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]]. 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, for example, A♯ is sharper than B♭ (in contrast to meantone where ~~A♯ is~~ flatter than ~~B♭,~~ or ~~12edo where they are identical)~~. An interesting coincidence is that 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.

'''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]]. 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. An interesting coincidence is that 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.

Such a temperament without the 5th harmonic is also called '''archy'''. If ~~the 5th harmonic is used at all~~, 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~~ has 3/2 ~~tempered narrow~~ so that intervals ~~of 5~~ are simple and intervals ~~of 7~~ are complex, while superpyth has 3/2 ~~tempered wide~~ so that intervals ~~of 7~~ are simple while intervals ~~of 5~~ are complex.

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, ~~so 11/8 is~~ a ~~double~~ augmented second (~~C-Dx~~), tempering out 100/99. Yet a simpler but reasonable way is to map it to -6 generators, ~~so 11/8 is~~ 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]].

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

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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]]. 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, for example, A♯ is sharper than B♭ (in contrast to meantone where A♯ is flatter than B♭, or 12edo where they are identical). An interesting coincidence is that 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.
+'''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]]. 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. An interesting coincidence is that 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.
-Such a temperament without the 5th harmonic is also called '''archy'''. If the 5th harmonic is used at all, 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 has 3/2 tempered narrow so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 3/2 tempered wide so that intervals of 7 are simple while intervals of 5 are complex.
+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, so 11/8 is a double augmented second (C-Dx), tempering out 100/99. Yet a simpler but reasonable way is to map it to -6 generators, so 11/8 is 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]].
 [[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22.