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]].''
{{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. 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;).
'''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 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.
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♯).
 
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 ==
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
| 704.2
| '''3/2'''
| '''3/2'''
|
|-
|-
| 2
| 2
| 208.6
| 208.5
| '''9/8'''
| '''9/8'''
| 17/15, 26/23
|-
|-
| 3
| 3
| 912.9
| 912.7
| 22/13, 27/16
| 22/13, 27/16
| 17/10
|-
|-
| 4
| 4
| 417.2
| 416.9
| 14/11
| 14/11, 33/26
| 23/18
|-
|-
| 5
| 5
| 1121.5
| 1121.2
| 21/11, 40/21
| 21/11, 40/21
| 23/12, 44/23
|-
|-
| 6
| 6
| 625.8
| 625.4
| 10/7, 13/9
| 10/7, 13/9
| '''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
|-
|-
| 9
| 9
| 338.6
| 338.1
| 11/9, 39/32, 40/33
| 11/9, 39/32, 40/33
| 17/14, 28/23
|-
|-
| 10
| 10
| 1042.9
| 1042.3
| 11/6, 20/11
| 11/6, 20/11
| 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
|-
|-
| 13
| 13
| 755.8
| 755.1
| 14/9, 20/13
| 14/9, 20/13
| 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
|-
|-
| 16
| 16
| 468.7
| 467.8
| '''21/16'''
| '''21/16'''
| 17/13, 30/23
|-
|-
| 17
| 17
| 1173.0
| 1172.0
| 63/32, 160/81
| 63/32, 160/81
| 45/23, 51/26
|-
|-
| 18
| 18
| 677.3
| 676.2
| 40/27
| 40/27
| 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
|-
|-
| 23
| 23
| 598.7
| 597.4
| 45/32
| 45/32
| 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 ===
{| 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
| 3/2
| 700.829
|
|-
|
| 24/19
| 701.110
|
|-
|
| 19/18
| 701.279
|
|-
|
| 4/3
| 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 178: Line 255:
|  
|  
| 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 196: 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 261: Line 313:
|-
|-
|  
|  
| 14/13
| 13/7
| 704.043
| 704.043
|  
|  
Line 271: Line 323:
|-
|-
|  
|  
| 22/17
| 17/11
| 704.126
| 704.126
|  
|  
Line 278: 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 291: 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 298: Line 360:
|  
|  
| 704.348
| 704.348
|  
| 17-odd-limit, no-19 21- and 23-odd-limit diamond monotone (singleton)
|-
|-
|  
|  
Line 306: Line 368:
|-
|-
|  
|  
| 14/11
| 11/7
| 704.377
| 704.377
|  
|  
Line 316: 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 333: Line 405:
| 11/8
| 11/8
| 704.665
| 704.665
|
|-
|
| 23/16
| 704.712
|  
|  
|-
|-
Line 338: Line 415:
|  
|  
| 704.762
| 704.762
|  
| 63c val
|-
|-
|  
|  
Line 346: Line 423:
|-
|-
|  
|  
| 12/11
| 11/6
| 704.936
| 704.936
|  
|  
Line 356: Line 433:
|-
|-
|  
|  
| 16/13
| 13/8
| 705.066
| 705.066
|
|-
|
| 23/12
| 705.264
|  
|  
|-
|-
Line 373: Line 455:
|  
|  
| 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
|-
|
| 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 390: Line 482:
|  
|  
|}
|}
<nowiki>*</nowiki> Besides the octave
<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: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]]

Latest revision as of 12:39, 6 June 2026

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);
121/120, 441/440, 686/675 (11-limit);
91/90, 121/120, 169/168, 352/351
(13-limit)
Reduced mapping ⟨1; 1 21 15 11 8]
ET join 29 & 46
Generators (CWE) ~3/2 = 704.2 ¢
MOS scales 2L 3s, 5L 2s, 5L 7s, 12L 5s
Ploidacot monocot
Minimax error 9-odd-limit: 8.53 ¢;
13-limit 21-odd-limit: 10.6 ¢
Target scale size 9-odd-limit: 29 notes;
13-limit 21-odd-limit: 29 notes
Not to be confused with calendar-based scales such as those in 293edo, 400edo, 353edo, or Irvian mode.

Leapday is a temperament based on the chain of fifths, but the fifth is tuned slightly sharp of just (approximately 704 ¢) 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𝄪).

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♯).

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 primes 17 and 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 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.

Leapday was named by Herman Miller in 2004[1][2].

See Hemifamity temperaments #Leapday for technical data.

Interval chain

In the following table, odd harmonics 1–23 are in bold.

# Cents* Approximate ratios
13-limit Additional ratios
of 17 and 23
0 0.0 1/1
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
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
21 388.9 5/4
22 1093.2 15/8 17/9
23 597.4 45/32 17/12

* In 13-limit CWE tuning, octave reduced

Tunings

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 704.2257 ¢ CWE: ~3/2 = 704.2504 ¢ POTE: ~3/2 = 704.2634 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 704.2924 ¢ CWE: ~3/2 = 704.2346 ¢ POTE: ~3/2 = 704.2138 ¢
No-19 23-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 704.3142 ¢ CWE: ~3/2 = 704.2450 ¢ POTE: ~3/2 = 704.2246 ¢

Tuning spectrum

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
3/2 701.955 Pythagorean tuning
24\41 702.439 41cc… val, lower bound of 5-odd-limit diamond monotone
15/14 702.778
7/5 702.915
21/20 703.107
15/11 703.359
15/13 703.410
17\29 703.448 29g val, lower bound of 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
11/10 703.500
13/10 703.522
13/11 703.597
23/15 703.750
21/13 703.782
23/20 703.869
21/11 703.893
44\75 704.000 75dfg val
15/8 704.012
17/14 704.014
17/13 704.027
13/7 704.043
5/4 704.110 5-odd-limit minimax
17/11 704.126
71\121 704.132 121defg val
5/3 704.218 7-, 15- and 17-odd-limit minimax
23/21 704.251
23/17 704.260
21/17 704.272
9/5 704.337 9-, 11- and 13-odd-limit minimax
27\46 704.348 17-odd-limit, no-19 21- and 23-odd-limit diamond monotone (singleton)
17/16 704.373
11/7 704.377
21/16 704.424
17/12 704.478
23/14 704.506
7/4 704.588
17/9 704.593
23/22 704.609
11/8 704.665
23/16 704.712
37\63 704.762 63c val
7/6 704.776
11/6 704.936
9/7 704.994
13/8 705.066
23/12 705.264
11/9 705.268
13/12 705.510
10\17 705.882 17cg val, upper bound of 5-, 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
23/18 706.091
13/9 706.103
23/13 706.127
17/10 706.214
17/15 708.343

* Besides the octave

Music

Claudi Meneghin

References and external links

  1. ↑ Yahoo! Tuning Group (Archive) | Some 13-limit temperaments
  2. ↑ Yahoo! Tuning Group (Archive) | 24 13-limit temperaments supported by 46
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