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 [[~~Circle-of-fifths notation|~~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. ~~The~~ [[~~Wikipedia:Plastic number~~|~~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~~.

'''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-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 (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–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 (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𝄪), by tempering out [[31213/31104]].

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

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== 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.

=== Prime-optimized tunings ===

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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&ndash;G and C&ndash;B♭ are harmonic sevenths). Since 3/2 is a generator we can use the same standard [[Circle-of-fifths notation|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. The [[Wikipedia:Plastic number|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.
+'''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&ndash;G and C&ndash;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-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 (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–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 (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𝄪), by tempering out [[31213/31104]].
 [[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22.
@@ Line 239: / Line 239: @@
 == 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.
 === Prime-optimized tunings ===
 {| class="wikitable mw-collapsible mw-collapsed"