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: ''Not to be confused with calendar-based scales such as those in [[293edo]], [[400edo]], [[353edo]] or [[Irvic scale|Irvian mode]].''
{{Infobox regtemp
| Title = Leapday
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[686/675]], [[5120/5103]] (7-limit); <br>[[121/120]], [[441/440]], [[686/675]] (11-limit); <br>[[91/90]], [[121/120]], [[169/168]], [[352/351]]<br>(13-limit)
| Edo join 1 = 29 | Edo join 2 = 46
| Mapping = 1; 1 21 15 11 8
| Generators = 3/2 | Generators tuning = 704.2 | Optimization method = CWE
| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[12L 5s]]
| Odd limit 1 = 9 | Mistuning 1 = 8.53 | Complexity 1 = 29
| Odd limit 2 = 13-limit 21 | Mistuning 2 = 10.6 | Complexity 2 = 29
}}
: ''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 19-limit JI. It has a fifth generator of ~3/2 = 704.2¢, eight of them makes ~13/8, eleven of them makes ~11/8, fifteen of them makes ~7/4, twenty-one of them makes ~5/4 and twenty-four of them makes ~17/16. Equivalently, the fifth of leapday in size is ~2.3 cents sharp of 3/2, 13/8 is represented by an augmented fifth, 11/8 is represented by an augmented third, 7/4 is represented by a double-augmented fifth, 5/4 is represented by a triple-augmented unison, and 17/16 is represented by a negative triple-diminished third.
'''Leapday''' is a [[regular temperament|temperament]] based on the [[chain of fifths]], but the fifth is tuned slightly sharp of just (approximately 704{{cent}}) so that 15 fifths give [[7/4]] and 21 fifths give [[5/4]]. In other words, the classical major third (5/4) is represented by a triple-augmented unison (C–C𝄪♯), and the harmonic seventh (7/4) is represented by a double-augmented fifth (C–G𝄪).  


The temperament 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>.  
Leapday can be easily extended to the [[13-limit]] by identifying [[14/11]] with the major third and [[13/11]] with the minor third. This implies 11/8 is represented by an augmented third (C–E♯) and 13/8 is represented by an augmented fifth (C–G♯).  


See [[Hemifamity temperaments #Leapday]] for more technical data.
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.


== Tuning spectrum ==
Further extensions for [[prime]]s [[17/1|17]] and [[23/1|23]] are available, where 17/16 is represented by an octave-reduced triple-augmented sixth (C–A𝄪♯), and 23/16 is represented by an augmented fourth (C–F♯).
Gencom: [2 3/2; 91/90 121/120 133/132 136/135 154/153 169/168]


Gencom mapping: [{{val| 1 1 -10 -6 -3 -1 -10 6 }}, {{val| 0 1 21 15 11 8 24 -3 }}]
Since ratios of 5 are complex, they can be omitted, and the [[2.3.7.11.13 subgroup|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.


{| class="wikitable center-all"
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]] for technical data.
 
== Interval chain ==
In the following table, odd harmonics 1–23 are in '''bold'''.
 
