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37edo: Difference between revisions

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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>
| de = 37-EDO
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-10-23 12:45:59 UTC</tt>.<br>
| en = 37edo
: The original revision id was <tt>172988221</tt>.<br>
| es =
: 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">37edo is the scale derived from dividing the octave into 37 microtonal steps of approximately 32.43 cents each. It offers close approximations to [[OverToneSeries|harmonics]] 5, 7, 11, and 13:
{{ED intro}}


12\37 = 389.2 cents
== Theory ==
30\37 = 973.0 cents
37edo has very accurate approximations of [[harmonic]]s [[5/1|5]], [[7/1|7]], [[11/1|11]] and [[13/1|13]], making it a good choice for a [[no-threes subgroup temperaments|no-threes]] approach. Harmonic 11 is particularly accurate, being only 0.03 cents sharp. A usable approximation of [[9/1|9]] is available at 6\37 (194.6 cents) as well, and the no-3 no-15 no-21 [[23-odd-limit]] is represented [[consistent]]ly.  
17\37 = 551.4 cents
26\37 = 843.2 cents


However, the just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo:
This means 37edo is useful in a number of ways. It is accurate on the 2.5.7.11.13 [[subgroup]], where it shares the same tuning as [[111edo]]. In fact, on the larger [[k*N subgroups|3*37 subgroup]], 2.27.5.7.11.13.51.57, it not only shares the same tuning as 19-limit 111edo, but tempers out the same commas. A simpler but less accurate approach is to use the 2*37-subgroup, 2.9.7.11.13.17.19, on which it has the same tuning and commas as [[74edo]]. The native [[3/2|perfect fifth]] at 22\37 (713.5 cents) can also be used, making it a sharp-tending full [[13-limit]] system, and there is the alternative, very flat fifth at 21\37 (681.1 cents), which generates an [[2L 5s|antidiatonic]] scale.


21\37 = 681.1 cents
=== Odd harmonics ===
22\37 = 713.5 cents
{{Harmonics in equal|37}}


37edo thus has the distinction of being the first [[edo]] which occupies two spaces on the syntonic spectrum.
=== As a tuning of other temperaments ===
Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of [[porcupine]] temperament. It is the [[optimal patent val]] for [[Porcupine family #Porcupinefish|porcupinefish]], which is about as accurate as 13-limit porcupine extensions will be. Using its alternative flat fifth, it tempers out [[16875/16384]], making it a [[negri]] tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth ([[gorgo]]/[[laconic]]).
 
It is a good tuning of the 2.5.11.13 subgroup temperament [[barton]], especially if it is desirable to avoid approximating the perfect fifth.
 
37edo is also a very accurate equal tuning for [[undecimation]] temperament, which has a [[generator]] of about 519 cents; 2 generators lead to 29/16; 3 generators to 32/13; 6 generators to a 10 cent sharp 6/1; 8 generators to a very accurate 11/1 and 10 generators to 20/1. It has a [[7L 2s]] enneatonic [[mos]], which in 37edo scale degrees is 0, 1, 6, 11, 16, 17, 22, 27, 32, a scale structure reminiscent of mavila; as well as a 16-note mos.
 
=== Subsets and supersets ===
37edo is the 12th [[prime edo]], following [[31edo]] and coming before [[41edo]].
 
[[74edo]], which doubles it, provides an alternative approximation to harmonic 3 that supports [[meantone]]. [[111edo]], which triples it, gives a very accurate approximation of harmonic 3, and manifests itself as a great higher-limit system. [[296edo]], which slices its step in eight, is a good 13-limit system.
 
