7 5 6 8 4 7 9 7 5 6 8 6 5 4 3 9 6 7 7 5
Music
References
- ↑ "Musical Mathematics: Meyer's Diamond", Chrysalis-Foundation.org.
- ↑ Kassel, R. (2001, January 20). Partch, Harry. Grove Music Online.
- ↑ 3.0 3.1 "Letter to John from ERV Wilson, 19 October 1964 - SH 5 Chalmers" (PDF). Anaphoria.com. Retrieved 2016-10-28.page 11
Further reading
- "Musical Mathematics: Meyer's Diamond" at Chrysalis-Foundation.org