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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 [[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]] 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.


== Tuning spectrum ==
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.
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 }}]
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>.


{| class="wikitable center-all"
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'''.
 
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents*
! colspan="2" | Approximate ratios
|-
! 13-limit
! Additional ratios<br />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
|
|-
|-
! EDO<br>generator
| 20
! [[eigenmonzo|eigenmonzo<br>(unchanged-interval)]]
| 885.8
! generator<br>(¢)
| 5/3
! comments
|
|-
| 21
| 390.1
| '''5/4'''
|
|-
| 22
| 1094.4
| '''15/8'''
| 17/9
|-
| 23
| 598.7
| 45/32
| 17/12
|}
<nowiki />* In 13-limit CTE tuning
 
== Tunings ==
=== Tuning spectrum ===
This spectrum assumes 19-limit leapday.
 
{| class="wikitable center-all left-4"
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|-
|  
|  
Line 25: Line 169:
|-
|-
|  
|  
| 24/19
| 19/12
| 701.110
| 701.110
|  
|  
Line 35: Line 179:
|-
|-
|  
|  
| 4/3
| 3/2
| 701.955
| 701.955
|  
|  
Line 42: Line 186:
|  
|  
| 702.439
| 702.439
|  
| 41cc… val, lower bound of 5-odd-limit diamond monotone
|-
|-
|  
|  
Line 72: Line 216:
|  
|  
| 703.448
| 703.448
|  
| 29g val, lower bound of 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
|-
|-
|  
|  
Line 95: 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 115: Line 259:
|-
|-
|  
|  
| 22/21
| 21/11
| 703.893
| 703.893
|  
|  
|-
|-
|  
|  
| 26/19
| 19/13
| 703.910
| 703.910
|  
|  
Line 137: Line 281:
|  
|  
| 704.000
| 704.000
|  
| 75dfgh val
|-
|-
|  
|  
| 16/15
| 15/8
| 704.012
| 704.012
|  
|  
Line 155: Line 299:
|-
|-
|  
|  
| 14/13
| 13/7
| 704.043
| 704.043
|  
|  
Line 165: Line 309:
|-
|-
|  
|  
| 22/17
| 17/11
| 704.126
| 704.126
|  
|  
Line 172: 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 185: 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 200: Line 344:
|-
|-
|  
|  
| 14/11
| 11/7
| 704.377
| 704.377
|  
|  
Line 210: 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 232: Line 376:
|  
|  
| 704.762
| 704.762
|  
| 63ch val
|-
|-
|  
|  
Line 240: Line 384:
|-
|-
|  
|  
| 12/11
| 11/6
| 704.936
| 704.936
|  
|  
Line 250: Line 394:
|-
|-
|  
|  
| 16/13
| 13/8
| 705.066
| 705.066
|  
|  
Line 267: 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 284: Line 428:
|  
|  
|}
|}
<nowiki/>* Besides the octave


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


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