# Superpyth

**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)

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

- 12-note MOS (5L 7s, borderline improper)

The boundary of propriety is 17edo.

## Tunings

### 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) |

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

- 12of22studyPentUp4thsMstr
^{[dead link]} - 12of22gamelan1b
^{[dead link]} - 12of22study3 (children's story)
^{[dead link]} - 12of22study7
^{[dead link]}

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

Both by Lillian Hearne in 22edo tuning