Superpyth: Difference between revisions
→Tunings: if you include 52edo, you must include 47edo. I don't think 52edo deserves a place here as it's too sharp even for ultrapyth. Similar story for 107edo tuning |
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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 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. | '''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. 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 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. | 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. | ||