Superpyth: Difference between revisions
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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 [[tempering out|tempered out]], so a stack of two generators [[Octave reduction|octave-reduced]] represents [[8/7]] in addition to [[9/8]] (in other words, intervals such as A–G and C–B♭ are harmonic sevenths). 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 sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[17edo|10\17]], [[22edo|13\22]], and [[27edo|16\27]] are typical tunings of the generator. | '''Superpyth''' is a [[temperament]] of the [[archytas clan]] where [[~]][[3/2]] is a [[generator]], and the Archytas comma [[64/63]] is [[tempering out|tempered out]], so a stack of two generators [[Octave reduction|octave-reduced]] represents [[8/7]] in addition to [[9/8]] (in other words, intervals such as A–G and C–B♭ are harmonic sevenths). 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 sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[17edo|10\17]], [[22edo|13\22]], and [[27edo|16\27]] are typical tunings of the generator. | ||
Such a temperament without the 5th harmonic is also called '''archy'''. If intervals of 5 are desired, it is mapped to +9 generators through tempering out [[245/243]], so C | Such a temperament without the 5th harmonic is also called '''archy'''. If intervals of 5 are desired, 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 (including [[12edo]]) has 3/2 tuned flat so that the 5th harmonic's intervals are simple and the 7th harmonic's intervals are complex, while superpyth has 3/2 tuned sharp so that the 7th harmonic's intervals are simple while the 5th harmonic's intervals are complex. | ||
If intervals of 11 are desired, the canonical way is to map 11/8 to +16 generators, or a doubly augmented second ( | If intervals of 11 are desired, the canonical way is to map 11/8 to +16 generators, or a doubly augmented second (C–D𝄪), tempering out 100/99. Yet a simpler but reasonable way is to map it to −6 generators, or a diminished fifth (C–G♭), by tempering out 99/98. The latter is called '''supra''', or '''suprapyth'''. The two mappings unite on [[22edo]]. | ||
If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth ( | If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C–F𝄪), by tempering out [[31213/31104]]. | ||
[[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22. | [[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22. | ||
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|+ style="font-size: 105%;" | Archy (2.3.7) | |+ style="font-size: 105%;" | Archy (2.3.7) | ||
|- | |- | ||
! # !! Cents* !! Approximate Ratios | ! # !! Cents* !! Approximate Ratios | ||
|- | |- | ||
| 0 || 0.0 || '''1/1''' | | 0 || 0.0 || '''1/1''' | ||
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|+ style="font-size: 105%;" | Supra (2.3.7.11) | |+ style="font-size: 105%;" | Supra (2.3.7.11) | ||
|- | |- | ||
! # !! Cents* !! Approximate Ratios | ! # !! Cents* !! Approximate Ratios | ||
|- | |- | ||
| 0 || 0.0 || '''1/1''' | | 0 || 0.0 || '''1/1''' | ||
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|+ style="font-size: 105%;" | Full 7-limit superpyth | |+ style="font-size: 105%;" | Full 7-limit superpyth | ||
|- | |- | ||
! rowspan="3" | # !! rowspan="3" | Cents* !! colspan="3" | Approximate Ratios | ! rowspan="3" | # !! rowspan="3" | Cents* !! colspan="3" | Approximate Ratios | ||
|- | |- | ||
! rowspan="2" | 7-limit !! colspan="2" | 11-limit Extension | ! rowspan="2" | 7-limit !! colspan="2" | 11-limit Extension | ||
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== Scales == | == Scales == | ||
; 5-note MOS ([[2L 3s]], proper) | ; 5-note MOS ([[2L 3s]], proper) | ||
* [[Archy5]] | * [[Archy5]] – archy in 472edo tuning | ||
; 7-note MOS ([[5L 2s]], improper) | ; 7-note MOS ([[5L 2s]], improper) | ||
* [[Archy7]] | * [[Archy7]] – archy in 472edo tuning | ||
* [[Supra7]] | * [[Supra7]] – supra in 56edo tuning | ||
In contrast to the meantone diatonic scale, the superpyth diatonic is improper. | In contrast to the meantone diatonic scale, the superpyth diatonic is improper. | ||
; 12-note MOS ([[5L 7s]], borderline improper) | ; 12-note MOS ([[5L 7s]], borderline improper) | ||
* [[Archy12]] | * [[Archy12]] – archy in 472edo tuning | ||
* [[Supra12]] | * [[Supra12]] – supra in 56edo tuning | ||
* [[12-22a]] | * [[12-22a]] – superpyth in 22edo tuning | ||
The boundary of propriety is [[17edo]]. | The boundary of propriety is [[17edo]]. | ||
== Tunings == | == Tunings == | ||
The {{w|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. This can be explained since superpyth equates [[21/16]] and [[4/3]], making the 9:12:16:21 chord evenly spaced by ~4/3, and when keeping ~9 + ~12 = ~21 the generator becomes the plastic number. | The {{w|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. This can be explained since superpyth equates [[21/16]] and [[4/3]], making the 9:12:16:21 chord evenly spaced by ~4/3, and when keeping {{nowrap|~9 + ~12 {{=}} ~21}} the generator becomes the plastic number. | ||
=== Prime-optimized tunings === | === Prime-optimized tunings === | ||
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[[Category:Temperaments]] | [[Category:Temperaments]] | ||
[[Category:Superpyth| ]] <!-- | [[Category:Superpyth| ]] <!-- Main article --> | ||
[[Category:Archytas clan]] | [[Category:Archytas clan]] | ||
[[Category:Sensamagic clan]] | [[Category:Sensamagic clan]] | ||
[[Category:Orwellismic temperaments]] | [[Category:Orwellismic temperaments]] | ||