Leapday: Difference between revisions

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Equivalently:

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

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

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

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

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

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

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

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

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.

If ratios of 5 are omitted, the 2.3.7.11.13 [[subgroup]] version of leapday is known as '''leapfrog''', 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 inaccurate.

If ratios of 5 are omitted, the 2.3.7.11.13 [[subgroup]] version of leapday is known as '''leapfrog''', notable as tempering [[parapyth]] (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 inaccurate.

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>.

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|-

! 13-limit

! ~~No-19~~ 23~~-limit extension~~

! Additional ratios of 17 and 23

|-

| 0

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

! Edo generator

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

! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*

! Generator (¢)

! Comments

@@ Line 4: / Line 4: @@
 Equivalently:
-* the classical major third is represented by a triply augmented unison (C–C𝄪♯),
+* 5/4, the classical major third, is represented by a triply augmented unison (C–C𝄪♯),
-* the harmonic seventh is represented by a doubly augmented fifth (C–G𝄪),
+* 7/4, the harmonic seventh, is represented by a doubly augmented fifth (C–G𝄪),
 * 11/8 is represented by an augmented third (C–E♯),
 * 13/8 is represented by an augmented fifth (C–G♯),
 * 17/16 is represented by an octave-reduced triply augmented sixth (C–A𝄪♯), and
 * 23/16 is represented by an augmented fourth (C–F♯).
 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.
-If ratios of 5 are omitted, the 2.3.7.11.13 [[subgroup]] version of leapday is known as '''leapfrog''', 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 inaccurate.
+If ratios of 5 are omitted, the 2.3.7.11.13 [[subgroup]] version of leapday is known as '''leapfrog''', notable as tempering [[parapyth]] (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 inaccurate.
 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>.
@@ Line 29: / Line 29: @@
 |-
 ! 13-limit
-! No-19 23-limit extension
+! Additional ratios<br />of 17 and 23
 |-
 | 0
@@ Line 159: / Line 159: @@
 {| class="wikitable center-all left-4"
 ! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
+! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
 ! Generator (¢)
 ! Comments