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55edo: Difference between revisions

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**Imported revision 211672942 - Original comment: **
→Selected just intervals by error: Add 55cf val — this one surprisingly improves consistency more than the other warts or combinations thereof
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en = 55edo
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-03-18 03:55:07 UTC</tt>.<br>
| de = 55-EDO
: The original revision id was <tt>211672942</tt>.<br>
| es = 55 EDO
: The revision comment was: <tt></tt><br>
| ja =
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
}}
<h4>Original Wikitext content:</h4>
{{Infobox ET}}
<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">=&lt;span style="color: #008023; font-size: 103%;"&gt;55 tone equal temperament&lt;/span&gt;=
{{ED intro}}
//55edo// divides the octave into 55 parts of 21.818 cents. It can be used for a meantone tuning, and is close to 1/6 comma [[meantone]] (and is almost exactly 10/57 comma meantone.) [[http://en.wikipedia.org/wiki/Georg_Philipp_Telemann|Telemann]] suggested it as a theoretical basis for analyzing the intervals of meantone, in which he was followed by [[http://en.wikipedia.org/wiki/Leopold_Mozart|Leopold]] and [[http://en.wikipedia.org/wiki/Wolfgang_Amadeus_Mozart|Wolfgang Mozart]]. It can also be used for [[Meantone family|mohajira and liese]] temperaments.


5-limit commas: 81/80, &lt;31 1 -14|
== Theory ==
55edo supports [[meantone]], and is close to [[1/6-comma meantone]]. {{w|Georg Philipp Telemann|Telemann}} suggested it as a theoretical basis for analyzing the [[meantone intervals|intervals of meantone]]. {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}} recommended 55edo or something close to it, with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.<ref>Chesnut, John (1977) ''Mozart's Teaching of Intonation'', '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]</ref> It can also be used for [[mohajira]] and [[liese]] temperaments. It also supports an extremely sharp tuning of [[huygens|huygens/undecimal meantone]] using the 55de [[val]], meaning that primes 7 and 11 are mapped very sharply to their second-best mapping. 55edo is a [[zeta valley edo]], suggesting it does not approximate the harmonic series very well for its size.


7-limit commas: 81/80, 686/675, 6144/6125


11-limit commas: 81/80, 121/120, 176/175, 686/675
=== Odd harmonics ===
{{Harmonics in equal|55}}


==Intervals==  
=== Subsets and supersets ===
|| Degrees of 55-EDO || Cents value ||
Since 55 factors into primes as {{nowrap| 5 × 11 }}, 55edo contains [[5edo]] and [[11edo]] as its subsets.
|| 0 || 0 ||
|| 1 || 21,818 ||
|| 2 || 43,636 ||
|| 3 || 65,455 ||
|| 4 || 87,273 ||
|| 5 || 109,091 ||
|| 6 || 130,909 ||
|| 7 || 152,727 ||
|| 8 || 174,545 ||
|| 9 || 196,364 ||
|| 10 || 218,182 ||
|| 11 || 240 ||
|| 12 || 261,818 ||
|| 13 || 283,636 ||
|| 14 || 305,455 ||
|| 15 || 327,273 ||
|| 16 || 349,091 ||
|| 17 || 370,909 ||
|| 18 || 392,727 ||
|| 19 || 414,545 ||
|| 20 || 436,364 ||
|| 21 || 458,182 ||
|| 22 || 480 ||
|| 23 || 501,818 ||
|| 24 || 523,636 ||
|| 25 || 545,455 ||
|| 26 || 567,273 ||
|| 27 || 589,091 ||
|| 28 || 610,909 ||
|| 29 || 632,727 ||
|| 30 || 654,545 ||
|| 31 || 676,364 ||
|| 32 || 698,182 ||
|| 33 || 720 ||
|| 34 || 741,818 ||
|| 35 || 763,636 ||
|| 36 || 785,455 ||
|| 37 || 807,273 ||
|| 38 || 829,091 ||
|| 39 || 850,909 ||
|| 40 || 872,727 ||
|| 41 || 894,545 ||
|| 42 || 916,364 ||
|| 43 || 938,182 ||
|| 44 || 960 ||
|| 45 || 981,818 ||
|| 46 || 1003,636 ||
|| 47 || 1025,455 ||
|| 48 || 1047,273 ||
|| 49 || 1069,091 ||
|| 50 || 1090,909 ||
|| 51 || 1112,727 ||
|| 52 || 1134,545 ||
|| 53 || 1156,364 ||
|| 54 || 1178,182 ||</pre></div>
<h4>Original HTML content:</h4>
<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;55edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x55 tone equal temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #008023; font-size: 103%;"&gt;55 tone equal temperament&lt;/span&gt;&lt;/h1&gt;
&lt;em&gt;55edo&lt;/em&gt; divides the octave into 55 parts of 21.818 cents. It can be used for a meantone tuning, and is close to 1/6 comma &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; (and is almost exactly 10/57 comma meantone.) &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Georg_Philipp_Telemann" rel="nofollow"&gt;Telemann&lt;/a&gt; suggested it as a theoretical basis for analyzing the intervals of meantone, in which he was followed by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Leopold_Mozart" rel="nofollow"&gt;Leopold&lt;/a&gt; and &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Wolfgang_Amadeus_Mozart" rel="nofollow"&gt;Wolfgang Mozart&lt;/a&gt;. It can also be used for &lt;a class="wiki_link" href="/Meantone%20family"&gt;mohajira and liese&lt;/a&gt; temperaments.&lt;br /&gt;
&lt;br /&gt;
5-limit commas: 81/80, &amp;lt;31 1 -14|&lt;br /&gt;
&lt;br /&gt;
7-limit commas: 81/80, 686/675, 6144/6125&lt;br /&gt;
&lt;br /&gt;
11-limit commas: 81/80, 121/120, 176/175, 686/675&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x55 tone equal temperament-Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Intervals&lt;/h2&gt;


