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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 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;)<ref group="note">This implies that adjacent accidental notes such as F&#x266F; and G&#x266D; are different pitches, with flats sitting significantly lower than their nearby sharps.</ref>, 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|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𝄪).  


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 the 5<sup>th</sup> harmonic's intervals are simple while the 7<sup>th</sup>, 11<sup>th</sup>, and 13<sup>th</sup> harmonics' intervals are complex, while Leapday has 3/2 tuned sharp so that the 7<sup>th</sup>, 11<sup>th</sup>, and 13<sup>th</sup> harmonics' intervals are (relatively) simple while the 5<sup>th</sup> harmonic's intervals are complex.
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♯).  


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 5/4 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.
 
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♯).
 
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.  


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


== Interval chain ==
== Interval chain ==
Line 16: Line 31:
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! rowspan="2" | &#35;
! rowspan="2" | #
! rowspan="2" | Cents*
! rowspan="2" | Cents*
! colspan="2" | Approximate ratios
! colspan="2" | Approximate ratios
|-
|-
! 13-limit
! 13-limit
! No-19 23-limit extension
! Additional ratios<br>of 17 and 23
|-
|-
| 0
| 0
Line 29: Line 44:
|-
|-
| 1
| 1
| 704.3
| 704.2
| '''3/2'''
| '''3/2'''
|  
|  
|-
|-
| 2
| 2
| 208.6
| 208.5
| '''9/8'''
| '''9/8'''
| 17/15, 26/23
| 17/15, 26/23
|-
|-
| 3
| 3
| 912.9
| 912.7
| 22/13, 27/16
| 22/13, 27/16
| 17/10
| 17/10
|-
|-
| 4
| 4
| 417.2
| 416.9
| 14/11
| 14/11, 33/26
| 23/18
| 23/18
|-
|-
| 5
| 5
| 1121.5
| 1121.2
| 21/11, 40/21
| 21/11, 40/21
| 23/12, 44/23
| 23/12, 44/23
|-
|-
| 6
| 6
| 625.8
| 625.4
| 10/7, 13/9
| 10/7, 13/9
| '''23/16'''
| '''23/16'''
|-
|-
| 7
| 7
| 130.0
| 129.6
| 13/12, 14/13, 15/14
| 13/12, 14/13, 15/14
|  
|  
|-
|-
| 8
| 8
| 834.3
| 833.9
| '''13/8''', 21/13
| '''13/8''', 21/13
| 34/21
| 34/21
|-
|-
| 9
| 9
| 338.6
| 338.1
| 11/9, 39/32, 40/33
| 11/9, 39/32, 40/33
| 17/14, 28/23
| 17/14, 28/23
|-
|-
| 10
| 10
| 1042.9
| 1042.3
| 11/6, 20/11
| 11/6, 20/11
| 42/23
| 42/23
|-
|-
| 11
| 11
| 547.2
| 546.6
| '''11/8''', 15/11
| '''11/8''', 15/11
|  
|  
|-
|-
| 12
| 12
| 51.5
| 50.8
| 28/27, 33/32, 40/39, 45/44
| 28/27, 33/32, 40/39, 45/44
| 34/33, 35/34
| 34/33, 35/34
|-
|-
| 13
| 13
| 755.8
| 755.1
| 14/9, 20/13
| 14/9, 20/13
| 17/11
| 17/11
|-
|-
| 14
| 14
| 260.1
| 259.3
| 7/6, 15/13
| 7/6, 15/13
|  
|  
|-
|-
| 15
| 15
| 964.4
| 963.5
| '''7/4'''
| '''7/4'''
| 40/23
| 40/23
|-
|-
| 16
| 16
| 468.7
| 467.8
| '''21/16'''
| '''21/16'''
| 17/13, 30/23
| 17/13, 30/23
|-
|-
| 17
| 17
| 1173.0
| 1172.0
| 63/32, 160/81
| 63/32, 160/81
| 45/23, 51/26
| 45/23, 51/26
|-
|-
| 18
| 18
| 677.3
| 676.2
| 40/27
| 40/27
| 34/23
| 34/23
|-
|-
| 19
| 19
| 181.6
| 180.5
| 10/9
| 10/9
|  
|  
|-
|-
| 20
| 20
| 885.8
| 884.7
| 5/3
| 5/3
|  
|  
|-
|-
| 21
| 21
| 390.1
| 388.9
| '''5/4'''
| '''5/4'''
|  
|  
|-
|-
| 22
| 22
| 1094.4
| 1093.2
| '''15/8'''
| '''15/8'''
| 17/9
| 17/9
|-
|-
| 23
| 23
| 598.7
| 597.4
| 45/32
| 45/32
| 17/12
| 17/12
|}
|}
<nowiki />* In 13-limit CTE tuning
<nowiki/>* In 13-limit CWE tuning, octave reduced


