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]].''

'''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~~𝄪♯).

'''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𝄪♯).

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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== Tunings ==

=== Tuning spectrum ===

This spectrum assumes 19-limit leapday.

{| class="wikitable center-all left-4"

@@ Line 1: / Line 1: @@
 : ''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.&nbsp;C&ndash;G&#x266F;), 11/8 is represented by an augmented third (e.g.&nbsp;C&ndash;E&#x266F;), the harmonic seventh is represented by a doubly augmented fifth (e.g.&nbsp;C&ndash;G&#x1D12A;), the classical major third is represented by a triply augmented unison (e.g.&nbsp;C&ndash;C&#x1D12A;&#x266F;), and 17/16 is represented by an octave-reduced triply augmented sixth (e.g.&nbsp;C&ndash;A&#x1D12A;&#x266F;).
+'''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;).
 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.
@@ Line 117: / Line 117: @@
 == Tunings ==
 === Tuning spectrum ===
+This spectrum assumes 19-limit leapday.
 {| class="wikitable center-all left-4"