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 [[regular temperament]] for the 7-, 11-, 13-, 17-, and 19-limit JI. 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 inverted triply diminished third (e.g.&nbsp;E&#x1D12B;&#x1D12B;&ndash;C).
'''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]].  


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>.  
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♯).


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.
 
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 1–13 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
| 704.2
| 704.3
| '''3/2'''
| '''3/2'''
|
|-
|-
| 2
| 2
| 208.5
| 208.6
| '''9/8'''
| '''9/8'''
| 17/15, 26/23
|-
|-
| 3
| 3
| 912.7
| 912.9
| 22/13, 27/16
| 22/13, 27/16
| 17/10
|-
|-
| 4
| 4
| 416.9
| 417.2
| 14/11
| 14/11, 33/26
| 23/18
|-
|-
| 5
| 5
| 1121.2
| 1121.5
| 21/11, 40/21
| 21/11, 40/21
| 23/12, 44/23
|-
|-
| 6
| 6
| 625.4
| 625.8
| 10/7
| 10/7, 13/9
| '''23/16'''
|-
|-
| 7
| 7
| 129.6
| 130.0
| 13/12, 14/13, 15/14
| 13/12, 14/13, 15/14
|
|-
|-
| 8
| 8
| 833.9
| 834.3
| '''13/8''', 21/13
| '''13/8''', 21/13
| 34/21
|-
|-
| 9
| 9
| 338.1
| 338.6
| 39/32, 40/33
| 11/9, 39/32, 40/33
| 17/14, 28/23
|-
|-
| 10
| 10
| 1042.3
| 1042.9
| 11/6, 20/11
| 11/6, 20/11
| 42/23
|-
|-
| 11
| 11
| 546.6
| 547.2
| '''11/8''', 15/11
| '''11/8''', 15/11
|
|-
|-
| 12
| 12
| 50.8
| 51.5
| 28/27, 33/32, 40/39, 45/44
| 28/27, 33/32, 40/39, 45/44
| 34/33, 35/34
|-
|-
| 13
| 13
| 755.0
| 755.8
| 14/9, 20/13
| 14/9, 20/13
| 17/11
|-
|-
| 14
| 14
| 259.3
| 260.1
| 7/6, 15/13
| 7/6, 15/13
|
|-
|-
| 15
| 15
| 963.5
| 964.4
| '''7/4'''
| '''7/4'''
| 40/23
|-
|-
| 16
| 16
| 467.8
| 468.7
| 21/16
| '''21/16'''
| 17/13, 30/23
|-
|-
| 17
| 17
| 1172.0
| 1173.0
| 63/32, 160/81
| 63/32, 160/81
| 45/23, 51/26
|-
|-
| 18
| 18
| 676.2
| 677.3
| 40/27
| 40/27
| 34/23
|-
|-
| 19
| 19
| 180.5
| 181.6
| 10/9
| 10/9
|
|-
|-
| 20
| 20
| 884.7
| 885.8
| 5/3
| 5/3
|
|-
|-
| 21
| 21
| 388.9
| 390.1
| '''5/4'''
| '''5/4'''
|
|-
|-
| 22
| 22
| 1093.2
| 1094.4
| 15/8
| '''15/8'''
| 17/9
|-
|-
| 23
| 23
| 597.4
| 598.7
| 45/32
| 45/32
| 17/12
|}
|}
<nowiki>*</nowiki> in 13-limit CTE tuning
<nowiki />* In 13-limit CTE tuning
 
== Tuning spectrum ==
Gencom: [2 3/2; 91/90 121/120 133/132 136/135 154/153 169/168]


Gencom mapping: {{mapping| 1 1 -10 -6 -3 -1 -10 6 | 0 1 21 15 11 8 24 -3 }}
== Tunings ==
=== Tuning spectrum ===
This spectrum assumes 19-limit leapday.


