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𝄪♯). | '''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. | |||
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 prime 5 is complex and the canonical mapping for prime 19 is fairly off. | ||
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{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
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! rowspan="2" | # | ! rowspan="2" | # | ||
! rowspan="2" | Cents* | ! rowspan="2" | Cents* | ||
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! 13-limit | ! 13-limit | ||
! | ! No-19 23-limit extension | ||
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| 0 | | 0 | ||