Superpyth: Difference between revisions

| Comma basis = [[64/63]] (2.3.7); <br> [[64/63]], [[245/243]] (2.3.5.7)

| Edo join 1 = 5 | Edo join 2 = 22

| Odd limit 1 = (2.3.7) 9 | Mistuning 1 = ? | Complexity 1 = 12

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

'''Superpyth''', often 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]].  

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.

If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so 5/4 is an augmented second (e.g. C–D♯). Therefore 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. A simpler way to map it is 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]].

For more technical data, see [[Archytas clan #Superpyth]].

|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7-subgroup prime-optimized tunings

|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings