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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]].''


{{ See also | No-5's leapday }}
'''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]].


'''Leapday''' is a [[regular temperament]] for the 7-, 11-, 13-, 17-, and 19-limit, [[no-5's leapday|or simply as a no-5's 13-limit temperament]], which 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>. It is based on the [[chain of fifths]], but here, the fifth is tuned slightly sharp of just so that 8 fifths give a 13/8, 11 fifths make an 11/8, 15 fifths give 7/4, twenty-one fifths give [[5/4]], and twenty-four of them makes ~17/16. Equivalently, the fifth in leapday is ~2.3 cents sharp of 3/2 (approximately 704{{cent}}), so that 13/8 is represented by an augmented fifth (e.g.&nbsp;C&ndash;G&#x266F;), 11/8 is represented by an augmented third (e.g.&nbsp;C&ndash;E&#x266F;), the harmonic seventh is represented by a doubly augmented fifth (e.g.&nbsp;C&ndash;G&#x1D12A;), the classical major third is represented by a triply augmented unison (e.g.&nbsp;C&ndash;C&#x1D12A;&#x266F;), and 17/16 is represented by an octave-reduced triply-augmented sixth (e.g.&nbsp;C&ndash;A&#x1D12A;&#x266F;).
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 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 the [[tetris comma]], and is a good combination of simplicity and efficiency, 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.


See [[Hemifamity temperaments #Leapday]] or [[no-5's leapday]] for more technical data.
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>.
 
See [[Hemifamity temperaments #Leapday]] or [[No-fives subgroup temperaments #Leapfrog]] for more technical data.


== Interval chain ==
== Interval chain ==
In the following table, odd harmonics 1–21 are in '''bold'''.  
In the following table, odd harmonics 1–23 are in '''bold'''.  


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! #
|-
! Cents*
! rowspan="2" | #
! Approximate Ratios
! rowspan="2" | Cents*
! colspan="2" | Approximate ratios
|-
! 13-limit
! Additional ratios<br />of 17 and 23
|-
|-
| 0
| 0
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|
|-
|-
| 1
| 1
| 704.3
| 704.3
| '''3/2'''
| '''3/2'''
|
|-
|-
| 2
| 2
| 208.6
| 208.6
| '''9/8'''
| '''9/8'''
| 17/15, 26/23
|-
|-
| 3
| 3
| 912.9
| 912.9
| 22/13, 27/16
| 22/13, 27/16
| 17/10
|-
|-
| 4
| 4
| 417.2
| 417.2
| 14/11
| 14/11, 33/26
| 23/18
|-
|-
| 5
| 5
| 1121.5
| 1121.5
| 21/11, 40/21
| 21/11, 40/21
| 23/12, 44/23
|-
|-
| 6
| 6
| 625.8
| 625.8
| 10/7, 13/9
| 10/7, 13/9
| '''23/16'''
|-
|-
| 7
| 7
| 130.0
| 130.0
| 13/12, 14/13, 15/14
| 13/12, 14/13, 15/14
|
|-
|-
| 8
| 8
| 834.3
| 834.3
| '''13/8''', 21/13
| '''13/8''', 21/13
| 34/21
|-
|-
| 9
| 9
| 338.6
| 338.6
| 11/9, 39/32, 40/33
| 11/9, 39/32, 40/33
| 17/14, 28/23
|-
|-
| 10
| 10
| 1042.9
| 1042.9
| 11/6, 20/11
| 11/6, 20/11
| 42/23
|-
|-
| 11
| 11
| 547.2
| 547.2
| '''11/8''', 15/11
| '''11/8''', 15/11
|
|-
|-
| 12
| 12
| 51.5
| 51.5
| 28/27, 33/32, 40/39, 45/44
| 28/27, 33/32, 40/39, 45/44
| 34/33, 35/34
|-
|-
| 13
| 13
| 755.8
| 755.8
| 14/9, 20/13
| 14/9, 20/13
| 17/11
|-
|-
| 14
| 14
| 260.1
| 260.1
| 7/6, 15/13
| 7/6, 15/13
|
|-
|-
| 15
| 15
| 964.4
| 964.4
| '''7/4'''
| '''7/4'''
| 40/23
|-
|-
| 16
| 16
| 468.7
| 468.7
| '''21/16'''
| '''21/16'''
| 17/13, 30/23
|-
|-
| 17
| 17
| 1173.0
| 1173.0
| 63/32, 160/81
| 63/32, 160/81
| 45/23, 51/26
|-
|-
| 18
| 18
| 677.3
| 677.3
| 40/27
| 40/27
| 34/23
|-
|-
| 19
| 19
| 181.6
| 181.6
| 10/9
| 10/9
|
|-
|-
| 20
| 20
| 885.8
| 885.8
| 5/3
| 5/3
|
|-
|-
| 21
| 21
| 390.1
| 390.1
| '''5/4'''
| '''5/4'''
|
|-
|-
| 22
| 22
| 1094.4
| 1094.4
| '''15/8'''
| '''15/8'''
| 17/9
|-
|-
| 23
| 23
| 598.7
| 598.7
| 45/32
| 45/32
| 17/12
|}
|}
<nowiki />* In 13-limit CTE tuning
<nowiki />* In 13-limit CTE tuning