&lt;table class="wiki_table"&gt;
== Intervals ==
    &lt;tr&gt;
{| class="wikitable center-1 right-2 left-3"
        &lt;td&gt;Degrees of 55-EDO&lt;br /&gt;
|-
&lt;/td&gt;
! [[Degree|#]]
        &lt;td&gt;Cents value&lt;br /&gt;
! [[Cent]]s
&lt;/td&gt;
! Approximate ratios
    &lt;/tr&gt;
! colspan="3" | [[Ups and downs notation]]
    &lt;tr&gt;
|-
        &lt;td&gt;0&lt;br /&gt;
| 0
&lt;/td&gt;
| 0.0
        &lt;td&gt;0&lt;br /&gt;
| [[1/1]]
&lt;/td&gt;
| P1
    &lt;/tr&gt;
| perfect 1sn
    &lt;tr&gt;
| D
        &lt;td&gt;1&lt;br /&gt;
|-
&lt;/td&gt;
| 1
        &lt;td&gt;21,818&lt;br /&gt;
| 21.8
&lt;/td&gt;
| [[65/64]], [[78/77]], [[99/98]], ''[[128/125]]''
    &lt;/tr&gt;
| ^1
    &lt;tr&gt;
| up 1sn
        &lt;td&gt;2&lt;br /&gt;
| ^D
&lt;/td&gt;
|-
        &lt;td&gt;43,636&lt;br /&gt;
| 2
&lt;/td&gt;
| 43.6
    &lt;/tr&gt;
| [[36/35]], ''[[64/63]]''
    &lt;tr&gt;
| ^^1
        &lt;td&gt;3&lt;br /&gt;
| dup 1sn
&lt;/td&gt;
| ^^D
        &lt;td&gt;65,455&lt;br /&gt;
|-
&lt;/td&gt;
| 3
    &lt;/tr&gt;
| 65.5
    &lt;tr&gt;
| [[28/27]]
        &lt;td&gt;4&lt;br /&gt;
| vvm2
&lt;/td&gt;
| dudminor 2nd
        &lt;td&gt;87,273&lt;br /&gt;
| vvEb
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 4
    &lt;tr&gt;
| 87.3
        &lt;td&gt;5&lt;br /&gt;
| [[21/20]], ''[[18/17]]'', ''[[25/24]]''
&lt;/td&gt;
| vm2
        &lt;td&gt;109,091&lt;br /&gt;
| downminor 2nd
&lt;/td&gt;
| vEb
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 5
        &lt;td&gt;6&lt;br /&gt;
| 109.1
&lt;/td&gt;
| [[16/15]], [[17/16]]
        &lt;td&gt;130,909&lt;br /&gt;
| m2
&lt;/td&gt;
| minor 2nd
    &lt;/tr&gt;
| Eb
    &lt;tr&gt;
|-
        &lt;td&gt;7&lt;br /&gt;
| 6
&lt;/td&gt;
| 130.9
        &lt;td&gt;152,727&lt;br /&gt;
| [[13/12]], [[14/13]]
&lt;/td&gt;
| ^m2
    &lt;/tr&gt;
| upminor 2nd
    &lt;tr&gt;
| ^Eb
        &lt;td&gt;8&lt;br /&gt;
|-
&lt;/td&gt;
| 7
        &lt;td&gt;174,545&lt;br /&gt;
| 152.7
&lt;/td&gt;
| [[12/11]], ''[[11/10]]''
    &lt;/tr&gt;
| ~2
    &lt;tr&gt;
| mid 2nd
        &lt;td&gt;9&lt;br /&gt;
| vvE
&lt;/td&gt;
|-
        &lt;td&gt;196,364&lt;br /&gt;
| 8
&lt;/td&gt;
| 174.5
    &lt;/tr&gt;
|
    &lt;tr&gt;
| vM2
        &lt;td&gt;10&lt;br /&gt;
| downmajor 2nd
&lt;/td&gt;
| vE
        &lt;td&gt;218,182&lt;br /&gt;
|-
&lt;/td&gt;
| 9
    &lt;/tr&gt;
| 196.4
    &lt;tr&gt;
| [[9/8]], ''[[10/9]]''
        &lt;td&gt;11&lt;br /&gt;
| M2
&lt;/td&gt;
| major 2nd
        &lt;td&gt;240&lt;br /&gt;
| E
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 10
    &lt;tr&gt;
| 218.2
        &lt;td&gt;12&lt;br /&gt;
| [[17/15]]
&lt;/td&gt;
| ^M2
        &lt;td&gt;261,818&lt;br /&gt;
| upmajor 2nd
&lt;/td&gt;
| ^E
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 11
        &lt;td&gt;13&lt;br /&gt;
| 240.0
&lt;/td&gt;
| [[8/7]]
        &lt;td&gt;283,636&lt;br /&gt;
| ^^M2
&lt;/td&gt;
| dupmajor 2nd
    &lt;/tr&gt;
| ^^E
    &lt;tr&gt;
|-
        &lt;td&gt;14&lt;br /&gt;
| 12
&lt;/td&gt;
| 261.8
        &lt;td&gt;305,455&lt;br /&gt;
| [[7/6]]
&lt;/td&gt;
| vvm3
    &lt;/tr&gt;
| dudminor 3rd
    &lt;tr&gt;
| vvF
        &lt;td&gt;15&lt;br /&gt;
|-
&lt;/td&gt;
| 13
        &lt;td&gt;327,273&lt;br /&gt;
| 283.6
&lt;/td&gt;
| [[13/11]]
    &lt;/tr&gt;
| vm3
    &lt;tr&gt;
| downminor 3rd
        &lt;td&gt;16&lt;br /&gt;
| vF
&lt;/td&gt;
|-
        &lt;td&gt;349,091&lt;br /&gt;
| 14
&lt;/td&gt;
| 305.5
    &lt;/tr&gt;
| [[6/5]]
    &lt;tr&gt;
| m3
        &lt;td&gt;17&lt;br /&gt;
| minor 3rd
&lt;/td&gt;
| F
        &lt;td&gt;370,909&lt;br /&gt;
|-
&lt;/td&gt;
| 15
    &lt;/tr&gt;
| 327.3
    &lt;tr&gt;
|
        &lt;td&gt;18&lt;br /&gt;
| ^m3
&lt;/td&gt;
| upminor 3rd
        &lt;td&gt;392,727&lt;br /&gt;
| ^F
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 16
    &lt;tr&gt;
| 349.1
        &lt;td&gt;19&lt;br /&gt;
| [[11/9]], [[27/22]]
&lt;/td&gt;
| ~3
        &lt;td&gt;414,545&lt;br /&gt;
| mid 3rd
&lt;/td&gt;
| ^^F
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 17
        &lt;td&gt;20&lt;br /&gt;
| 370.9
&lt;/td&gt;
| [[26/21]], ''[[16/13]]''
        &lt;td&gt;436,364&lt;br /&gt;
| vM3
&lt;/td&gt;
| downmajor 3rd
    &lt;/tr&gt;
| vF#
    &lt;tr&gt;
|-
        &lt;td&gt;21&lt;br /&gt;
| 18
&lt;/td&gt;
| 392.7
        &lt;td&gt;458,182&lt;br /&gt;
| [[5/4]]
&lt;/td&gt;
| M3
    &lt;/tr&gt;
| major 3rd
    &lt;tr&gt;
| F#
        &lt;td&gt;22&lt;br /&gt;
|-
&lt;/td&gt;
| 19
        &lt;td&gt;480&lt;br /&gt;
| 414.5
&lt;/td&gt;
| [[14/11]]
    &lt;/tr&gt;
| ^M3
    &lt;tr&gt;
| upmajor 3rd
        &lt;td&gt;23&lt;br /&gt;
| ^F#
&lt;/td&gt;
|-
        &lt;td&gt;501,818&lt;br /&gt;
| 20
&lt;/td&gt;
| 436.4
    &lt;/tr&gt;
| [[9/7]]
    &lt;tr&gt;
| ^^M3
        &lt;td&gt;24&lt;br /&gt;
| dupmajor 3rd
&lt;/td&gt;
| ^^F#
        &lt;td&gt;523,636&lt;br /&gt;
|-
&lt;/td&gt;
| 21
    &lt;/tr&gt;
| 458.2
    &lt;tr&gt;
| ''[[21/16]]''
        &lt;td&gt;25&lt;br /&gt;
| vv4
&lt;/td&gt;
| dud 4th
        &lt;td&gt;545,455&lt;br /&gt;
| vvG
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 22
    &lt;tr&gt;
| 480.0
        &lt;td&gt;26&lt;br /&gt;
|
&lt;/td&gt;
| v4
        &lt;td&gt;567,273&lt;br /&gt;
| down 4th
&lt;/td&gt;
| vG
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 23
        &lt;td&gt;27&lt;br /&gt;
| 501.8
&lt;/td&gt;
| [[4/3]], ''[[27/20]]''
        &lt;td&gt;589,091&lt;br /&gt;
| P4
&lt;/td&gt;
| perfect 4th
    &lt;/tr&gt;
| G
    &lt;tr&gt;
|-
        &lt;td&gt;28&lt;br /&gt;
| 24
&lt;/td&gt;
| 523.6
        &lt;td&gt;610,909&lt;br /&gt;
|
&lt;/td&gt;
| ^4
    &lt;/tr&gt;
| up 4th
    &lt;tr&gt;
| ^G
        &lt;td&gt;29&lt;br /&gt;
|-
&lt;/td&gt;
| 25
        &lt;td&gt;632,727&lt;br /&gt;
| 545.5
&lt;/td&gt;
| [[11/8]], [[15/11]]
    &lt;/tr&gt;
| ~4
    &lt;tr&gt;
| mid 4th
        &lt;td&gt;30&lt;br /&gt;
| ^^G
&lt;/td&gt;
|-
        &lt;td&gt;654,545&lt;br /&gt;
| 26
&lt;/td&gt;
| 567.3
    &lt;/tr&gt;
| [[7/5]], [[18/13]]
    &lt;tr&gt;
| vA4
        &lt;td&gt;31&lt;br /&gt;
| downaug 4th
&lt;/td&gt;
| vG#
        &lt;td&gt;676,364&lt;br /&gt;
|-
&lt;/td&gt;
| 27
    &lt;/tr&gt;
| 589.1
    &lt;tr&gt;
| [[24/17]]
        &lt;td&gt;32&lt;br /&gt;
| A4, vd5
&lt;/td&gt;
| aug 4th, downdim 5th
        &lt;td&gt;698,182&lt;br /&gt;
| G#, vAb
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 28
    &lt;tr&gt;
| 610.9
        &lt;td&gt;33&lt;br /&gt;
| [[17/12]]
&lt;/td&gt;
| ^A4, d5
        &lt;td&gt;720&lt;br /&gt;
| upaug 4th, dim 5th
&lt;/td&gt;
| ^G#, Ab
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 29
        &lt;td&gt;34&lt;br /&gt;
| 632.7
&lt;/td&gt;
| [[10/7]], [[13/9]]
        &lt;td&gt;741,818&lt;br /&gt;
| ^d5
&lt;/td&gt;
| updim 5th
    &lt;/tr&gt;
| ^Ab
    &lt;tr&gt;
|-
        &lt;td&gt;35&lt;br /&gt;
| 30
&lt;/td&gt;
| 654.5
        &lt;td&gt;763,636&lt;br /&gt;
| [[16/11]], [[22/15]]
&lt;/td&gt;
| ~5
    &lt;/tr&gt;
| mid 5th
    &lt;tr&gt;
| vvA
        &lt;td&gt;36&lt;br /&gt;
|-
&lt;/td&gt;
| 31
        &lt;td&gt;785,455&lt;br /&gt;
| 676.4
&lt;/td&gt;
|
    &lt;/tr&gt;
| v5
    &lt;tr&gt;
| down 5th
        &lt;td&gt;37&lt;br /&gt;
| vA
&lt;/td&gt;
|-
        &lt;td&gt;807,273&lt;br /&gt;
| 32
&lt;/td&gt;
| 698.2
    &lt;/tr&gt;
| [[3/2]], ''[[40/27]]''
    &lt;tr&gt;
| P5
        &lt;td&gt;38&lt;br /&gt;
| perfect 5th
&lt;/td&gt;
| A
        &lt;td&gt;829,091&lt;br /&gt;
|-
&lt;/td&gt;
| 33
    &lt;/tr&gt;
| 720.0
    &lt;tr&gt;
|
        &lt;td&gt;39&lt;br /&gt;
| ^5
&lt;/td&gt;
| up 5th
        &lt;td&gt;850,909&lt;br /&gt;
| ^A
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 34
    &lt;tr&gt;
| 741.8
        &lt;td&gt;40&lt;br /&gt;
| ''[[32/21]]''
&lt;/td&gt;
| ^^5
        &lt;td&gt;872,727&lt;br /&gt;
| dup 5th
&lt;/td&gt;
| ^^A
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 35
        &lt;td&gt;41&lt;br /&gt;
| 763.6
&lt;/td&gt;
| [[14/9]]
        &lt;td&gt;894,545&lt;br /&gt;
| vvm6
&lt;/td&gt;
| dudminor 6th
    &lt;/tr&gt;
| vvBb
    &lt;tr&gt;
|-
        &lt;td&gt;42&lt;br /&gt;
| 36
&lt;/td&gt;
| 785.5
        &lt;td&gt;916,364&lt;br /&gt;
| [[11/7]]
&lt;/td&gt;
| vm6
    &lt;/tr&gt;
| downminor 6th
    &lt;tr&gt;
| vBb
        &lt;td&gt;43&lt;br /&gt;
|-
&lt;/td&gt;
| 37
        &lt;td&gt;938,182&lt;br /&gt;
| 807.3
&lt;/td&gt;
| [[8/5]]
    &lt;/tr&gt;
| m6
    &lt;tr&gt;
| minor 6th
        &lt;td&gt;44&lt;br /&gt;
| Bb
&lt;/td&gt;
|-
        &lt;td&gt;960&lt;br /&gt;
| 38
&lt;/td&gt;
| 829.1
    &lt;/tr&gt;
| [[21/13]], ''[[13/8]]''
    &lt;tr&gt;
| ^m6
        &lt;td&gt;45&lt;br /&gt;
| upminor 6th
&lt;/td&gt;
| ^Bb
        &lt;td&gt;981,818&lt;br /&gt;
|-
&lt;/td&gt;
| 39
    &lt;/tr&gt;
| 850.9
    &lt;tr&gt;
| [[18/11]], [[44/27]]
        &lt;td&gt;46&lt;br /&gt;
| ~6
&lt;/td&gt;
| mid 6th
        &lt;td&gt;1003,636&lt;br /&gt;
| vvB
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 40
    &lt;tr&gt;
| 872.7
        &lt;td&gt;47&lt;br /&gt;
|
&lt;/td&gt;
| vM6
        &lt;td&gt;1025,455&lt;br /&gt;
| downmajor 6th
&lt;/td&gt;
| vB
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 41
        &lt;td&gt;48&lt;br /&gt;
| 894.5
&lt;/td&gt;
| [[5/3]]
        &lt;td&gt;1047,273&lt;br /&gt;
| M6
&lt;/td&gt;
| major 6th
    &lt;/tr&gt;
| B
    &lt;tr&gt;
|-
        &lt;td&gt;49&lt;br /&gt;
| 42
&lt;/td&gt;
| 916.4
        &lt;td&gt;1069,091&lt;br /&gt;
| [[22/13]]
&lt;/td&gt;
| ^M6
    &lt;/tr&gt;
| upmajor 6th
    &lt;tr&gt;
| ^B
        &lt;td&gt;50&lt;br /&gt;
|-
&lt;/td&gt;
| 43
        &lt;td&gt;1090,909&lt;br /&gt;
| 938.2
&lt;/td&gt;
| [[12/7]]
    &lt;/tr&gt;
| ^^M6
    &lt;tr&gt;
| dupmajor 6th
        &lt;td&gt;51&lt;br /&gt;
| ^^B
&lt;/td&gt;
|-
        &lt;td&gt;1112,727&lt;br /&gt;
| 44
&lt;/td&gt;
| 960.0
    &lt;/tr&gt;
| [[7/4]]
    &lt;tr&gt;
| vvm7
        &lt;td&gt;52&lt;br /&gt;
| dudminor 7th
&lt;/td&gt;
| vvC
        &lt;td&gt;1134,545&lt;br /&gt;
|-
&lt;/td&gt;
| 45
    &lt;/tr&gt;
| 981.8
    &lt;tr&gt;
| [[30/17]]
        &lt;td&gt;53&lt;br /&gt;
| vm7
&lt;/td&gt;
| downminor 7th
        &lt;td&gt;1156,364&lt;br /&gt;
| vC
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 46
    &lt;tr&gt;
| 1003.6
        &lt;td&gt;54&lt;br /&gt;
| [[16/9]], ''[[9/5]]''
&lt;/td&gt;
| m7
        &lt;td&gt;1178,182&lt;br /&gt;
| minor 7th
&lt;/td&gt;
| C
    &lt;/tr&gt;
|-
&lt;/table&gt;
| 47
| 1025.5
|
| ^m7
| upminor 7th
| ^C
|-
| 48
| 1047.3
| [[11/6]], ''[[20/11]]''
| ~7
| mid 7th
| ^^C
|-
| 49
| 1069.1
| [[13/7]], [[24/13]]
| vM7
| downmajor 7th
| vC#
|-
| 50
| 1090.9
| [[15/8]], ''[[32/17]]''
| M7
| major 7th
| C#
|-
| 51
| 1112.7
| [[40/21]], ''[[17/9]]'', ''[[48/25]]''
| ^M7
| upmajor 7th
| ^C#
|-
| 52
| 1134.5
| [[56/27]]
| ^^M7
| dupmajor 7th
| ^^C#
|-
| 53
| 1156.4
| [[35/18]], ''[[63/32]]''
| vv8
| dud 8ve
| vvD
|-
| 54
| 1178.2
| [[128/65]], [[77/39]], [[196/99]], ''[[125/64]]''
| v8
| down 8ve
| vD
|-
| 55
| 1200.0
| [[2/1]]
| P8
| perfect 8ve
| D
|}
<nowiki />* 55f val (tending flat), inconsistent intervals labeled in ''italic''


