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 (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 give [[17/16]].

Equivalently:

* ~~The minor third (C–E♭) represents [[13/11]],~~

* the classical major third is represented by a triply augmented unison (C–C𝄪♯),

* The major third ~~(C–E) represents [[14/11]],~~

* the harmonic seventh is represented by a doubly augmented fifth (C–G𝄪),

* The chromatic semitone (C–C♯) represents [[14/13]] (as in [[parapyth]]),

* 11/8 is represented by an augmented third (C–E♯),

* 23/16 is represented by an augmented ~~fourth~~ (~~C–F♯~~),

* 13/8 is represented by an augmented fifth (C–G♯),

* ~~13/8~~ is represented by an augmented fifth (~~C–G♯~~),

* 17/16 is represented by an octave-reduced triply augmented sixth (C–A𝄪♯), and

* 11/8 is represented by an augmented third (C–E♯),

* 23/16 is represented by an augmented fourth (C–F♯).

* 7/~~4, the harmonic seventh,~~ is represented by ~~a doubly~~ augmented fifth (~~C–G𝄪~~),

* 5/~~4, the classical major third,~~ is represented by a triply augmented ~~unison~~ (~~C–C𝄪♯),~~

* 6/5, the classical minor third, is a triply diminished fifth (C–G♭♭♭), and

* ~~If intervals of 17 are desired, 17~~/16 is represented by an ~~octave-reduced triply~~ augmented ~~sixth~~ (~~C–A𝄪♯~~).

As a result, leapday is very much the "opposite" of meantone in many respects, similar to [[superpyth]]: meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of harmonics 7, 11, and 13 are complex, while leapday has the fifth tuned sharp so that intervals of 7, 11, and 13 are relatively simple while intervals of 5 are complex.

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| 4

| 417.2

| 14/11, 81/64

| 14/11, 33/26

| 23/18

|-

@@ 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 (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 give [[17/16]].
 Equivalently:
-* The minor third (C–E♭) represents [[13/11]],
+* the classical major third is represented by a triply augmented unison (C–C𝄪♯),
-* The major third (C–E) represents [[14/11]],
+* the harmonic seventh is represented by a doubly augmented fifth (C–G𝄪),
-* The chromatic semitone (C–C♯) represents [[14/13]] (as in [[parapyth]]),
+* 11/8 is represented by an augmented third (C–E♯),
-* 23/16 is represented by an augmented fourth (C–F♯),
+* 13/8 is represented by an augmented fifth (C–G♯),
-* 13/8 is represented by an augmented fifth (C–G♯),
+* 17/16 is represented by an octave-reduced triply augmented sixth (C–A𝄪♯), and
-* 11/8 is represented by an augmented third (C–E♯),
+* 23/16 is represented by an augmented fourth (C–F♯).
-* 7/4, the harmonic seventh, is represented by a doubly augmented fifth (C–G𝄪),
-* 5/4, the classical major third, is represented by a triply augmented unison (C–C𝄪♯),
-* 6/5, the classical minor third, is a triply diminished fifth (C–G♭♭♭), and
-* If intervals of 17 are desired, 17/16 is represented by an octave-reduced triply augmented sixth (C–A𝄪♯).
 As a result, leapday is very much the "opposite" of meantone in many respects, similar to [[superpyth]]: meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of harmonics 7, 11, and 13 are complex, while leapday has the fifth tuned sharp so that intervals of 7, 11, and 13 are relatively simple while intervals of 5 are complex.
@@ Line 57: / Line 53: @@
 | 4
 | 417.2
-| 14/11, 81/64
+| 14/11, 33/26
 | 23/18
 |-