=== Dual fifths ===
The just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo:
 
The flat fifth is 21\37 = 681.1 cents (37b val)
 
The sharp fifth is 22\37 = 713.5 cents


21\37 generates an anti-diatonic, or mavila, scale: 5 5 6 5 5 5 6
21\37 generates an anti-diatonic, or mavila, scale: 5 5 6 5 5 5 6
"minor third" = 10\37 = 324.3 cents
"minor third" = 10\37 = 324.3 cents
"major third" = 11\37 = 356.8 cents
"major third" = 11\37 = 356.8 cents


22\37 generates an extreme superpythagorean scale: 7 7 1 7 7 7 1
22\37 generates an extreme superpythagorean scale: 7 7 1 7 7 7 1
"minor third" = 8\37 = 259.5 cents
"minor third" = 8\37 = 259.5 cents
"major third" = 14\37 = 454.1 cents
"major third" = 14\37 = 454.1 cents


37edo has great potential as a xenharmonic system, which high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions.</pre></div>
If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of [[The Biosphere|Oceanfront]] temperament.
<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;37edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;37edo is the scale derived from dividing the octave into 37 microtonal steps of approximately 32.43 cents each. It offers close approximations to &lt;a class="wiki_link" href="/OverToneSeries"&gt;harmonics&lt;/a&gt; 5, 7, 11, and 13:&lt;br /&gt;
37edo can only barely be considered as "dual-fifth", because the sharp fifth is 12 cents sharp of 3/2, has a regular diatonic scale, and can be interpreted as somewhat accurate regular temperaments like [[archy]] and the aforementioned oceanfront. In contrast, the flat fifth is 21 cents flat and the only low-limit interpretation is as the very inaccurate [[mavila]].
&lt;br /&gt;
 
12\37 = 389.2 cents&lt;br /&gt;
Since both fifths do not support [[meantone]], the "major thirds" of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in 37edo.
30\37 = 973.0 cents&lt;br /&gt;
 
17\37 = 551.4 cents&lt;br /&gt;
37edo has great potential as a near-just xenharmonic system, with high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions. The 9/8 approximation is usable but introduces error. One may choose to treat either of the intervals close to 3/2 as 3/2, introducing additional approximations with considerable error (see interval table below).
26\37 = 843.2 cents&lt;br /&gt;
 
&lt;br /&gt;
=== Miscellaneous properties ===
However, the just &lt;a class="wiki_link" href="/perfect%20fifth"&gt;perfect fifth&lt;/a&gt; of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo:&lt;br /&gt;
37edo has the sharpest fifth of any edo that can possibly be [[diamond monotone]] in the [[15-odd-limit]]. The sharpest mapping of [[7/4]] where [[9/8]] is mapped no wider than [[8/7]] is 30\37, and the sharpest possible mapping of [[15/8]] where diamond monotone is achieveable is 34\37, where [[15/14]] is equated with [[14/13]][[~]][[13/12]] to half of [[7/6]]. Here [[5/4]] is mapped to 12\37, and [[10/9]] is mapped to 5\37. Equating both [[11/10]] and [[12/11]] with 10/9 makes the mappings for 9/8, 10/9, 11/10, and 12/11 add up to [[3/2]]. If the fifth was any sharper, then [[7/4]] and [[15/8]] would have to be flatter. Then 5/4 would have to be flatter, and therefore 10/9 as well, and at least one of 11/10 and 12/11 would have to be mapped wider than 10/9 for 9/8, 10/9, 11/10, and 12/11 to add up to 3/2. 37edo is, in fact, diamond monotone in the 15-odd-limit (see [[Monotonicity limits of small EDOs]]). Therefore, 22\37 is the sharpest fifth where 15-odd-limit diamond monotone is possible. The flattest fifth where 15-odd-limit diamond monotone is possible is [[19edo#Miscellaneous properties|11\19]].
&lt;br /&gt;
 
21\37 = 681.1 cents&lt;br /&gt;
== Intervals ==
22\37 = 713.5 cents&lt;br /&gt;
Inconsistent intervals are in ''italics''.
&lt;br /&gt;
{| class="wikitable center-1 right-2"
37edo thus has the distinction of being the first &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt; which occupies two spaces on the syntonic spectrum.&lt;br /&gt;
|-
&lt;br /&gt;
! #
21\37 generates an anti-diatonic, or mavila, scale: 5 5 6 5 5 5 6&lt;br /&gt;
! Cents
&amp;quot;minor third&amp;quot; = 10\37 = 324.3 cents&lt;br /&gt;
! Approximate ratios<br>of 2.27.5.7.11.13 subgroup
&amp;quot;major third&amp;quot; = 11\37 = 356.8 cents&lt;br /&gt;
! Additional ratios of 3<br>with a sharp 3/2
&lt;br /&gt;
! Additional ratios of 3<br>with a flat 3/2
22\37 generates an extreme superpythagorean scale: 7 7 1 7 7 7 1&lt;br /&gt;
! Additional ratios of 9<br>with 194.59 ¢ 9/8
&amp;quot;minor third&amp;quot; = 8\37 = 259.5 cents&lt;br /&gt;
|-
&amp;quot;major third&amp;quot; = 14\37 = 454.1 cents&lt;br /&gt;
| 0
&lt;br /&gt;
| 0.0
37edo has great potential as a xenharmonic system, which high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions.&lt;/body&gt;&lt;/html&gt;</pre></div>
| 1/1
|
|
|
|-
| 1
| 32.4
| [[55/54]], [[56/55]]
|
|
|
|-
| 2
| 64.9
| [[27/26]], [[28/27]]
|
|
|
|-
| 3
| 97.3
| [[128/121]], [[55/52]]
| [[16/15]]
|
|
|-
| 4
| 129.7
| [[14/13]]
| [[13/12]], [[15/14]]
| ''[[12/11]]''
|
|-
| 5
| 162.2
| [[11/10]]
| ''[[10/9]]'', [[12/11]]
| ''[[13/12]]''
|
|-
| 6
| 194.6
| [[28/25]]
|
|
| [[9/8]], [[10/9]]
|-
| 7
| 227.0
| [[8/7]]
| ''[[9/8]]''
|
|
|-
| 8
| 259.5
|
| [[7/6]], [[15/13]]
|
|
|-
| 9
| 291.9
| [[13/11]], [[32/27]]
|
| ''[[6/5]]'', ''[[7/6]]''
|
|-
| 10
| 324.3
|
| [[6/5]], ''[[11/9]]''
|
|
|-
| 11
| 356.8
| [[16/13]], [[27/22]]
|
|
| [[11/9]]
|-
| 12
| 389.2
| [[5/4]]
|
|
|
|-
| 13
| 421.6
| [[14/11]], [[32/25]]
|
|
| [[9/7]]
|-
| 14
| 454.1
| [[13/10]]
| ''[[9/7]]''
|
|
|-
| 15
| 486.5
|
| [[4/3]]
|
|
|-
| 16
| 518.9
| [[27/20]]
|
| ''[[4/3]]''
|
|-
| 17
| 551.4
| [[11/8]]
| [[15/11]]
|
| [[18/13]]
|-
| 18
| 583.8
| [[7/5]]
| ''[[18/13]]''
|
|
|-
| 19
| 616.2
| [[10/7]]
| ''[[13/9]]''
|
|
|-
| 20
| 648.6
| [[16/11]]
| [[22/15]]
|
| [[13/9]]
|-
| 21
| 681.1
| [[40/27]]
|
| ''[[3/2]]''
|
|-
| 22
| 713.5
|
| [[3/2]]
|
|
|-
| 23
| 745.9
| [[20/13]]
| ''[[14/9]]''
|
|
|-
| 24
| 778.4
| [[11/7]], [[25/16]]
|
|
| [[14/9]]
|-
| 25
| 810.8
| [[8/5]]
|
|
|
|-
| 26
| 843.2
| [[13/8]], [[44/27]]
|
|
| [[18/11]]
|-
| 27
| 875.7
|
| [[5/3]], ''[[18/11]]''
|
|
|-
| 28
| 908.1
| [[22/13]], [[27/16]]
|
| ''[[5/3]], [[12/7]]''
|
|-
| 29
| 940.5
|
| [[12/7]], [[26/15]]
|
|
|-
| 30
| 973.0
| [[7/4]]
| ''[[16/9]]''
|
|
|-
| 31
| 1005.4
| [[25/14]]
|
|
| [[16/9]], [[9/5]]
|-
| 32
| 1037.8
| [[20/11]]
| ''[[9/5]]'', [[11/6]]
|
|
|-
| 33
| 1070.3
| [[13/7]]
| [[24/13]], [[28/15]]
| ''[[11/6]]''
|
|-
| 34
| 1102.7
| [[121/64]], [[104/55]]
| [[15/8]]
|
|
|-
| 35
| 1135.1
| [[27/14]], [[52/27]]
|
|
|
|-
| 36
| 1167.6
|
|
|
|
|-
| 37
| 1200.0
| [[2/1]]
|
|
|
|}
 
=== Proposed interval names and solfèges ===
{| class="wikitable center-all right-2 left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Table of proposed interval names and solfèges
|-
! #
! Cents
! colspan="3" | [[Kite's ups and downs notation|Ups and downs notation]]
|-
| 0
| 0.0
| Perfect 1sn
| P1
| D
|-
| 1
| 32.4
| Minor 2nd
| m2
| Eb
|-
| 2
| 64.9
| Upminor 2nd
| ^m2
| ^Eb
|-
| 3
| 97.3
| Downmid 2nd
| v~2
| ^^Eb
|-
| 4
| 129.7
| Mid 2nd
| ~2
| Ed
|-
| 5
| 162.2
| Upmid 2nd
| ^~2
| vvE
|-
| 6
| 194.6
| Downmajor 2nd
| vM2
| vE
|-
| 7
| 227.0
| Major 2nd
| M2
| E
|-
| 8
| 259.5
| Minor 3rd
| m3
| F
|-
| 9
| 291.9
| Upminor 3rd
| ^m3
| ^F
|-
| 10
| 324.3
| Downmid 3rd
| v~3
| ^^F
|-
| 11
| 356.8
| Mid 3rd
| ~3
| Ft
|-
| 12
| 389.2
| Upmid 3rd
| ^~3
| vvF#
|-
| 13
| 421.6
| Downmajor 3rd
| vM3
| vF#
|-
| 14
| 454.1
| Major 3rd
| M3
| F#
|-
| 15
| 486.5
| Perfect 4th
| P4
| G
|-
| 16
| 518.9
| Up 4th, dim 5th
| ^4, d5
| ^G, Ab
|-
| 17
| 551.4
| Downmid 4th, updim 5th
| v~4, ^d5
| ^^G, ^Ab
|-
| 18
| 583.8
| Mid 4th, downmid 5th
| ~4, v~5
| Gt, ^^Ab
|-
| 19
| 616.2
| Mid 5th, upmid 4th
| ~5, ^~4
| Ad, vvG#
|-
| 20
| 648.6
| Upmid 5th, downaug 5th
| ^~5, vA4
| vvA, vG#
|-
| 21
| 681.1
| Down 5th, aug 4th
| v5, A4
| vA, G#
|-
| 22
| 713.5
| Perfect 5th
| P5
| A
|-
| 23
| 745.9
| Minor 6th
| m6
| Bb
|-
| 24
| 778.4
| Upminor 6th
| ^m6
| ^Bb
|-
| 25
| 810.8
| Downmid 6th
| v~6
| ^^Bb
|-
| 26
| 843.2
| Mid 6th
| ~6
| Bd
|-
| 27
| 875.7
| Upmid 6th
| ^~6
| vvB
|-
| 28
| 908.1
| Downmajor 6th
| vM6
| vB
|-
| 29
| 940.5
| Major 6th
| M6
| B
|-
| 30
| 973.0
| Minor 7th
| m7
| C
|-
| 31
| 1005.4
| Upminor 7th
| ^m7
| ^C
|-
| 32
| 1037.8
| Downmid 7th
| v~7
| ^^C
|-
| 33
| 1070.3
| Mid 7th
| ~7
| Ct
|-
| 34
| 1102.7
| Upmid 7th
| ^~7
| vvC#
|-
| 35
| 1135.1
| Downmajor 7th
| vM7
| vC#
|-
| 36
| 1167.6
| Major 7th
| M7
| C#
|-
| 37
| 1200.0
| Perfect 8ve
| P8
| D
|}
 
== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}
 
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt-szg}}
 
=== Kite's ups and downs notation ===
37edo can also be notated using [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
{{Sharpness-sharp6a}}
 
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
 
=== Ivan Wyschnegradsky's notation ===
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
{{Sharpness-sharp6-iw}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as edos [[23edo #Second-best fifth notation|23b]], [[30edo #Sagittal notation|30]], and [[44edo #Sagittal notation|44]].
 
==== Evo and Revo flavors ====
<imagemap>
File:37-EDO_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 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:37-EDO_Sagittal.svg]]
</imagemap>
 
==== Alternative Evo flavor ====
<imagemap>
File:37-EDO_Alternative_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 639 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:37-EDO_Alternative_Evo_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:37-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 623 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:37-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals|37}}
 
== 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]] (%)
|-
| 2.5
| {{Monzo| 86 -37 }}
| {{Mapping| 37 86 }}
| −0.619
| 0.619
| 1.91
|-
| 2.5.7
| 3136/3125, 4194304/4117715
| {{Mapping| 37 86 104 }}
| −0.905
| 0.647
| 2.00
|-
| 2.5.7.11
| 176/175, 1375/1372, 65536/65219
| {{Mapping| 37 86 104 128 }}
| −0.681
| 0.681
| 2.10
|-
| 2.5.7.11.13
| 176/175, 640/637, 847/845, 1375/1372
| {{Mapping| 37 86 104 128 137 }}
| −0.692
| 0.610
| 1.88
|}
* 37et is most prominent in the no-3 11-, 13-, 17-, 19- and 23-limit subgroups. The next equal temperaments doing better in these subgroups are 109, 581, 103, 124 and 93, respectively.
 
=== Rank-2 temperaments ===
* [[List of 37et rank two temperaments by badness]]
 
{| class="wikitable center-1 center-2"
|-
! Generator*
! Cents*
! In patent val
! In 37b val
|-
| 1\37
| 32.4
|
|
|-
| 2\37
| 64.9
| [[Sycamore]]
|
|-
| 3\37
| 97.3
| [[Passion]]
|
|-
| 4\37
| 129.7
| [[Twothirdtonic]]
| [[Negri]] (37bd, out-of-tune)
|-
| 5\37
| 162.2
| [[Porcupine]] / [[porcupinefish]]
|
|-
| 6\37
| 194.6
| [[Hemiwürschmidt]] / [[hemiwur]]
| [[Hemithirds]] (37b, out-of-tune)
|-
| 7\37
| 227.0
| [[Semaja]] / [[gorgik]]
| [[Gorgo]] (37b)
|-
| 8\37
| 259.5
|
| [[Semaphore]] (37bd, out-of-tune)
|-
| 9\37
| 291.9
| [[Quasitemp]]
|
|-
| 10\37
| 324.3
| [[Hyperkleismic]]
| [[Superkleismic]] (37bc, out-of-tune)
|-
| 11\37
| 356.8
| [[Beatles]]
|
|-
| 12\37
| 389.2
| [[Würschmidt]] (out-of-tune)
|
|-
| 13\37
| 421.6
| [[Skwares]] (37dd, out-of-tune)
|
|-
| 14\37
| 454.1
| [[Ammonite]]
|
|-
| 15\37
| 486.5
| [[Ultrapyth]]
|
|-
| 16\37
| 518.9
| [[Undecimation]]
| [[Shallowtone]] (37b)
|-
| 17\37
| 551.4
| [[Freivald]], [[emka]]
|
|-
| 18\37
| 583.8
| [[Cotritone]]
|
|}
<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
 
== Octave stretch or compression ==
37edo's [[prime]]s 3, 5, 7, 11 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]]. Some compressed-octave 37edo tunings (least to most compressed) include [[161zpi]], [[ed5|86ed5]], [[ed7|104ed7]], [[ed12|133ed12]] or [[ed6|96ed6]].
 
