Leapday: Difference between revisions

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♯~~), 13/8 is represented by an augmented fifth (~~C–G♯~~), 11/8 is represented by an augmented third (~~C–E♯~~)<ref group="note" name="enharmonics">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𝄪~~)~~<ref group="note" name="enharmonics" />~~, 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 (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 ~~the~~ 5~~th harmonic's intervals~~ are simple while the 7~~th~~, 11~~th~~, and 13~~th harmonics' intervals~~ are complex, while ~~Leapday~~ has 3/2 tuned sharp so that ~~the~~ 7~~th~~, 11~~th~~, and 13~~th harmonics' intervals~~ are (relatively) simple while ~~the~~ 5~~th harmonic's intervals~~ are complex.

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.

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]])4, 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.

Line 16:

{| class="wikitable center-1 right-2"

|-

! rowspan="2" | &#~~35;~~

! rowspan="2" | #

! rowspan="2" | Cents*

! colspan="2" | Approximate ratios

Line 150:

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

! Edo generator

! Edo generator

! [[Eigenmonzo|Eigenmonzo (unchanged-interval)]]*

! [[Eigenmonzo|Eigenmonzo (unchanged-interval)]]*

! Generator (¢)

! Comments

Line 420:

|

|}

<nowiki />* Besides the octave

<nowiki/>* Besides the octave

~~== Notes ==~~

~~<references group="note" />~~

== References and external links ==

[[Category:Leapday| ]]

@@ 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;)<ref group="note" name="enharmonics">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;)<ref group="note" name="enharmonics" />, 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 (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 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.
+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.
 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.
@@ Line 16: / Line 16: @@
 {| class="wikitable center-1 right-2"
 |-
-! rowspan="2" | &#35;
+! rowspan="2" | #
 ! rowspan="2" | Cents*
 ! colspan="2" | Approximate ratios
@@ Line 150: / Line 150: @@
 {| class="wikitable center-all left-4"
-! Edo<br />generator
+! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*
+! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
 ! Generator (¢)
 ! Comments
@@ Line 420: / Line 420: @@
 |
 |}
-<nowiki />* Besides the octave
+<nowiki/>* Besides the octave
-== Notes ==
-<references group="note" />
 == References and external links ==
-<references />
+<references/>
 [[Category:Leapday| ]] <!-- Main article -->