== Tuning spectrum ==
== Tunings ==
Gencom: [2 3/2; 91/90 121/120 133/132 136/135 154/153 169/168]
=== Tuning spectrum ===
 
This spectrum assumes 19-limit leapday.
Gencom mapping: {{mapping| 1 1 -10 -6 -3 -1 -10 6 | 0 1 21 15 11 8 24 -3 }}


{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Edo<br />Generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 133: Line 169:
|-
|-
|  
|  
| 24/19
| 19/12
| 701.110
| 701.110
|  
|  
Line 143: Line 179:
|-
|-
|  
|  
| 4/3
| 3/2
| 701.955
| 701.955
|  
|  
Line 150: Line 186:
|  
|  
| 702.439
| 702.439
|  
| 41cc… val, lower bound of 5-odd-limit diamond monotone
|-
|-
|  
|  
Line 180: Line 216:
|  
|  
| 703.448
| 703.448
| Lower bound of 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
| 29g val, lower bound of 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
|-
|-
|  
|  
Line 203: Line 239:
|-
|-
|  
|  
| 20/19
| 19/10
| 703.700
| 703.700
|  
|  
|-
|-
|  
|  
| 26/21
| 21/13
| 703.782
| 703.782
|  
|  
|-
|-
|  
|  
| 22/19
| 19/11
| 703.843
| 703.843
|  
|  
Line 223: Line 259:
|-
|-
|  
|  
| 22/21
| 21/11
| 703.893
| 703.893
|  
|  
|-
|-
|  
|  
| 26/19
| 19/13
| 703.910
| 703.910
|  
|  
Line 245: Line 281:
|  
|  
| 704.000
| 704.000
|  
| 75dfgh val
|-
|-
|  
|  
| 16/15
| 15/8
| 704.012
| 704.012
|  
|  
Line 263: Line 299:
|-
|-
|  
|  
| 14/13
| 13/7
| 704.043
| 704.043
|  
|  
Line 273: Line 309:
|-
|-
|  
|  
| 22/17
| 17/11
| 704.126
| 704.126
|  
|  
Line 280: Line 316:
|  
|  
| 704.132
| 704.132
|  
| 121defgh val
|-
|-
|  
|  
| 6/5
| 5/3
| 704.218
| 704.218
| 7-, 15- and 17-odd-limit minimax
| 7-, 15- and 17-odd-limit minimax
Line 293: Line 329:
|-
|-
|  
|  
| 10/9
| 9/5
| 704.337
| 704.337
| 9-, 11- and 13-odd-limit minimax
| 9-, 11- and 13-odd-limit minimax
Line 308: Line 344:
|-
|-
|  
|  
| 14/11
| 11/7
| 704.377
| 704.377
|  
|  
Line 318: Line 354:
|-
|-
|  
|  
| 24/17
| 17/12
| 704.478
| 704.478
|  
|  
|-
|-
|  
|  
| 8/7
| 7/4
| 704.588
| 704.588
|  
|  
|-
|-
|  
|  
| 18/17
| 17/9
| 704.593
| 704.593
|  
|  
Line 340: Line 376:
|  
|  
| 704.762
| 704.762
|  
| 63ch val
|-
|-
|  
|  
Line 348: Line 384:
|-
|-
|  
|  
| 12/11
| 11/6
| 704.936
| 704.936
|  
|  
Line 358: Line 394:
|-
|-
|  
|  
| 16/13
| 13/8
| 705.066
| 705.066
|  
|  
Line 375: Line 411:
|  
|  
| 705.882
| 705.882
| Upper bound of 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
| 17cg val, upper bound of 5-, 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
|-
|-
|  
|  
| 18/13
| 13/9
| 706.103
| 706.103
|  
|  
|-
|-
|  
|  
| 20/17
| 17/10
| 706.214
| 706.214
|  
|  
Line 392: Line 428:
|  
|  
|}
|}
<nowiki />* Besides the octave
<nowiki/>* Besides the octave


== Notes ==
== References and external links ==
<references/>


[[Category:Leapday| ]] <!-- main article -->
[[Category:Leapday| ]] <!-- Main article -->
[[Category:Temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Sengic temperaments]]
[[Category:Sengic temperaments]]
[[Category:Hemifamity temperaments]]
[[Category:Hemifamity temperaments]]
Retrieved from "https://en.xen.wiki/w/Leapday"