== Scales ==
''See also: [[MOS Scales of 37edo]], [[Chromatic pairs#Roulette|Roulette scales]]''
 
=== [[MOS scale]]s ===
* [[Ammonite]][21]: 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1
* [[Beatles]][7]: 4 7 4 7 4 7 4
* Beatles[10]: 4 3 4 4 3 4 4 4 3 4
* Beatles[17]: 3 1 3 1 3 3 1 3 1 3 1 3 3 1 3 1 3
* Ultrapyth[5] (quasi-[[equipentatonic]]): 7 8 7 8 7 (''recommended mode: 8 7 7 8 7'')
* Ultrapyth[7]: 7 1 7 7 7 1 7
* Ultrapyth[12]: 1 6 1 6 1 6 1 1 6 1 6 1
* Ultrapyth[17]: 1 5 1 1 1 5 1 1 5 1 1 5 1 1 1 5 1 (''great as a [[dual-fifth]] scale'')
* Ultrapyth[22]: 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1 1 4 1 1 (''great as a [[dual-fifth]] scale'')
* Passion[9]: 13 3 3 3 3 3 3 3 3
* Passion[12]: 3 3 3 3 3 3 4 3 3 3 3 3
* Passion[25]: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 (''great as a [[dual-fifth]] scale'')
* Porcupine[5]: 5 17 5 5 5
* Porcupine[6]: 12 5 5 5 5 5
* Porcupine[7]: 5 5 5 7 5 5 5
* Porcupine[15]: 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2
* Porcupine[22]: 2 1 2 2 1 2 2 1 2 2 1 2 2 2 1 2 2 1 2 2 1 2
* Twothirdtonic[7]: 13 4 4 4 4 4 4
* Twothirdtonic[8]: 9 4 4 4 4 4 4 4
* Twothirdtonic[10]: 4 4 4 4 1 4 4 4 4 4
* Twothirdtonic[19]: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1
 
=== Scales by individuals ===
{| class="wikitable mw-collapsible mw-collapsed"
|+[[Budjarn Lambeth]]'s scales
|'''Contains [[Template:Idiosyncratic|idiosyncratic terms]].'''
 
 
* Opalised ammonite{{idio}} (modmos of Ammonite[8]): 5 4 6 5 2 5 4 6
* [[User:BudjarnLambeth/Antechinus|Antechinus]]{{idio}} (''nonoctave period'')
* [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]]{{idio}} (''octave-reduced ver.: 5 3 13 9 7'')
* Approximated [[pelog]] lima: 4 5 12 4 12
 
 
* Flattened ionian pentatonic: 12 3 6 12 4
* Flattened major: 6 6 3 6 6 6 4
* Flattened major pentatonic: 6 6 9 6 10
* Sharpened natural minor: 7 3 6 6 3 6 6
* Sharpened harmonic minor: 7 3 6 6 3 9 3
* Sharpened pentatonic minor: 10 6 6 9 6
* Superharmonic minor pentatatonic I: 7 3 12 13 2
* Superharmonic minor pentatatonic II: 10 6 6 13 2
* Flattened hexatonic minor: 6 3 6 6 9 7
* Flattened phrygian dominant: 2 9 4 6 3 6 7
* Sharpened blues aeolian hexatonic: 10 6 3 3 3 12
* Flattened blues aeolian pentatonic: 9 6 6 3 13
* Sharpened blues aeolian pentatonic: 10 12 3 6 6
* Sharpened blues dorian hexatonic: 10 6 6 6 3 6
* Extrasharp blues dorian hexatonic: 10 6 6 6 4 5
* Roughened augmented: 10 2 10 2 11 2
* Flattened cosmic: 15 6 3 6 7 (''approximated from [[32afdo]]'')
* Sharpened Hirajoshi: 7 3 12 3 12
* Sharpened Akebono I: 7 3 12 6 9
* Roughened Javanese pentachordal: 2 8 9 2 16
* Sharpened underpass: 10 12 7 2 6 (''approximated from [[10afdo]]'')
 