{| class="wikitable center-all"
{| 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 131: Line 169:
|-
|-
|  
|  
| 24/19
| 19/12
| 701.110
| 701.110
|  
|  
Line 141: Line 179:
|-
|-
|  
|  
| 4/3
| 3/2
| 701.955
| 701.955
|  
|  
Line 148: Line 186:
|  
|  
| 702.439
| 702.439
|  
| 41cc… val, lower bound of 5-odd-limit diamond monotone
|-
|-
|  
|  
Line 178: Line 216:
|  
|  
| 703.448
| 703.448
|  
| 29g val, lower bound of 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
|-
|-
|  
|  
Line 201: 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 221: Line 259:
|-
|-
|  
|  
| 22/21
| 21/11
| 703.893
| 703.893
|  
|  
|-
|-
|  
|  
| 26/19
| 19/13
| 703.910
| 703.910
|  
|  
Line 243: Line 281:
|  
|  
| 704.000
| 704.000
|  
| 75dfgh val
|-
|-
|  
|  
| 16/15
| 15/8
| 704.012
| 704.012
|  
|  
Line 261: Line 299:
|-
|-
|  
|  
| 14/13
| 13/7
| 704.043
| 704.043
|  
|  
Line 271: Line 309:
|-
|-
|  
|  
| 22/17
| 17/11
| 704.126
| 704.126
|  
|  
Line 278: 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 291: 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 306: Line 344:
|-
|-
|  
|  
| 14/11
| 11/7
| 704.377
| 704.377
|  
|  
Line 316: 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 338: Line 376:
|  
|  
| 704.762
| 704.762
|  
| 63ch val
|-
|-
|  
|  
Line 346: Line 384:
|-
|-
|  
|  
| 12/11
| 11/6
| 704.936
| 704.936
|  
|  
Line 356: Line 394:
|-
|-
|  
|  
| 16/13
| 13/8
| 705.066
| 705.066
|  
|  
Line 373: Line 411:
|  
|  
| 705.882
| 705.882
|  
| 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 390: Line 428:
|  
|  
|}
|}
<nowiki>*</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]]

Latest revision as of 06:56, 21 June 2025

Not to be confused with calendar-based scales such as those in 293edo, 400edo, 353edo, or 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 (approximately 704 ¢) 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.

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♯).

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.

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)4, 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[1][2].

See Hemifamity temperaments #Leapday or No-fives subgroup temperaments #Leapfrog for more 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
1 704.3 3/2
2 208.6 9/8 17/15, 26/23
3 912.9 22/13, 27/16 17/10
4 417.2 14/11, 33/26 23/18
5 1121.5 21/11, 40/21 23/12, 44/23
6 625.8 10/7, 13/9 23/16
7 130.0 13/12, 14/13, 15/14
8 834.3 13/8, 21/13 34/21
9 338.6 11/9, 39/32, 40/33 17/14, 28/23
10 1042.9 11/6, 20/11 42/23
11 547.2 11/8, 15/11
12 51.5 28/27, 33/32, 40/39, 45/44 34/33, 35/34
13 755.8 14/9, 20/13 17/11
14 260.1 7/6, 15/13
15 964.4 7/4 40/23
16 468.7 21/16 17/13, 30/23
17 1173.0 63/32, 160/81 45/23, 51/26
18 677.3 40/27 34/23
19 181.6 10/9
20 885.8 5/3
21 390.1 5/4
22 1094.4 15/8 17/9
23 598.7 45/32 17/12

* In 13-limit CTE tuning

Tunings

Tuning spectrum

This spectrum assumes 19-limit leapday.

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
19/16 700.829
19/12 701.110
19/18 701.279
3/2 701.955
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
19/15 703.630
19/10 703.700
21/13 703.782
19/11 703.843
21/19 703.856
21/11 703.893
19/13 703.910
19/14 703.962
19/17 703.979 19- and 21-odd-limit minimax
44\75 704.000 75dfgh 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 121defgh val
5/3 704.218 7-, 15- and 17-odd-limit minimax
21/17 704.272
9/5 704.337 9-, 11- and 13-odd-limit minimax
27\46 704.348
17/16 704.373
11/7 704.377
21/16 704.424
17/12 704.478
7/4 704.588
17/9 704.593
11/8 704.665
37\63 704.762 63ch val
7/6 704.776
11/6 704.936
9/7 704.994
13/8 705.066
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
13/9 706.103
17/10 706.214
17/15 708.343

* Besides the octave

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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