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Leapday: Difference between revisions

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**Imported revision 246780659 - Original comment: **
 
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox regtemp
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| Title = Leapday
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-08-18 15:56:06 UTC</tt>.<br>
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
: The original revision id was <tt>246780659</tt>.<br>
| 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)
: The revision comment was: <tt></tt><br>
| Edo join 1 = 29 | Edo join 2 = 46
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Mapping = 1; 1 21 15 11 8
<h4>Original Wikitext content:</h4>
| Generators = 3/2 | Generators tuning = 704.2 | Optimization method = CWE
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">See [[Hemifamity temperaments#Leapday]].</pre></div>
| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[12L 5s]]
<h4>Original HTML content:</h4>
| Odd limit 1 = 9 | Mistuning 1 = 8.53 | Complexity 1 = 29
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Leapday&lt;/title&gt;&lt;/head&gt;&lt;body&gt;See &lt;a class="wiki_link" href="/Hemifamity%20temperaments#Leapday"&gt;Hemifamity temperaments&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
| 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|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𝄪).
 
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>.
 
See [[Hemifamity temperaments #Leapday]] for 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
| 704.2
| '''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
|}
<nowiki/>* In 13-limit CWE tuning, octave reduced
 
== 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 ===
{| class="wikitable center-all left-4"
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(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
|
|}
<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)
 
== References and external links ==
<references/>
 
[[Category:Leapday| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Sengic temperaments]]
[[Category:Aberschismic temperaments]]
Retrieved from "https://en.xen.wiki/w/Leapday"