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]].''

: ''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 (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]].

~~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 5 are simple while intervals of 7~~, 11, ~~and~~ 13 ~~are complex~~, ~~while Leapday has 3~~/~~2 tuned sharp so that intervals of 7, 11~~, and ~~13 are~~ (~~relatively~~) ~~simple while intervals of 5 are complex~~.

Equivalently:

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

* 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♯).

~~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 ~~prime~~ 5 is complex and the canonical mapping for prime 19 is fairly ~~off~~.

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

|-

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

! rowspan="2" | #

! rowspan="2" | Cents*

! colspan="2" | Approximate ratios

|-

! 13-limit

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

! Additional ratios of 17 and 23

|-

| 0

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| 4

| 417.2

| 14/11

| 14/11, 33/26

| 23/18

|-

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

! Edo generator

! Edo generator

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

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

! Generator (¢)

! Comments

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|

|}

<nowiki />* Besides the octave

<nowiki/>* Besides the octave

== ~~Notes~~ ==

== References and external links ==

[[Category:Leapday| ]]

[[Category:~~Temperaments~~]]

[[Category:Rank-2 temperaments]]

[[Category:Sengic temperaments]]

[[Category:Hemifamity temperaments]]

@@ Line 1: / Line 1: @@
-: ''Not to be confused with calendar-based scales such as those in [[293edo]], [[400edo]], [[353edo]] or [[Irvic scale|Irvian mode]].''
+: ''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 (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]].
-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 5 are simple while intervals of 7, 11, and 13 are complex, while Leapday has 3/2 tuned sharp so that intervals of 7, 11, and 13 are (relatively) simple while intervals of 5 are complex.
+Equivalently:
+* 5/4, the classical major third, is represented by a triply augmented unison (C–C𝄪♯),
+* 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♯).
-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.
+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 [[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 16: / Line 24: @@
 {| class="wikitable center-1 right-2"
 |-
-! rowspan="2" | &#35;
+! rowspan="2" | #
 ! rowspan="2" | Cents*
 ! colspan="2" | Approximate ratios
 |-
 ! 13-limit
-! No-19 23-limit extension
+! Additional ratios<br />of 17 and 23
 |-
 | 0
@@ Line 45: / Line 53: @@
 | 4
 | 417.2
-| 14/11
+| 14/11, 33/26
 | 23/18
 |-
@@ Line 150: / Line 158: @@
 {| class="wikitable center-all left-4"
-! Edo<br />generator
+! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*
+! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
 ! Generator (¢)
 ! Comments
@@ Line 420: / Line 428: @@
 |
 |}
-<nowiki />* Besides the octave
+<nowiki/>* Besides the octave
-== Notes ==
+== References and external links ==
-<references />
+<references/>
 [[Category:Leapday| ]] <!-- Main article -->
-[[Category:Temperaments]]
+[[Category:Rank-2 temperaments]]
 [[Category:Sengic temperaments]]
 [[Category:Hemifamity temperaments]]