Superpyth: Difference between revisions

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{{Infobox regtemp|Comma basis=[[64/63]] (2.3.7); <br> [[64/63]], [[245/243]] (2.3.5.7)|Edo join 1=5|Edo join 2=22|Generator=3/2|Generator tuning=712.6|Optimization method=DKW|Mapping=1; 1 9 -2

{{Infobox regtemp

| Comma basis = [[64/63]] (2.3.7);<br />[[64/63]], [[245/243]] (2.3.5.7)

| Edo join 1 = 5

| Edo join 2 = 22

| Generator = 3/2

| Generator tuning = 712.6

| Optimization method = DKW

| Mapping = 1; 1 9 -2

| Pergen = (P8, P5)

| Color name = Ruti

|MOS scales=[[2L 3s]], [[5L 2s]], [[5L 7s]]|Subgroups=2.3.7, 2.3.5.7|Title=Archy; superpyth|Odd limit 1=(2.3.7) 9|Mistuning 1=?|Odd limit 2=9|Mistuning 2=?|Complexity 1=12|Complexity 2=27}}'''Superpyth,''' often called '''archy''' in the no-fives subgroup, is a [[regular temperament|temperament]] where the [[generator]] is [[4/3]], tuned flat so that stacking two of them gives ~~the~~ interval [[7/4]]. This means that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, two [[3/2]] perfect fifths [[octave reduction|octave-reduced]] gives [[8/7]]~~. Three~~ fourths reach a subminor third that approximates [[7/6]], ~~while~~ four fifths reach a supermajor third that approximates [[9/7]]. This means that the septimal comma ([[64/63]]) is [[tempering out|tempered out]].

| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]]

| Subgroups = 2.3.7, 2.3.5.7

| Title = Archy and superpyth

| Odd limit 1 = (2.3.7) 9

| Mistuning 1 = ?

| Odd limit 2 = 9

| Mistuning 2 = ?

| Complexity 1 = 12

| Complexity 2 = 27

}}

'''Superpyth,''' often called '''archy''' in the no-fives subgroup, is a [[regular temperament|temperament]] where the [[generator]] is [[4/3]], tuned flat so that stacking two of them gives an interval representing both [[7/4]] and [[16/9]]. This means that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, two [[3/2]] perfect fifths [[octave reduction|octave-reduced]] gives a major whole tone representing both [[9/8]] and [[8/7]], three fourths reach a subminor third that approximates [[7/6]], and four fifths reach a supermajor third that approximates [[9/7]]. This means that the septimal comma ([[64/63]]) is [[tempering out|tempered out]].

Since the generator is a perfect fourth or perfect fifth, archy can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], 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. [[22edo|13\22]] and [[27edo|16\27]] are ~~typical~~ tunings of the generator.

Since the generator is a perfect fourth or perfect fifth, archy and superpyth can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], 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. [[22edo|13\22]] (~{{frac|1|4}}-septimal comma) and [[27edo|16\27]] (~{{frac|1|3}}-septimal comma) are the most common tunings of the generator.

If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so C–D♯ (an augmented second or limma-flat major third) is 5/4, leading to the full 7-limit superpyth temperament. Superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of 7 are complex, while superpyth has the fifth tuned sharp 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, or a doubly augmented second (C–Dx), 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 11 are desired, the canonical way is to map 11/8 to +16 generators, or a doubly augmented second (C–Dx), tempering out 100/99. A simpler way to map it is 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 (C–F𝄪), by tempering out [[31213/31104]].

If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C–F𝄪, as in [[27edo]]), by tempering out [[31213/31104]].

[[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22.

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== Scales ==

; 5-note mos ([[2L 3s]], proper)

; 5-note mos ([[2L 3s]], proper)

* [[Archy5]] – archy in 472edo tuning

; 7-note mos ([[5L 2s]], improper)

; 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)

; 12-note mos ([[5L 7s]], borderline improper)

* [[Archy12]] – archy in 472edo tuning

* [[Supra12]] – supra in 56edo tuning

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=== Prime-optimized tunings ===