&lt;/body&gt;&lt;/html&gt;</pre></div>
== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp4-szg}}
 
=== Kite's ups and downs notation ===
55edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
{{Ups and downs sharpness}}
 
=== Sagittal notation ===
==== Evo flavor ====
<imagemap>
File:55-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 615 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[896/891]]
rect 160 80 280 106 [[33/32]]
default [[File:55-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:55-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[896/891]]
rect 160 80 280 106 [[33/32]]
default [[File:55-EDO_Revo_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:55-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[896/891]]
rect 160 80 280 106 [[33/32]]
default [[File:55-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
=== 31-tone subset ===
The 31-out-of-55edo subset can be notated entirely with the standard notation of 7 each of naturals/sharps/flats, and 5 each of doublesharps/doubleflats, as a 31-tone chain-of-5ths from Gbb to Ax.
 
[[File:Monzo55Notation.jpeg|400px|frameless|alt=Diagram of 31-tone subset of 55edo using plain Western notation, by Joe Monzo.|Diagram of 31-tone subset of 55edo using plain Western notation, by [[Joe Monzo]].]]
 
== Approximation to JI ==
[[File:55ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 55edo]]
 
=== Selected just intervals by error ===
{{Q-odd-limit intervals|55}}
{{Q-odd-limit intervals|55.05|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55d val mapping}}
{{Q-odd-limit intervals|54.95|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55f val mapping}}
{{Q-odd-limit intervals|54.91|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55cf val mapping}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -87 55 }}
| {{mapping| 55 87 }}
| +1.31
| 1.19
| 7.21
|-
| 2.3.5
| 81/80, {{monzo| 31 1 -14 }}
| {{mapping| 55 87 128 }}
| −0.13
| 2.10
| 9.63
|}
 
=== Uniform maps ===
{{Uniform map|edo=55}}
 
=== Commas ===
{{Todo|cleanup|inline=true}}
 
'''5-limit commas''': [[81/80]], [[Quintosec_family|{{monzo| 47 -15 -10 }}]], {{monzo| 31 1 -14 }}, {{monzo| 27 5 -15 }}
 
'''7-limit commas''': 31104/30625, [[6144/6125]], 81648/78125, 16128/15625, 28672/28125, 33075/32768, 83349/80000, 1029/1000, [[686/675]], [[10976/10935]], [[Cloudy comma|16807/16384]], 84035/82944
 
'''11-limit commas''': 59049/58564, 74088/73205, 46656/46585, 21609/21296, 12005/11979, 19683/19360, [[243/242]], 3087/3025, 5488/5445, 19683/19250, 1944/1925, 45927/45056, 2835/2816, 35721/34375, 7056/6875, 12544/12375, 7203/7040, 2401/2376, 24057/24010, 72171/70000, 891/875, [[176/175]], 2079/2048, [[385/384]], 3234/3125, 17248/16875, 26411/25600, 26411/2592, 26411/262404, 88209/87808, 30976/30625, 3267/3200, [[121/120]], 81312/78125, 41503/40000, 41503/40500, 35937/35000, 2662/2625, 42592/42525, 83853/81920, 9317/9216, 65219/62500, 43923/43904, 14641/14400, [[14641/14580]]
 
