Superpyth

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Superpyth, a member of the archytas clan, has 4/3 as a generator, and the Archytas comma 64/63 is tempered out, so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for meantone and 12edo, with the understanding that, for example, A♯ is sharper than B♭ (in contrast to meantone where A♯ is flatter than B♭, or 12edo where they are identical). An interesting coincidence is that the plastic number has a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth.

Such a temperament without the 5th harmonic is also called archy. If the 5th harmonic is used at all, it is mapped to -9 generators through tempering out 245/243, so C-D♯ is 5/4. So superpyth is the "opposite" of septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex.

If intervals of 11 are desired, the canonical way is to map 11/8 to -16 generators, so 11/8 is a double augmented second (C-Dx), tempering out 100/99. Yet a simpler but reasonable way is to map it to +6 generators, so 11/8 is a diminished fifth (C-G♭), by tempering out 99/98. The latter is called supra, or suprapyth. The two mappings unite on 22edo.

MOS scales of superpyth have cardinalities of 5, 7, 12, 17, or 22.

Temperament data

Main article: Archytas clan #Superpyth

Interval chains

Archy (2.3.7)
1146.61 437.29 927.97 218.64 709.32 0 490.68 981.36 272.03 762.71 53.39
27/14 9/7 12/7 9/8~8/7 3/2 1/1 4/3 7/4~16/9 7/6 14/9 28/27
Full 7-limit superpyth
613.20 1102.91 392.62 882.33 172.04 661.75 1151.46 441.16 930.87 220.58 710.29 0 489.71 979.42 269.13 758.84 48.54 538.25 1027.96 317.67 807.38 97.09 586.80
10/7 15/8 5/4 5/3 10/9 27/14 9/7 12/7 9/8~8/7 3/2 1/1 4/3 7/4~16/9 7/6 14/9 28/27 9/5 6/5 8/5 16/15 7/5
Supra (2.3.7.11)
857.54 150.35 643.15 1135.96 428.77 921.58 214.38 707.19 0 492.81 985.62 278.42 771.23 64.04 556.85 1049.65 342.46
18/11 12/11 16/11 27/14 14/11~9/7 12/7 9/8~8/7 3/2 1/1 4/3 7/4~16/9 7/6 14/9~11/7 33/32~28/27 11/8 11/6 11/9
Full 11-limit suprapyth
604.44 1094.94 385.45 875.96 166.46 656.97 1147.47 437.98 928.48 218.99 709.49 0 490.51 981.01 271.52 762.02 52.53 543.03 1033.54 324.04 814.55 105.06 595.56
10/7 15/8 5/4 18/11~5/3 12/11~10/9 16/11 27/14 14/11~9/7 12/7 9/8~8/7 3/2 1/1 4/3 7/4~16/9 7/6 14/9~11/7 33/32~28/27 11/8 9/5~11/6 6/5~11/9 8/5 16/15 7/5

Scales

5-note MOS (2L 3s, proper)
  • Archy5 – archy in 472edo tuning
7-note MOS (5L 2s, improper)
  • Archy7 – archy in 472edo tuning
  • Supra7 – supra in 56edo tuning

In contrast to the meantone diatonic scale, the superpyth diatonic is improper.

12-note MOS (5L 7s, borderline improper)
  • Archy12 – archy in 472edo tuning
  • Supra12 – supra in 56edo tuning
  • 12-22a – superpyth in 22edo tuning

The boundary of propriety is 17edo.

Tuning spectrum

ET
Generator
Eigenmonzo
(Unchanged Interval)
Generator
(¢)
Comments
4/3 701.955 Pythagorean tuning
10\17 705.882 Lower bound of 7- and 9-odd-limit diamond monotone
28/27 707.408 1/5 comma
23\39 707.692
9/7 708.771 1/4 comma, 1.3.7.9 minimax
16/15 708.807
13\22 709.091
5/4 709.590 9-odd-limit minimax
54/49 709.745 2/7 comma
25/24 710.040
29\49 710.204
6/5 710.545
7/5 710.681 7-odd-limit minimax
7/6 711.043 1/3 comma, 1.3.7 minimax
16\27 711.111
10/9 711.772
49/48 712.861 2/5 comma
8/7 715.587 1/2 comma
3\5 720.000 Upper bound of 7- and 9-odd-limit diamond monotone

Music

By Joel Grant Taylor, all in Superpyth[12] in 22edo tuning.

Both by Lillian Hearne in 22edo tuning

See also