Superpyth

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Superpyth is a temperament of the archytas clan where ~3/2 is a generator, and the Archytas comma 64/63 is tempered out, so a stack of two generators octave-reduced represents 8/7 in addition to 9/8. Since 3/2 is a generator we can use the same standard chain-of-fifths 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 (~4/3) 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 3/2 tempered narrow so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 3/2 tempered wide 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.

For more technical data, see 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.

Tunings

Prime-optimized tunings

2.3.7 Prime-Optimized Tunings
Weight-skew\Order Euclidean
Tenney CTE
~3/2 = 709.5948¢
Tenney-Weil CTWE
~3/2 = 709.3901¢
Equilateral CEE
~3/2 = 712.8606¢
Eigenmonzo basis (unchanged-interval basis): 2.49/3 (2/5-comma tuning)
Equilateral-Weil CEWE
~3/2 = 711.9997¢
Eigenmonzo basis (unchanged-interval basis): 2.823543/243 (7/19-comma tuning)
Benedetti CBE
~3/2 = 707.7286¢
Eigenmonzo basis (unchanged-interval basis): 2.[0 -49 0 18 (18/85-comma tuning)
Benedetti-Weil CBWE
~3/2 = 707.9869¢
Eigenmonzo basis (unchanged-interval basis): 2.[0 -63 25 (25/113-comma tuning)
7-limit Prime-Optimized Tunings
Weight-skew\Order Euclidean
Tenney CTE
~3/2 = 709.5907¢
Tenney-Weil CTWE
~3/2 = 710.1193¢
Equilateral CEE
~3/2 = 709.7805¢
Eigenmonzo basis (unchanged-interval basis): 2.5859375/49
Equilateral-Weil CEWE
~3/2 = 710.2428¢
Eigenmonzo basis (unchanged-interval basis): 2.[0 3 -37 18
Benedetti CBE
~3/2 = 709.4859¢
Eigenmonzo basis (unchanged-interval basis): 2.[0 -1225 -3969 450
Benedetti-Weil CBWE
~3/2 = 710.0321¢
Eigenmonzo basis (unchanged-interval basis): 2.[0 665 -15771 5160

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, 2.3.7 CEE tuning
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