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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 temperament for the 7, 11, 13, 17, and 19 [[Harmonic limit|prime limit]]s. It is a member of [[High badness temperaments #Leapday|trisayo family]], sengic temperaments and [[hemifamity temperaments]]. 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]] 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.


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>.  
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>.  
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Gencom: [2 3/2; 91/90 121/120 133/132 136/135 154/153 169/168]
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}}]
Gencom mapping: [{{val| 1 1 -10 -6 -3 -1 -10 6 }}, {{val| 0 1 21 15 11 8 24 -3 }}]


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! | ET<br>generator
! Edo<br>generator
! | [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged Interval)]]
! | fifth<br>(¢)
! Generator<br>(¢)
! | comments
! Comments
|-
|-
| |  
|  
| | 19/16
| 19/16
| | 700.829
| 700.829
| |  
|  
|-
|-
| |  
|  
| | 24/19
| 24/19
| | 701.110
| 701.110
| |  
|  
|-
|-
| |  
|  
| | 19/18
| 19/18
| | 701.279
| 701.279
| |  
|  
|-
|-
| |  
|  
| | 4/3
| 4/3
| | 701.955
| 701.955
| |  
|  
|-
|-
| | 24\41
| 24\41
| |  
|  
| | 702.439
| 702.439
| |  
|  
|-
|-
| |  
|  
| | 15/14
| 15/14
| | 702.778
| 702.778
| |  
|  
|-
|-
| |  
|  
| | 7/5
| 7/5
| | 702.915
| 702.915
| |  
|  
|-
|-
| |  
|  
| | 21/20
| 21/20
| | 703.107
| 703.107
| |  
|  
|-
|-
| |  
|  
| | 15/11
| 15/11
| | 703.359
| 703.359
| |  
|  
|-
|-
| |  
|  
| | 15/13
| 15/13
| | 703.410
| 703.410
| |  
|  
|-
|-
| | 17\29
| 17\29
| |  
|  
| | 703.448
| 703.448
| |  
|  
|-
|-
| |  
|  
| | 11/10
| 11/10
| | 703.500
| 703.500
| |  
|  
|-
|-
| |  
|  
| | 13/10
| 13/10
| | 703.522
| 703.522
| |  
|  
|-
|-
| |  
|  
| | 13/11
| 13/11
| | 703.597
| 703.597
| |  
|  
|-
|-
| |  
|  
| | 19/15
| 19/15
| | 703.630
| 703.630
| |  
|  
|-
|-
| |  
|  
| | 20/19
| 20/19
| | 703.700
| 703.700
| |  
|  
|-
|-
| |  
|  
| | 26/21
| 26/21
| | 703.782
| 703.782
| |  
|  
|-
|-
| |  
|  
| | 22/19
| 22/19
| | 703.843
| 703.843
| |  
|  
|-
|-
| |  
|  
| | 21/19
| 21/19
| | 703.856
| 703.856
| |  
|  
|-
|-
| |  
|  
| | 22/21
| 22/21
| | 703.893
| 703.893
| |  
|  
|-
|-
| |  
|  
| | 26/19
| 26/19
| | 703.910
| 703.910
| |  
|  
|-
|-
| |  
|  
| | 19/14
| 19/14
| | 703.962
| 703.962
| |  
|  
|-
|-
| |  
|  
| | 19/17
| 19/17
| | 703.979
| 703.979
| | 19 and 21-odd-limit minimax
| 19- and 21-odd-limit minimax
|-
|-
| | 44\75
| 44\75
| |  
|  
| | 704.000
| 704.000
| |  
|  
|-
|-
| |  
|  
| | 16/15
| 16/15
| | 704.012
| 704.012
| |  
|  
|-
|-
| |  
|  
| | 17/14
| 17/14
| | 704.014
| 704.014
| |  
|  
|-
|-
| |  
|  
| | 17/13
| 17/13
| | 704.027
| 704.027
| |  
|  
|-
|-
| |  
|  
| | 14/13
| 14/13
| | 704.043
| 704.043
| |  
|  
|-
|-
| |  
|  
| | 5/4
| 5/4
| | 704.110
| 704.110
| | 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| |  
|  
| | 22/17
| 22/17
| | 704.126
| 704.126
| |  
|  
|-
|-
| | 71\121
| 71\121
| |  
|  
| | 704.132
| 704.132
| |  
|  
|-
|-
| |  
|  
| | 6/5
| 6/5
| | 704.218
| 704.218
| | 7, 15 and 17-odd-limit minimax
| 7-, 15- and 17-odd-limit minimax
|-
|-
| |  
|  
| | 21/17
| 21/17
| | 704.272
| 704.272
| |  
|  
|-
|-
| |  
|  
| | 10/9
| 10/9
| | 704.337
| 704.337
| | 9, 11 and 13-odd-limit minimax
| 9-, 11- and 13-odd-limit minimax
|-
|-
| | 27\46
| 27\46
| |  
|  
| | 704.348
| 704.348
| |  
|  
|-
|-
| |  
|  
| | 17/16
| 17/16
| | 704.373
| 704.373
| |  
|  
|-
|-
| |  
|  
| | 14/11
| 14/11
| | 704.377
| 704.377
| |  
|  
|-
|-
| |  
|  
| | 21/16
| 21/16
| | 704.424
| 704.424
| |  
|  
|-
|-
| |  
|  
| | 24/17
| 24/17
| | 704.478
| 704.478
| |  
|  
|-
|-
| |  
|  
| | 8/7
| 8/7
| | 704.588
| 704.588
| |  
|  
|-
|-
| |  
|  
| | 18/17
| 18/17
| | 704.593
| 704.593
| |  
|  
|-
|-
| |  
|  
| | 11/8
| 11/8
| | 704.665
| 704.665
| |  
|  
|-
|-
| | 37\63
| 37\63
| |  
|  
| | 704.762
| 704.762
| |  
|  
|-
|-
| |  
|  
| | 7/6
| 7/6
| | 704.776
| 704.776
| |  
|  
|-
|-
| |  
|  
| | 12/11
| 12/11
| | 704.936
| 704.936
| |  
|  
|-
|-
| |  
|  
| | 9/7
| 9/7
| | 704.994
| 704.994
| |  
|  
|-
|-
| |  
|  
| | 16/13
| 16/13
| | 705.066
| 705.066
| |  
|  
|-
|-
| |  
|  
| | 11/9
| 11/9
| | 705.268
| 705.268
| |  
|  
|-
|-
| |  
|  
| | 13/12
| 13/12
| | 705.510
| 705.510
| |  
|  
|-
|-
| | 10\17
| 10\17
| |  
|  
| | 705.882
| 705.882
| |  
|  
|-
|-
| |  
|  
| | 18/13
| 18/13
| | 706.103
| 706.103
| |  
|  
|-
|-
| |  
|  
| | 20/17
| 20/17
| | 706.214
| 706.214
| |  
|  
|-
|-
| |  
|  
| | 17/15
| 17/15
| | 708.343
| 708.343
| |  
|  
|}
|}


Retrieved from "https://en.xen.wiki/w/Leapday"