'''13-limit commas''': 59535/57122, 29400/28561, 29568/28561, 29645/28561, 24576/24167, 99225/96668, 24500/24167, 50421/48334, 45927/43940, 2268/2197, 2240/2197, 57624/54925, 61875/61516, 57024/54925, 11264/10985, 72765/70304, 13475/13182, 22869/21970, 6776/6591, 20736/20449, 20480/20449, 84035/81796, 91125/91091, 65536/65065, 15309/14872, 1890/1859, 5600/5577, 9604/9295, 59049/57967, 58320/57967, 4374/4225, 864/845, [[512/507]], 11025/10816, 6125/6084, 21952/21125, 16807/16224, 84035/82134, 66825/66248, 90112/88725, 56133/54080, 693/676, 1540/1521, 26411/25350, 58806/57967, 58080/57967, 88209/84500, 4356/4225, 7744/7605, 88935/86528, 33275/33124, 27951/27040, 9317/9126, 58564/57967, 43923/42250, 17496/17303, 87808/86515, 55296/55055, 25515/25168, [[1575/1573]], 64827/62920, 4802/4719, 98415/98098, 59049/57200, 729/715, [[144/143]], 18375/18304, 18522/17875, 10976/10725, 84035/82368, 59049/56875, 11664/11375, 2304/2275, [[4096/4095]], 1701/1664, [[105/104]], 42336/40625, 25088/24375, 21609/20800, 2401/2340, 9604/9477, 72171/71344, 2673/2600, [[66/65]], [[352/351]], 13475/13312, 33957/32500, 15092/14625, 81675/81536, 58806/56875, 11616/11375, 61952/61425, 68607/66560, 847/832, 4235/4212, 35937/35672, 1331/1300, 5324/5265, 58564/56875, 85293/85184, 13377/13310, 85293/84700, 15288/15125, 31213/30976, 67392/67375, 28431/28160, 34944/34375, 4459/4400, 4459/4455, 28431/28000, [[351/350]], 79872/78125, 66339/65536, 51597/50000, 637/625, 10192/10125, 31213/30720, [[31213/31104]], 30888/30625, 1287/1280, 81081/78125, 16016/15625, 49049/48000, 49049/48600, 14157/14000, 33033/32768, 77077/75000, 51909/51200, 17303/17280, 75712/75625, 8281/8250, 41067/40960, 31941/31250, 9464/9375, 57967/57600, 91091/90000, 61347/61250, 79092/78125
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 6\55
| 130.9
| 14/13
| [[Twothirdtonic]] (55f)
|-
| 1
| 8\55
| 174.5
| [[10/9]]~[[11/10]]
| [[Tetracot]] (55c)
|-
| 1
| 16\55
| 349.1
| 11/9
| [[Mohaha]]
|-
| 1
| 23\55
| 501.8
| 4/3
| [[Meantone]] (55d)
|-
| 1
| 26\55
| 567.3
| 7/5
| [[Liese]] (55)
|-
| 1
| 27\55
| 589.1
| 45/32
| [[Untriton]] (55d) / [[aufo]] (55)
|-
| 5
| 17\55<br>(5\55)
| 370.9<br>(109.1)
| 99/80<br>(16/15)
| [[Quintosec]]
|-
| 11
| 23\55<br>(3\55)
| 501.8<br>(65.5)
| 4/3<br>(36/35)
| [[Hendecatonic (temperament)|Hendecatonic]] (55)
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
== Scales ==
; Subsets of Twothirdtonic[37]
* Undecimal otonal-like pentatonic: 17 8 7 12 11
 
; Subsets of Hendecatonic[33]
* Septimal pentatonic-like: 10 13 9 13 10
* Septimal minor blues-like: 13 10 4 5 13 10
* Septimal heptatonic blues-like: 13 10 4 5 8 5 10
 
; Others
* Sakura-like scale containing [[phi]]: 9 6 18 5 17
* Quasi-[[equiheptatonic]] scale: 8 8 7 9 7 9 7
 
== Instruments ==
* [[Lumatone mapping for 55edo]]
 
== Music ==
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=oymJKnYzzOw "Jesus bleibet meine Freude" from ''Herz und Mund und Tat und Leben'', BWV 147] (1723) – arranged for two organs, rendered by Claudi Meneghin (2021)
* [https://www.youtube.com/watch?v=xoCNOIsjfeU "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by [[Claudi Meneghin]] (2024)
* "Ricercar a 6" from ''The Musical Offering'', BWV 1079 (1747) – rendered by Claudi Meneghin
** [https://www.youtube.com/watch?v=OkRVNo19guo harpsichord rendition] (2025)
** [https://www.youtube.com/watch?v=X_qROPtHf9g fortepiano rendition] (2025)
** [https://www.youtube.com/watch?v=X9SexO03MTw organ rendition] (2026)
* [https://www.youtube.com/watch?v=Y5sIjh_Te40 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=QOPxqNgkVWM "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
 
; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=OfOt3nOp-f8 ''Prelude in E Minor "The Great"''] – rendered by [[Claudi Meneghin]] (2023)
* [https://www.youtube.com/watch?v=tuIPIhSxUPs ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)
 
; {{W|John Bull (composer)|John Bull}}
* [https://www.youtube.com/watch?v=I2d-540RuoM ''Fantasia «Ut Re Mi Fa Sol La»''] (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – rendered by Claudi Meneghin (2026)
 
; {{W|Louis Couperin}}
* [https://www.youtube.com/shorts/bcgY3k-6rao ''La Piémontoise''] (1658?) – rendered by Claudi Meneghin (2026)
 
