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 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♯)<ref group="note">This implies that adjacent accidental notes such as F♯ and G♭ are different pitches, with flats sitting significantly lower than their nearby sharps.</ref>, 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 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.

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== Notes ==

== References and external links ==

@@ 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 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;).
+'''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;)<ref group="note">This implies that adjacent accidental notes such as F&#x266F; and G&#x266D; are different pitches, with flats sitting significantly lower than their nearby sharps.</ref>, 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 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.
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 == Notes ==
+<references group="note" />
+== References and external links ==
 <references />