Leapday: Difference between revisions

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'''Leapday''' is a [[regular temperament]] for the 7-, 11-, 13-, 17-, and no-19 23-limit. It is based on the [[chain of fifths]], but here, the fifth is tuned slightly sharp of just so that 6 fifths give [[23/16]], 8 fifths give [[13/8]], 11 fifths give [[11/8]], 15 fifths give [[7/4]], 21 fifths give [[5/4]], and 24 fifths make [[17/16]]. Equivalently, the fifth in leapday is ~2.3 cents sharp of 3/2 (approximately 704{{cent}}), so that 23/16 is represented by an augmented fourth (C–F♯), 13/8 is represented by an augmented fifth (C–G♯), 11/8 is represented by an augmented third (C–E♯), the harmonic seventh is represented by a doubly augmented fifth (C–G𝄪), the classical major third is represented by a triply augmented unison (C–C𝄪♯), and 17/16 is represented by an octave-reduced triply augmented sixth (C–A𝄪♯).

Like [[superpyth]], Leapday goes in a completely different direction than meantone despite being based on the circle of fifths: ~~meantone~~ (including [[12edo]]) has 3/2 tuned flat so that intervals ~~of 5~~ are simple while ~~intervals of~~ 7, 11, and 13 are complex, while Leapday has 3/2 tuned sharp so that ~~intervals of~~ 7, 11, and 13 are (relatively) simple while intervals ~~of 5~~ are complex.

Like [[superpyth]], Leapday goes in a completely different direction than meantone despite being based on the circle of fifths: Meantone (including [[12edo]]) has 3/2 tuned flat so that the 5th harmonic's intervals are simple while the 7th, 11th, and 13th harmonics' intervals are complex, while Leapday has 3/2 tuned sharp so that the 7th, 11th, and 13th harmonics' intervals are (relatively) simple while the 5th harmonic's intervals are complex.

The no-5's 13-limit version of leapday, known as '''leapfrog''', is notable as tempering [[parapythic]] (a rank-3 temperament of the 2.3.7.11.13 subgroup) to rank 2 by finding [[~]][[13/8]] at ([[~]][[9/8]])4, that is, by tempering out the [[tetris comma]], and is a good combination of simplicity and accuracy, as ~~prime~~ 5 is complex and the canonical mapping for prime 19 is fairly off.

The no-5's 13-limit version of leapday, known as '''leapfrog''', is notable as tempering [[parapythic]] (a rank-3 temperament of the 2.3.7.11.13 subgroup) to rank 2 by finding [[~]][[13/8]] at ([[~]][[9/8]])4, that is, by tempering out the [[tetris comma]], and is a good combination of simplicity and accuracy, as 5/4 is complex and the canonical mapping for prime 19 is fairly off.

Leapday was named by [[Herman Miller]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10589.html Yahoo! Tuning Group (Archive) | ''Some 13-limit temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10604.html Yahoo! Tuning Group (Archive) | ''24 13-limit temperaments supported by 46'']</ref>.

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 '''Leapday''' is a [[regular temperament]] for the 7-, 11-, 13-, 17-, and no-19 23-limit. It is based on the [[chain of fifths]], but here, the fifth is tuned slightly sharp of just so that 6 fifths give [[23/16]], 8 fifths give [[13/8]], 11 fifths give [[11/8]], 15 fifths give [[7/4]], 21 fifths give [[5/4]], and 24 fifths make [[17/16]]. Equivalently, the fifth in leapday is ~2.3 cents sharp of 3/2 (approximately 704{{cent}}), so that 23/16 is represented by an augmented fourth (C–F&#x266F;), 13/8 is represented by an augmented fifth (C–G&#x266F;), 11/8 is represented by an augmented third (C–E&#x266F;), the harmonic seventh is represented by a doubly augmented fifth (C–G&#x1D12A;), the classical major third is represented by a triply augmented unison (C&ndash;C&#x1D12A;&#x266F;), and 17/16 is represented by an octave-reduced triply augmented sixth (C–A&#x1D12A;&#x266F;).
-Like [[superpyth]], Leapday goes in a completely different direction than meantone despite being based on the circle of fifths: meantone (including [[12edo]]) has 3/2 tuned flat so that intervals of 5 are simple while intervals of 7, 11, and 13 are complex, while Leapday has 3/2 tuned sharp so that intervals of 7, 11, and 13 are (relatively) simple while intervals of 5 are complex.
+Like [[superpyth]], Leapday goes in a completely different direction than meantone despite being based on the circle of fifths: Meantone (including [[12edo]]) has 3/2 tuned flat so that the 5<sup>th</sup> harmonic's intervals are simple while the 7<sup>th</sup>, 11<sup>th</sup>, and 13<sup>th</sup> harmonics' intervals are complex, while Leapday has 3/2 tuned sharp so that the 7<sup>th</sup>, 11<sup>th</sup>, and 13<sup>th</sup> harmonics' intervals are (relatively) simple while the 5<sup>th</sup> harmonic's intervals are complex.
-The no-5's 13-limit version of leapday, known as '''leapfrog''', is notable as tempering [[parapythic]] (a rank-3 temperament of the 2.3.7.11.13 subgroup) to rank 2 by finding [[~]][[13/8]] at ([[~]][[9/8]])<sup>4</sup>, that is, by tempering out the [[tetris comma]], and is a good combination of simplicity and accuracy, as prime 5 is complex and the canonical mapping for prime 19 is fairly off.
+The no-5's 13-limit version of leapday, known as '''leapfrog''', is notable as tempering [[parapythic]] (a rank-3 temperament of the 2.3.7.11.13 subgroup) to rank 2 by finding [[~]][[13/8]] at ([[~]][[9/8]])<sup>4</sup>, that is, by tempering out the [[tetris comma]], and is a good combination of simplicity and accuracy, as 5/4 is complex and the canonical mapping for prime 19 is fairly off.
 Leapday was named by [[Herman Miller]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10589.html Yahoo! Tuning Group (Archive) | ''Some 13-limit temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10604.html Yahoo! Tuning Group (Archive) | ''24 13-limit temperaments supported by 46'']</ref>.