; {{W|Georg Frideric Handel}}
* [https://www.youtube.com/watch?v=rDvKPuzsno8 ''Fugue'' from "Suite in E minor", HWV 429] (1720) – arranged for Baroque ensemble and drums, rendered by Claudi Meneghin (2025)
 
; {{W|Scott Joplin}}
* ''Maple Leaf Rag'' (1899) – arranged for harpsichord and rendered by [[Claudi Meneghin]] ([https://www.youtube.com/watch?v=GbhpuoIJgxk 2024 version]; [https://www.youtube.com/shorts/3Y9y9I6q1as 2026 version])
 
; {{W|Mladen Milićević}}
* [https://www.youtube.com/shorts/1BhIK0s2-vs "The Room" from ''The Room''] (2003) – covered by [[Bryan Deister]] (2026)
 
; {{W|Wolfgang Amadeus Mozart}}
* [https://www.youtube.com/watch?v=C_AML6XW-2g ''Rondo alla Turca'' from the Piano Sonata No. 11, K. 331] (1778) – rendered by Francium (2023)
* [https://www.youtube.com/watch?v=XgRksdk6zyQ ''Fugue in G minor'', K. 401] (1782) – rendered by Francium (2023)
* [http://www.seraph.it/dep/int/AdagioKV540.mp3 ''Adagio in B minor'', K. 540] (1788) – rendered by Carlo Serafini (2011) ([http://www.seraph.it/blog_files/706c4662272db7703def4d57edfcb955-119.html blog entry])
* [https://www.youtube.com/watch?v=pFjJCj2MBTM ''Allegro'' from the Piano Sonata No. 16, K. 545] (1788) – rendered by Francium (2023)
* ''Kleine Gigue in G'', K. 574 (1789) – rendered by [[Claudi Meneghin]]
** [https://www.youtube.com/watch?v=p88MWgdio14&list=PLC6ZSKWKnVz0mOTLQkCUi9ydWGLpBP8gZ&index=2 Fortepiano rendition] (2025)
** [https://www.youtube.com/shorts/7jt-jzziDBg Harpsichord rendition] (2026)
** [https://www.youtube.com/shorts/9J1DkS5UGzE Organ rendition] (2026)
 
; {{W|Keiichi Okabe}}
* [https://www.youtube.com/watch?v=L24G4Y7tZgI ''Yuutsu no Yuutsu''] (2006) – rendered by MortisTheneRd (2024)
 
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/l62rb8ULCXs ''55edo improv''] (2025)
* [https://www.youtube.com/watch?v=kVmToKkZU88 ''Waltz in 55edo''] (2025)
* [https://www.youtube.com/shorts/eQHpMFLrjjQ ''55edo prelude''] (2025)
 
; [[James Kukula]]
* [https://app.box.com/s/8hq89cb3rqqkrhvkxgvqtppa255kcqrq?fbclid=IwY2xjawISjSlleHRuA2FlbQIxMAABHcl5t8n_C7QUJqdEnwSaWBc5u3BpldmcAjhQQljsQIPl1qJ-zdCr9T8NMw_aem_Ez0m-Ls_ZqI0-c0Ld-28Yg ''55edo Melted Syntonic''] (2025)
 
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=9c5MtrZFNhA ''Improvisation One in 55edo''] (2025)
* [https://www.youtube.com/watch?v=ggFGUn1Ya2A ''Improvisation Two in 55edo''] (2025)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=AgsJCTyxqiM ''Double Fugue on "We Wish You a Merry Christmas" for String Quartet''] (2020)
* [https://www.youtube.com/watch?v=rAbbvyotIr4 ''Canon at the Diatonic Semitone on an Ancient Lombard Theme''] (2021)
* [https://www.youtube.com/watch?v=hCUIx1RzvEk ''Chacony "Lament & Deception"'' for Two Violins and Cello] (2021), [https://www.youtube.com/watch?v=abJP4euMlsg for Baroque Wind Ensemble] (2023)
* [https://www.youtube.com/watch?v=9zfWeO0eJdA Fantasy "Almost a Fugue" on a Theme by Giuliani, for String Quartet] (2021)
* [https://www.youtube.com/watch?v=jOiub14Cskw ''Double Fugue on "Old McDonald" + "Shave & a Haircut"''] (2024)
 
; [[Herman Miller]]
* [https://soundcloud.com/morphosyntax-1/road-trip-to-nowhere ''Road Trip to Nowhere''] (2021)
* [https://soundcloud.com/morphosyntax-1/migration ''Migration''] (2025)
 
== External links ==
* ''[http://tonalsoft.com/monzo/55edo/55edo.aspx Mozart's tuning: 55-edo and its close relative, 1/6-comma meantone]'' (containing another listening example) on [[Tonalsoft Encyclopedia]]
 
== References ==
<references />
 
[[Category:Meantone]]
[[Category:Historical]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/55edo"