Leapday: Difference between revisions

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: ''Not to be confused with calendar-based scales such as those in [[293edo]], [[400edo]], [[353edo]] or [[Irvic scale|Irvian mode]].''

: ''Not to be confused with calendar-based scales such as those in [[293edo]], [[400edo]], [[353edo]], or [[Irvic scale|Irvian mode]].''

'''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 (approximately 704{{cent}}) 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, 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 harmonic 5 are simple while ~~the~~ intervals of harmonics 7, 11, and 13 are complex, ~~while~~ leapday ~~has 3/2~~ tuned sharp so that intervals of ~~harmonics~~ 7, 11, and 13 are relatively simple while intervals of ~~harmonic~~ 5 are complex.

As a result, leapday is very much the "opposite" of meantone in many respects, similar to [[superpyth]]: while in meantone (including [[12edo]]) the fifth is tuned flat so that intervals of harmonic 5 are simple while intervals of harmonics 7, 11, and 13 are complex, in leapday the fifth is tuned sharp so that intervals of 7, 11, and 13 are relatively simple while intervals of 5 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 5/4 is complex and the canonical mapping for prime 19 is fairly ~~off~~.

If ratios of 5 are omitted, the 2.3.7.11.13 [[subgroup]] version of leapday is known as '''leapfrog''', 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 inaccurate.

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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-: ''Not to be confused with calendar-based scales such as those in [[293edo]], [[400edo]], [[353edo]] or [[Irvic scale|Irvian mode]].''
+: ''Not to be confused with calendar-based scales such as those in [[293edo]], [[400edo]], [[353edo]], or [[Irvic scale|Irvian mode]].''
 '''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 (approximately 704{{cent}}) 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, 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 harmonic 5 are simple while the intervals of harmonics 7, 11, and 13 are complex, while leapday has 3/2 tuned sharp so that intervals of harmonics 7, 11, and 13 are relatively simple while intervals of harmonic 5 are complex.
+As a result, leapday is very much the "opposite" of meantone in many respects, similar to [[superpyth]]: while in meantone (including [[12edo]]) the fifth is tuned flat so that intervals of harmonic 5 are simple while intervals of harmonics 7, 11, and 13 are complex, in leapday the fifth is tuned sharp so that intervals of 7, 11, and 13 are relatively simple while intervals of 5 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 5/4 is complex and the canonical mapping for prime 19 is fairly off.
+If ratios of 5 are omitted, the 2.3.7.11.13 [[subgroup]] version of leapday is known as '''leapfrog''', 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 inaccurate.
 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>.