Xenharmonic Wiki

Leapday: Difference between revisions

← Older edit
VisualWikitext
+disambiguation and history
m Text replacement - "Eigenmonzo<br>(unchanged-interval)" to "Unchanged interval<br>(eigenmonzo)"
Tags: Mobile edit Mobile web edit
 
(38 intermediate revisions by 5 users not shown)
Line 1: Line 1:
: ''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 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
|-
|-
! | ET<br>generator
| 18
! | [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
| 677.3
! | fifth<br>(¢)
| 40/27
! | comments
| 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
|}
<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
|-
|-
| |  
|  
| | 19/16
| 19/16
| | 700.829
| 700.829
| |  
|  
|-
|-
| |  
|  
| | 24/19
| 19/12
| | 701.110
| 701.110
| |  
|  
|-
|-
| |  
|  
| | 19/18
| 19/18
| | 701.279
| 701.279
| |  
|  
|-
|-
| |  
|  
| | 4/3
| 3/2
| | 701.955
| 701.955
| |  
|  
|-
|-
| | 24\41
| 24\41
| |  
|  
| | 702.439
| 702.439
| |
| 41cc… val, lower bound of 5-odd-limit diamond monotone
|-
|-
| |  
|  
| | 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
| |
| 29g val, lower bound of 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
|-
|-
| |  
|  
| | 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
| 19/10
| | 703.700
| 703.700
| |  
|  
|-
|-
| |  
|  
| | 26/21
| 21/13
| | 703.782
| 703.782
| |  
|  
|-
|-
| |  
|  
| | 22/19
| 19/11
| | 703.843
| 703.843
| |  
|  
|-
|-
| |  
|  
| | 21/19
| 21/19
| | 703.856
| 703.856
| |  
|  
|-
|-
| |  
|  
| | 22/21
| 21/11
| | 703.893
| 703.893
| |  
|  
|-
|-
| |  
|  
| | 26/19
| 19/13
| | 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
| |
| 75dfgh val
|-
|-
| |  
|  
| | 16/15
| 15/8
| | 704.012
| 704.012
| |  
|  
|-
|-
| |  
|  
| | 17/14
| 17/14
| | 704.014
| 704.014
| |  
|  
|-
|-
| |  
|  
| | 17/13
| 17/13
| | 704.027
| 704.027
| |  
|  
|-
|-
| |  
|  
| | 14/13
| 13/7
| | 704.043
| 704.043
| |  
|  
|-
|-
| |  
|  
| | 5/4
| 5/4
| | 704.110
| 704.110
| | 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| |  
|  
| | 22/17
| 17/11
| | 704.126
| 704.126
| |  
|  
|-
|-
| | 71\121
| 71\121
| |  
|  
| | 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
|-
|-
| |  
|  
| | 21/17
| 21/17
| | 704.272
| 704.272
| |  
|  
|-
|-
| |  
|  
| | 10/9
| 9/5
| | 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
| 11/7
| | 704.377
| 704.377
| |  
|  
|-
|-
| |  
|  
| | 21/16
| 21/16
| | 704.424
| 704.424
| |  
|  
|-
|-
| |  
|  
| | 24/17
| 17/12
| | 704.478
| 704.478
| |  
|  
|-
|-
| |  
|  
| | 8/7
| 7/4
| | 704.588
| 704.588
| |  
|  
|-
|-
| |  
|  
| | 18/17
| 17/9
| | 704.593
| 704.593
| |  
|  
|-
|-
| |  
|  
| | 11/8
| 11/8
| | 704.665
| 704.665
| |  
|  
|-
|-
| | 37\63
| 37\63
| |  
|  
| | 704.762
| 704.762
| |
| 63ch val
|-
|-
| |  
|  
| | 7/6
| 7/6
| | 704.776
| 704.776
| |  
|  
|-
|-
| |  
|  
| | 12/11
| 11/6
| | 704.936
| 704.936
| |  
|  
|-
|-
| |  
|  
| | 9/7
| 9/7
| | 704.994
| 704.994
| |  
|  
|-
|-
| |  
|  
| | 16/13
| 13/8
| | 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
| |
| 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
| |  
|  
|-
|-
| |  
|  
| | 17/15
| 17/15
| | 708.343
| 708.343
| |  
|  
|}
|}
<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"