Leapday: Difference between revisions
m Correction (tempering -> tempering out, "triple-augmented" (with hyphen) or "triply augmented" (without hyphen)) |
No-5 leapday has been named as leapfrog |
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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 19-limit. It is based on the [[chain of fifths]], but here, the fifth is tuned slightly sharp of just so that 8 fifths give a 13/8, 11 fifths make an 11/8, 15 fifths give 7/4, twenty-one fifths give [[5/4]], and twenty-four of them makes ~17/16. Equivalently, the fifth in leapday is ~2.3 cents sharp of 3/2 (approximately 704{{cent}}), so that 13/8 is represented by an augmented fifth (e.g. C–G♯), 11/8 is represented by an augmented third (e.g. C–E♯), the harmonic seventh is represented by a doubly augmented fifth (e.g. C–G𝄪), the classical major third is represented by a triply augmented unison (e.g. C–C𝄪♯), and 17/16 is represented by an octave-reduced triply augmented sixth (e.g. C–A𝄪♯). | |||
''' | 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. | ||
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>. | |||
See [[Hemifamity temperaments #Leapday]] or [[ | See [[Hemifamity temperaments #Leapday]] or [[No-fives subgroup temperaments #Leapfrog]] for more technical data. | ||
== Interval chain == | == Interval chain == | ||