Superpyth: Difference between revisions

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

{{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=4/3|Generator tuning=487.4|Optimization method=DKW|Mapping=1; 1 9 -2|MOS scales=[[3L 2s]], [[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]].

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. ~~[[17edo|10\17]],~~ [[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 C–D♯ is 5/4. ~~So 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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-'''Superpyth''' is a [[regular temperament|temperament]] where [[~]][[3/2]] is a [[generator]], and the septimal comma ([[64/63]]) is [[tempering out|tempered out]], so that a stack of two perfect fifths [[octave reduction|octave-reduced]] gives a major whole tone that represents both [[9/8]] and [[8/7]], and likewise, two perfect fourths give a minor seventh that represents both [[7/4]] and [[16/9]], so intervals such as A–G and C–B♭ are harmonic sevenths. Moreover, three fourths reach a subminor third that approximates [[7/6]], while four fifths reach a supermajor third that approximates [[9/7]].
+{{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=4/3|Generator tuning=487.4|Optimization method=DKW|Mapping=1; 1 9 -2|MOS scales=[[3L 2s]], [[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]].
-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. [[17edo|10\17]], [[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 C–D♯ is 5/4. So 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 ==

Superpyth: Difference between revisions

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