Superpyth: Difference between revisions

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{{Infobox regtemp

{{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

~~| Title = Archy; superpyth~~

~~| Subgroups = 2.3.7, 2.3.5.7~~

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

~~| Mapping = 1; 1 9 -2~~

| Edo join 1 = 5 | Edo join 2 = 22

| Generator = 3/2

| Generator tuning = 712.6

| Optimization method = DKW

| Pergen = (P8, P5)

| Color name = Ruti

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

|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}}'''Archy''' 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]].

| Odd limit 1 = (2.3.7) 9 | Mistuning 1 = ? ~~| Complexity 1 = 12~~

| Odd limit 2 = 9 | Mistuning 2 = ? | Complexity 2 = 27

}}

'''~~Superpyth~~''' is a [[regular temperament|temperament]] where the [[generator]] is a [[3~~/2|perfect fifth~~]], tuned ~~sharp such~~ that ~~a stack~~ of ~~two perfect fifths [[octave reduction|octave-reduced]]~~ gives ~~a whole tone that represents both [[9/8]] and [[8/7]], [[tempering out]]~~ the ~~septimal comma, [[64/63]]. Likewise, two perfect fourths give a minor seventh that represents both~~ [[7/4]] ~~and [[16/9]], so~~ that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, ~~three~~ fourths reach a ~~minor~~ third that approximates [[7/6]], while four fifths reach a ~~major~~ third that approximates [[9/7]].

Since the generator is a perfect fifth, ~~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]] and [[27edo|16\27]] are typical tunings of the generator.

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.

~~Such a temperament without the [[5/1|5th harmonic]] is also called '''archy'''.~~ If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so 5/4 is an ~~augmented second (e~~.~~g. C–D♯). Therefore 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 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 an extension often called '''superpyth'''. 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–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 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 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C–F𝄪), by tempering out [[31213/31104]].

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

For more technical data, see [[Archytas clan #Superpyth]].

For more technical data, see [[Archytas clan #Superpyth|Archytas clan.]]

== Interval chains ==

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 }}
-{{Infobox regtemp
+{{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
-| Title = Archy; superpyth
-| Subgroups = 2.3.7, 2.3.5.7
-| Comma basis = [[64/63]] (2.3.7); <br> [[64/63]], [[245/243]] (2.3.5.7)
-| Mapping = 1; 1 9 -2
-| Edo join 1 = 5 | Edo join 2 = 22
-| Generator = 3/2
-| Generator tuning = 712.6
-| Optimization method = DKW
 | Pergen = (P8, P5)
 | Color name = Ruti
-| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]]
+|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}}'''Archy''' 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]].
-| Odd limit 1 = (2.3.7) 9 | Mistuning 1 = ? | Complexity 1 = 12
-| Odd limit 2 = 9 | Mistuning 2 = ? | Complexity 2 = 27
-}}
-'''Superpyth''' is a [[regular temperament|temperament]] where the [[generator]] is a [[3/2|perfect fifth]], tuned sharp such that a stack of two perfect fifths [[octave reduction|octave-reduced]] gives a whole tone that represents both [[9/8]] and [[8/7]], [[tempering out]] the septimal comma, [[64/63]]. Likewise, two perfect fourths give a minor seventh that represents both [[7/4]] and [[16/9]], so that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, three fourths reach a minor third that approximates [[7/6]], while four fifths reach a major third that approximates [[9/7]].
-Since the generator is a perfect fifth, 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]] and [[27edo|16\27]] are typical tunings of the generator.
+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.
-Such a temperament without the [[5/1|5th harmonic]] is also called '''archy'''. If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so 5/4 is an augmented second (e.g. C–D♯). Therefore 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 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 an extension often called '''superpyth'''. 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–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 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 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C–F𝄪), by tempering out [[31213/31104]].
@@ Line 32: / Line 20: @@
 [[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22.
-For more technical data, see [[Archytas clan #Superpyth]].
+For more technical data, see [[Archytas clan #Superpyth|Archytas clan.]]
 == Interval chains ==