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 [[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.

'''Superpyth''', also known as '''superpythagorean''', 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.

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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.

[[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]].

== Interval chains ==

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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 [[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.
+'''Superpyth''', also known as '''superpythagorean''', 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&ndash;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.
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 If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C&ndash;F𝄪), by tempering out [[31213/31104]].
-[[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22.
+[[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]].
 == Interval chains ==