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Superpyth | Suprapyth | Music

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 Bb (in contrast to meantone where A# is flatter than Bb, or 12edo where they are identical). An interesting coincidence is that the plastic numberhas a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth.If the 5th harmonic is used at all, it is mapped to -9 generators, 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 simplest reasonable way is to map 11/8 to 6 generators (so 11/8 is a "diminished fifth"), by tempering out 99/98.

This temperament is called "

supra", or "suprapyth" if you include 5 as well.MOSes include 5, 7, 12, 17, and 22.

## Superpyth

Commas: 64/63, 245/243POTE generator: ~3/2 = 710.291

Map: [<1 0 -12 6|, <0 1 9 -2|]

Wedgie: <<1 9 -2 12 -6 -30||

EDOs: 5, 17, 22, 27, 49

Badness: 0.0323

## 11-limit

Commas: 64/63, 100/99, 245/243POTE generator: ~3/2 = 710.175

Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]

EDOs: 22, 27e, 49

Badness: 0.0250

## 13-limit

Commas: 64/63, 78/77, 91/90, 100/99POTE generator: ~3/2 = 710.479

Map: [<1 0 -12 6 -22 -17|, <0 1 9 -2 16 13|]

EDOs: 22, 27e, 49, 76bcde

Badness: 0.0247

## Suprapyth

Commas: 55/54, 64/63, 99/98POTE generator: ~3/2 = 709.495

Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|]

EDOs: 5, 17, 22

Badness: 0.0328

## Interval chains

## Basic superpyth (2.3.7)

## Full 7-limit superpyth

## Supra (2.3.7.11)

## Full 11-limit suprapyth

## MOSes

## 5-note (LsLss, proper)

See 2L 3s.## 7-note (LLLsLLs, improper)

See 5L 2s. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.## 12-note (LsLsLssLsLss, borderline improper)

See 5L 7s. The boundary of propriety is 17edo.## Music

12of22studyPentUp4thsMstr12of22gamelan1b

12of22study3 (children's story)

12of22study7

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