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7 Subgroup ~~Prime~~-~~Optimized Tunings~~

|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7 Subgroup prime-optimized tunings

|-

! rowspan="2" |

@@ Line 5: / Line 5: @@
 | ja =
 }}
-{{Infobox regtemp|Comma basis=[[64/63]] (2.3.7); <br> [[64/63]], [[245/243]] (2.3.5.7)|Edo join 1=5|Edo join 2=22|Generator=3/2|Generator tuning=712.6|Optimization method=DKW|Mapping=1; 1 9 -2
+{{Infobox regtemp
+| Comma basis = [[64/63]] (2.3.7);<br />[[64/63]], [[245/243]] (2.3.5.7)
+| Edo join 1 = 5
+| Edo join 2 = 22
+| Generator = 3/2
+| Generator tuning = 712.6
+| Optimization method = DKW
+| Mapping = 1; 1 9 -2
 | Pergen = (P8, P5)
 | Color name = Ruti
-|MOS scales=[[2L 3s]], [[5L 2s]], [[5L 7s]]|Subgroups=2.3.7, 2.3.5.7|Title=Archy; superpyth|Odd limit 1=(2.3.7) 9|Mistuning 1=?|Odd limit 2=9|Mistuning 2=?|Complexity 1=12|Complexity 2=27}}'''Superpyth,''' often called '''archy''' in the no-fives subgroup, is a [[regular temperament|temperament]] where the [[generator]] is [[4/3]], tuned flat so that stacking two of them gives the interval [[7/4]]. This means that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, two [[3/2]] perfect fifths [[octave reduction|octave-reduced]] gives [[8/7]]. Three fourths reach a subminor third that approximates [[7/6]], while four fifths reach a supermajor third that approximates [[9/7]]. This means that the septimal comma ([[64/63]]) is [[tempering out|tempered out]].
+| MOS scales = [[2L&nbsp;3s]], [[5L&nbsp;2s]], [[5L&nbsp;7s]]
+| Subgroups = 2.3.7, 2.3.5.7
+| Title = Archy and superpyth
+| Odd limit 1 = (2.3.7) 9
+| Mistuning 1 = ?
+| Odd limit 2 = 9
+| Mistuning 2 = ?
+| Complexity 1 = 12
+| Complexity 2 = 27
+}}
+'''Superpyth,''' often called '''archy''' in the no-fives subgroup, is a [[regular temperament|temperament]] where the [[generator]] is [[4/3]], tuned flat so that stacking two of them gives an interval representing both [[7/4]] and [[16/9]]. This means that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, two [[3/2]] perfect fifths [[octave reduction|octave-reduced]] gives a major whole tone representing both [[9/8]] and [[8/7]], three fourths reach a subminor third that approximates [[7/6]], and four fifths reach a supermajor third that approximates [[9/7]]. This means that the septimal comma ([[64/63]]) is [[tempering out|tempered out]].
-Since the generator is a perfect fourth or perfect fifth, archy can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], 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. [[22edo|13\22]] and [[27edo|16\27]] are typical tunings of the generator.
+Since the generator is a perfect fourth or perfect fifth, archy and superpyth can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], 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. [[22edo|13\22]] (~{{frac|1|4}}-septimal comma) and [[27edo|16\27]] (~{{frac|1|3}}-septimal comma) are the most common tunings of the generator.
 If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so C–D♯ (an augmented second or limma-flat major third) is 5/4, leading to the full 7-limit superpyth temperament. Superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of 7 are complex, while superpyth has the fifth tuned sharp 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, or a doubly augmented second (C–Dx), 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 11 are desired, the canonical way is to map 11/8 to +16 generators, or a doubly augmented second (C–Dx), tempering out 100/99. A simpler way to map it is 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 (C–F𝄪), by tempering out [[31213/31104]].
+If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C–F𝄪, as in [[27edo]]), by tempering out [[31213/31104]].
 [[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22.
@@ Line 123: / Line 140: @@
 == Scales ==
-; 5-note mos ([[2L 3s]], proper)
+; 5-note mos ([[2L&nbsp;3s]], proper)
 * [[Archy5]] – archy in 472edo tuning
-; 7-note mos ([[5L 2s]], improper)
+; 7-note mos ([[5L&nbsp;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)
+; 12-note mos ([[5L&nbsp;7s]], borderline improper)
 * [[Archy12]] – archy in 472edo tuning
 * [[Supra12]] – supra in 56edo tuning
@@ Line 142: / Line 159: @@
 === Prime-optimized tunings ===
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7 Subgroup Prime-Optimized Tunings
+|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7 Subgroup prime-optimized tunings
 |-
 ! rowspan="2" |