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents*
! colspan="2" | Approximate ratios
|-
|-
! EDO<br>generator
! 13-limit
! [[eigenmonzo|eigenmonzo<br>(unchanged-interval)]]
! Additional ratios<br>of 17 and 23
! generator<br>(¢)
! comments
|-
|-
| 0
| 0.0
| '''1/1'''
|  
|  
| 19/16
|-
| 700.829
| 1
| 704.2
| '''3/2'''
|  
|  
|-
|-
| 2
| 208.5
| '''9/8'''
| 17/15, 26/23
|-
| 3
| 912.7
| 22/13, 27/16
| 17/10
|-
| 4
| 416.9
| 14/11, 33/26
| 23/18
|-
| 5
| 1121.2
| 21/11, 40/21
| 23/12, 44/23
|-
| 6
| 625.4
| 10/7, 13/9
| '''23/16'''
|-
| 7
| 129.6
| 13/12, 14/13, 15/14
|  
|  
| 24/19
|-
| 701.110
| 8
| 833.9
| '''13/8''', 21/13
| 34/21
|-
| 9
| 338.1
| 11/9, 39/32, 40/33
| 17/14, 28/23
|-
| 10
| 1042.3
| 11/6, 20/11
| 42/23
|-
| 11
| 546.6
| '''11/8''', 15/11
|
|-
| 12
| 50.8
| 28/27, 33/32, 40/39, 45/44
| 34/33, 35/34
|-
| 13
| 755.1
| 14/9, 20/13
| 17/11
|-
| 14
| 259.3
| 7/6, 15/13
|
|-
| 15
| 963.5
| '''7/4'''
| 40/23
|-
| 16
| 467.8
| '''21/16'''
| 17/13, 30/23
|-
| 17
| 1172.0
| 63/32, 160/81
| 45/23, 51/26
|-
| 18
| 676.2
| 40/27
| 34/23
|-
| 19
| 180.5
| 10/9
|  
|  
|-
|-
| 20
| 884.7
| 5/3
|  
|  
| 19/18
|-
| 701.279
| 21
| 388.9
| '''5/4'''
|  
|  
|-
| 22
| 1093.2
| '''15/8'''
| 17/9
|-
| 23
| 597.4
| 45/32
| 17/12
|}
<nowiki/>* In 13-limit CWE tuning, octave reduced
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 704.2257{{c}}
| CWE: ~3/2 = 704.2504{{c}}
| POTE: ~3/2 = 704.2634{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 704.2924{{c}}
| CWE: ~3/2 = 704.2346{{c}}
| POTE: ~3/2 = 704.2138{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | No-19 23-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 704.3142{{c}}
| CWE: ~3/2 = 704.2450{{c}}
| POTE: ~3/2 = 704.2246{{c}}
|}
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|-
|  
|  
| 4/3
| 3/2
| 701.955
| 701.955
|  
| Pythagorean tuning
|-
|-
| 24\41
| 24\41
|  
|  
| 702.439
| 702.439
|  
| 41cc… val, lower bound of 5-odd-limit diamond monotone
|-
|-
|  
|  
Line 72: Line 255:
|  
|  
| 703.448
| 703.448
|  
| 29g val, lower bound of 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
|-
|-
|  
|  
Line 90: Line 273:
|-
|-
|  
|  
| 19/15
| 23/15
| 703.630
| 703.750
|
|-
|
| 20/19
| 703.700
|  
|  
|-
|-
|  
|  
| 26/21
| 21/13
| 703.782
| 703.782
|  
|  
|-
|-
|  
|  
| 22/19
| 23/20
| 703.843
| 703.869
|
|-
|
| 21/19
| 703.856
|  
|  
|-
|-
|  
|  
| 22/21
| 21/11
| 703.893
| 703.893
|  
|  
|-
|
| 26/19
| 703.910
|
|-
|
| 19/14
| 703.962
|
|-
|
| 19/17
| 703.979
| 19- and 21-odd-limit minimax
|-
|-
| 44\75
| 44\75
|  
|  
| 704.000
| 704.000
|  
| 75dfg val
|-
|-
|  
|  
| 16/15
| 15/8
| 704.012
| 704.012
|  
|  
Line 155: Line 313:
|-
|-
|  
|  
| 14/13
| 13/7
| 704.043
| 704.043
|  
|  
Line 165: Line 323:
|-
|-
|  
|  
| 22/17
| 17/11
| 704.126
| 704.126
|  
|  
Line 172: Line 330:
|  
|  
| 704.132
| 704.132
|  
| 121defg val
|-
|-
|  
|  
| 6/5
| 5/3
| 704.218
| 704.218
| 7-, 15- and 17-odd-limit minimax
| 7-, 15- and 17-odd-limit minimax
|-
|
| 23/21
| 704.251
|
|-
|
| 23/17
| 704.260
|
|-
|-
|  
|  
Line 185: Line 353:
|-
|-
|  
|  
| 10/9
| 9/5
| 704.337
| 704.337
| 9-, 11- and 13-odd-limit minimax
| 9-, 11- and 13-odd-limit minimax
Line 192: Line 360:
|  
|  
| 704.348
| 704.348
|  
| 17-odd-limit, no-19 21- and 23-odd-limit diamond monotone (singleton)
|-
|-
|  
|  
Line 200: Line 368:
|-
|-
|  
|  
| 14/11
| 11/7
| 704.377
| 704.377
|  
|  
Line 210: Line 378:
|-
|-
|  
|  
| 24/17
| 17/12
| 704.478
| 704.478
|  
|  
|-
|-
|  
|  
| 8/7
| 23/14
| 704.506
|
|-
|
| 7/4
| 704.588
| 704.588
|  
|  
|-
|-
|  
|  
| 18/17
| 17/9
| 704.593
| 704.593
|
|-
|
| 23/22
| 704.609
|  
|  
|-
|-
Line 227: Line 405:
| 11/8
| 11/8
| 704.665
| 704.665
|
|-
|
| 23/16
| 704.712
|  
|  
|-
|-
Line 232: Line 415:
|  
|  
| 704.762
| 704.762
|  
| 63c val
|-
|-
|  
|  
Line 240: Line 423:
|-
|-
|  
|  
| 12/11
| 11/6
| 704.936
| 704.936
|  
|  
Line 250: Line 433:
|-
|-
|  
|  
| 16/13
| 13/8
| 705.066
| 705.066
|
|-
|
| 23/12
| 705.264
|  
|  
|-
|-
Line 267: Line 455:
|  
|  
| 705.882
| 705.882
| 17cg val, upper bound of 5-, 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
|-
|
| 23/18
| 706.091
|  
|  
|-
|-
|  
|  
| 18/13
| 13/9
| 706.103
| 706.103
|  
|  
|-
|-
|  
|  
| 20/17
| 23/13
| 706.127
|
|-
|
| 17/10
| 706.214
| 706.214
|  
|  
Line 284: Line 482:
|  
|  
|}
|}
<nowiki/>* Besides the octave
== Music ==
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=TgD7cN8a5D8 ''Lytel Twyelyghte Musicke (Little Twilight Music), for Brass, Winds, Strings, and Timpani, in 80-equal division of the octave, as the linear temperament generated by its fifth''] (2025)


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


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