== Tunings ==
== 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 ===
=== Tuning spectrum ===
This spectrum assumes 19-limit leapday.
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! Edo<br />generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|
| 19/16
| 700.829
|
|-
|
| 19/12
| 701.110
|
|-
|
| 19/18
| 701.279
|
|-
|-
|  
|  
| 3/2
| 3/2
| 701.955
| 701.955
|  
| Pythagorean tuning
|-
|-
| 24\41
| 24\41
Line 226: Line 273:
|-
|-
|  
|  
| 19/15
| 23/15
| 703.630
| 703.750
|
|-
|
| 19/10
| 703.700
|  
|  
|-
|-
Line 241: Line 283:
|-
|-
|  
|  
| 19/11
| 23/20
| 703.843
| 703.869
|
|-
|
| 21/19
| 703.856
|  
|  
|-
|-
Line 254: Line 291:
| 703.893
| 703.893
|  
|  
|-
|
| 19/13
| 703.910
|
|-
|
| 19/14
| 703.962
|
|-
|
| 19/17
| 703.979
| 19- and 21-odd-limit minimax
|-
|-
| 44\75
| 44\75
|  
|  
| 704.000
| 704.000
| 75dfgh val
| 75dfg val
|-
|-
|  
|  
Line 308: Line 330:
|  
|  
| 704.132
| 704.132
| 121defgh val
| 121defg val
|-
|-
|  
|  
Line 314: Line 336:
| 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 328: Line 360:
|  
|  
| 704.348
| 704.348
|  
| 17-odd-limit, no-19 21- and 23-odd-limit diamond monotone (singleton)
|-
|-
|  
|  
Line 348: Line 380:
| 17/12
| 17/12
| 704.478
| 704.478
|
|-
|
| 23/14
| 704.506
|  
|  
|-
|-
Line 358: Line 395:
| 17/9
| 17/9
| 704.593
| 704.593
|
|-
|
| 23/22
| 704.609
|  
|  
|-
|-
Line 363: Line 405:
| 11/8
| 11/8
| 704.665
| 704.665
|
|-
|
| 23/16
| 704.712
|  
|  
|-
|-
Line 368: Line 415:
|  
|  
| 704.762
| 704.762
| 63ch val
| 63c val
|-
|-
|  
|  
Line 388: Line 435:
| 13/8
| 13/8
| 705.066
| 705.066
|
|-
|
| 23/12
| 705.264
|  
|  
|-
|-
Line 404: Line 456:
| 705.882
| 705.882
| 17cg val, upper bound of 5-, 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
|-
|
| 23/18
| 706.091
|
|-
|-
|  
|  
| 13/9
| 13/9
| 706.103
| 706.103
|
|-
|
| 23/13
| 706.127
|  
|  
|-
|-
Line 420: Line 482:
|  
|  
|}
|}
<nowiki />* Besides the octave
<nowiki/>* Besides the octave


== Notes ==
== Music ==
<references group="note" />
; [[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)


== References and external links ==
== References and external links ==
<references />
<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:Aberschismic temperaments]]
Retrieved from "https://en.xen.wiki/w/Leapday"