Superpyth: Difference between revisions

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

}}

'''Superpyth''', sometimes called '''archy''' in the ~~no-5~~ [[subgroup]], 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]].

'''Superpyth''', sometimes called '''archy''' in the [[2.3.7 subgroup]], 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]] (~1/4 septimal comma) and [[27edo|16\27]] (~1/3 septimal comma) are the most common tunings of the generator.

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== Tunings ==

The fifth of superpyth is supposed to be tuned sharp of just for the accuracy of the overall temperament. Roughly speaking, it ranges from as flat as [[Pythagorean tuning|Pythagorean]] (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, close to [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of full 7-limit superpyth's optimal range.

The fifth of superpyth is accurately tuned sharp of just. Roughly speaking, it ranges from as flat as [[Pythagorean tuning|Pythagorean]] (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, between [[52edo|52b-edo]] and [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of full 7-limit superpyth's optimal range.

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If we focus purely on the 2.3.7 subgroup for now, and as a starting point adopt an approach based on the example of [[quarter-comma meantone]], treating archy's harmonic 7 as analogous to 5 in meantone, 1/3-comma, where the whole tone leans towards 8/7 a bit, so that the 7 is as sharp as the 3 and that the 7/6 minor third is tuned just, emerges as a logical solution, due to being the [[minimax]] tuning for the [[tonality diamond]] formed by odds 3 and 7; 27edo is very close to a closed system of 1/3-comma archy. In general, however, we would want to treat 3 somewhat more importantly than 7; in meantone, similar principles imply than an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]], and in archy, these place it in between 1/3-comma and 1/4-comma. This is the most common approach to optimizing archy, and is supported by the standard [[CTE]] and [[CWE]] metrics. From that lens, 22edo can be viewed as a closed form of 1/4-comma archy, where the whole tone is midway between 8/7 and 9/8, so that the 7 is as sharp as the 9 and that the 9/7 major third is tuned just; it is, furthermore, slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the correct direction given the above discussion.

~~The prominence of this set of tunings is a reason why the mapping of 5/4 at +9 generators is chosen as the canonical extension;~~ 27edo is the point where superpyth tunes 5/4 to the familiar 400 cents of [[12edo]], and where in sharper tunings, there are different mappings of 5/4 with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to 19edo (which represents 1/3-comma meantone and is on the edge between septimal meantone and flattone). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the 11-limit extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5. There is an alternative extension, suprapyth, that works for tunings in the range of 17edo to 22edo, however.

27edo is also the point where superpyth tunes 5/4 to the familiar 400 cents of [[12edo]], and where in sharper tunings, there are different mappings of 5/4 with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to [[19edo]] (which represents [[1/3-comma meantone]] and is on the edge between septimal meantone and [[flattone]]). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the 11-limit extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5. There is an alternative extension, suprapyth, that works for tunings in the range of 17edo to 22edo, however.

However, a case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6, a similar logic to Zarlino's preference for [[2/7-comma meantone]]. This would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning; reading off the comma 64/63, both 3 and 7 must be sharp, and because the comma has two 3's and one 7 in the denominator, 3 is tuned twice as sharp as 7 in the CEE tuning. In this range of archy, the main extension to prime 5 is in fact not superpyth, but ultrapyth.

@@ Line 20: / Line 20: @@
 | Odd limit 2 = 9 | Mistuning 2 = ? | Complexity 2 = 27
 }}
-'''Superpyth''', sometimes called '''archy''' in the no-5 [[subgroup]], 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]].
+'''Superpyth''', sometimes called '''archy''' in the [[2.3.7 subgroup]], 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]] (~1/4 septimal comma) and [[27edo|16\27]] (~1/3 septimal comma) are the most common tunings of the generator.
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 == Tunings ==
+The fifth of superpyth is supposed to be tuned sharp of just for the accuracy of the overall temperament. Roughly speaking, it ranges from as flat as [[Pythagorean tuning|Pythagorean]] (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, close to [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of full 7-limit superpyth's optimal range.
 The fifth of superpyth is accurately tuned sharp of just. Roughly speaking, it ranges from as flat as [[Pythagorean tuning|Pythagorean]] (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, between [[52edo|52b-edo]] and [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of full 7-limit superpyth's optimal range.
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 If we focus purely on the 2.3.7 subgroup for now, and as a starting point adopt an approach based on the example of [[quarter-comma meantone]], treating archy's harmonic 7 as analogous to 5 in meantone, 1/3-comma, where the whole tone leans towards 8/7 a bit, so that the 7 is as sharp as the 3 and that the 7/6 minor third is tuned just, emerges as a logical solution, due to being the [[minimax]] tuning for the [[tonality diamond]] formed by odds 3 and 7; 27edo is very close to a closed system of 1/3-comma archy. In general, however, we would want to treat 3 somewhat more importantly than 7; in meantone, similar principles imply than an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]], and in archy, these place it in between 1/3-comma and 1/4-comma. This is the most common approach to optimizing archy, and is supported by the standard [[CTE]] and [[CWE]] metrics. From that lens, 22edo can be viewed as a closed form of 1/4-comma archy, where the whole tone is midway between 8/7 and 9/8, so that the 7 is as sharp as the 9 and that the 9/7 major third is tuned just; it is, furthermore, slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the correct direction given the above discussion.
-The prominence of this set of tunings is a reason why the mapping of 5/4 at +9 generators is chosen as the canonical extension; 27edo is the point where superpyth tunes 5/4 to the familiar 400 cents of [[12edo]], and where in sharper tunings, there are different mappings of 5/4 with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to 19edo (which represents 1/3-comma meantone and is on the edge between septimal meantone and flattone). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the 11-limit extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5. There is an alternative extension, suprapyth, that works for tunings in the range of 17edo to 22edo, however.
+edo is also the point where superpyth tunes 5/4 to the familiar 400 cents of [[12edo]], and where in sharper tunings, there are different mappings of 5/4 with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to [[19edo]] (which represents [[1/3-comma meantone]] and is on the edge between septimal meantone and [[flattone]]). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the 11-limit extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5. There is an alternative extension, suprapyth, that works for tunings in the range of 17edo to 22edo, however.
 However, a case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6, a similar logic to Zarlino's preference for [[2/7-comma meantone]]. This would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning; reading off the comma 64/63, both 3 and 7 must be sharp, and because the comma has two 3's and one 7 in the denominator, 3 is tuned twice as sharp as 7 in the CEE tuning. In this range of archy, the main extension to prime 5 is in fact not superpyth, but ultrapyth.

Superpyth: Difference between revisions

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