 
* ''The scales listed in: [[User:BudjarnLambeth/Quasipelog theory]]''
* ''The scales listed in: [[Oceanfront scales]]'' (not all Budjarn's)
|}
 
=== Equally spaced scales ===
* [[37ed4]] (''every 2 steps''): 2 2 2...
* [[Square root of 13 over 10]] (''every 7 steps''): 7 7 7...
* ''Every 8 steps (see below)''
 
=== Every 8 steps of 37edo ===
{| class="wikitable center-1 right-2"
|+
!Degrees
!Cents
!Approximate Ratios<br>of 6.7.11.20.27 subgroup
!Additional Ratios
|-
|0
|0.000
|[[1/1]]
|
|-
|1
|259.46
|[[7/6]]
|
|-
|2
|518.92
|[[27/20]]
|
|-
|3
|778.38
|[[11/7]]
|
|-
|4
|1037.84
|[[20/11]], [[11/6]]
|
|-
|5
|1297.30
|
|[[19/9]]
|-
|6
|1556.76
|[[27/11]]
|
|-
|7
|1816.22
|[[20/7]]
|
|-
|8
|2075.68
|[[10/3]]
|
|-
|9
|2335.14
|[[27/7]]
|
|-
|10
|2594.59
|[[9/2]]
|
|-
|11
|2854.05
|
|[[26/5]]
|-
|12
|3113.51
|[[6/1]]
|
|-
|13
|3372.97
|[[7/1]]
|
|-
|14
|3632.43
|
|
|-
|15
|3891.89
|
|[[19/2]]
|-
|16
|4151.35
|[[11/1]]
|
|-
|17
|4410.81
|
|
|-
|18
|4670.27
|
|
|-
|19
|4929.73
|
|
|-
|20
|5189.19
|[[20/1]]
|
|-
|21
|5448.65
|
|
|-
|22
|5708.11
|[[27/1]]
|
|}
 
== Instruments ==
 
; Lumatone
* [[Lumatone mapping for 37edo]]
 
; Fretted instruments
* [[Skip fretting system 37 2 7]]
 
== Music ==
=== Modern renderings ===
; {{W|Alessandro Marcello}} and {{w|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=HTAobydvC20 ''Oboe Concerto in D minor'', BWV 974] (1715) – arranged for oboe & organ by [[Claudi Meneghin]] (2022)
 
; {{W|Pietro Domenico Paradies}}
* [https://www.youtube.com/watch?v=AJ2sa-fRqbE "Toccata" from ''Harpsichord Sonata in A major''] – arranged for organ by Claudi Meneghin (2023)
 
=== 21st century ===
; [[Beheld]]
* [https://www.youtube.com/watch?v=IULi2zSdatA ''Mindless vibe''] (2023)
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/TEzitpGJvt0 ''37edo''] (2023)
* [https://www.youtube.com/shorts/e7dLJTsS3PQ ''37edo''] (2025)
* [https://www.youtube.com/shorts/m9hmiH8zong ''37edo jam''] (2025)
* [https://www.youtube.com/shorts/mVRbcB2hoBU ''37edo prelude''] (2026)
* [https://www.youtube.com/shorts/Jt6_r6r3lGY ''37edo improv''] (2026)
 
; [[Francium]]
* [https://www.youtube.com/watch?v=jpPjVouoq3E ''5 days in''] (2023)
* [https://www.youtube.com/watch?v=ngxSiuVadls ''A Dark Era Arises''] (2023) – in Porcupine[15], 37edo tuning
* [https://www.youtube.com/watch?v=U93XFJJ1aXw ''Two Faced People''] (2025) – in Twothirdtonic[10], 37edo tuning
 
; [[groundfault]]
* From ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/album/souvenirs-of-the-affliction Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0 YouTube]
** "The Life Unreachable"
** "Not This Time"
 
; [[Andrew Heathwaite]]
* From [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)
** "Shorn Brown"
** "Jellybear"
 
; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3 ''Toccata Bianca 37EDO'']{{dead link}}
 
; [[JUMBLE]]
* [https://www.youtube.com/watch?v=taT1DClJ2KM ''Tyrian and Gold''] (2024)
 
; [[Fitzgerald Lee]]
* [https://www.youtube.com/watch?v=Nr0cUJcL4SU ''Bittersweet End''] (2025)
 
; [[Mandrake]]
* [https://www.youtube.com/watch?v=iL_4nRZBJDc ''What if?''] (2023)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=7dU8eyGbt9I ''Deck The Halls''] (2022)
* [https://www.youtube.com/watch?v=hpjZZXFM_Fk ''Little Fugue on Happy Birthday''] (2022) – in Passion, 37edo tuning
* [https://www.youtube.com/watch?v=SgHY3snZ5bs ''Fugue on an Original Theme''] (2022)
 
; [[Micronaive]]
* [https://www.youtube.com/watch?v=TMVRYLvg_cA No.27.50] (2022)
 
; [[Herman Miller]]
* [https://soundcloud.com/morphosyntax-1/luck-of-the-draw ''Luck of the Draw''] (2023)
 
; [[Joseph Monzo]]
* [https://youtube.com/watch?v=QERRKsbbWUQ ''The Kog Sisters''] (2014)
* [https://www.youtube.com/watch?v=BfP8Ig94kE0 ''Afrikan Song''] (2016)
 
; [[Mundoworld]]
* ''Reckless Discredit'' (2021) – [https://www.youtube.com/watch?v=ovgsjSoHOkg YouTube] · [https://mundoworld.bandcamp.com/track/reckless-discredit Bandcamp]
 
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=8reCr2nDGbw ''Porcupine Lullaby''] (2020) – in Porcupine, 37edo tuning
* [https://www.youtube.com/watch?v=j8C9ECvfyQM ''Fugue for Brass in 37EDO sssLsss "Dingoian"''] (2022) – in Porcupine[7], 37edo tuning
* [https://www.youtube.com/watch?v=_xfvNKUu8gY ''Fugue for Klezmer Band in 37EDO Porcupine<nowiki>[</nowiki>7<nowiki>]</nowiki> sssssLs "Lemurian"''] (2023) – in Porcupine[7], 37edo tuning
 
; [[Phanomium]]
* [https://www.youtube.com/watch?v=2otxZqUrvHc ''Elevated Floors''] (2025)
* [https://www.youtube.com/watch?v=BbexOU-9700 ''cat jam 37''] (2025)
 
; [[Togenom]]
* "Canals of Mars" from ''Xenharmonics, Vol. 5'' (2024) – [https://open.spotify.com/track/7v2dpCjiRKUfVVBZw8aWSf Spotify] |[https://togenom.bandcamp.com/track/canals-of-mars Bandcamp] | [https://www.youtube.com/watch?v=qPcEl_bifC0 YouTube]
 
; [[Uncreative Name]]
* [https://www.youtube.com/watch?v=rE9L56yZ1Kw ''Winter''] (2025)
 
; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=71yBnSVBsJk ''Leap Day Cloo''] (2025)
 
; [[Xeno*n*]]
* [https://www.youtube.com/watch?v=_m5u4VviMXw ''Galantean Drift''] (2025)
 
== See also ==
* [[User:Unque/37edo Composition Theory|Unque's approach]]
 
== External links ==
* [http://tonalsoft.com/enc/number/37-edo/37edo.aspx 37-edo / 37-et / 37-tone equal-temperament] on [[Tonalsoft Encyclopedia]]
 
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/37edo"