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<span style="display: block; text-align: right;">[[:de:37edo|Deutsch]]</span>
{{interwiki
{{Infobox ET
| de = 37-EDO
| Step size = 32.432¢
| en = 37edo
| Fifth type = 20\34 = 705.88¢ = [[17edo]]
| es =  
| Major 2nd = 7\37 = 227¢
| ja =  
| Minor 2nd = 1\37 = 32¢
| Augmented 1sn = 6\37 = 195¢
}}
}}
'''37edo''' is a scale derived from dividing the octave into 37 equal steps. It is the 12th [[prime_numbers|prime]] edo, following [[31edo]] and coming before [[41edo]].
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
{| class="wikitable center-all"
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.  
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
! prime 23
|-
! rowspan="2" |Error
! absolute (¢)
| 0
|  +11.56
|  +2.9
|  +4.1
|  +0.03
|  +2.7
| -7.7
|  -5.6
|  -12.1
|-
![[Relative error|relative]] (%)
| 0
|  +36
|  +9
|  +13
|  +0.1
| +8
|  -24
|  -17
|  -37
|-
! colspan="2" |[[nearest edomapping]]
|37
|22
|12
|30
|17
|26
|3
|9
|19
|}
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" will be. Using its alternative flat fifth, it tempers out 16875/16384, making it a [[Negri|negri]] tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth ([[gorgo]]/[[laconic]]).


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 nonatonic MOS, which in 37-edo 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.
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.  


=== Subgroups ===
=== Odd harmonics ===
37edo offers close approximations to [[Overtone series|harmonics]] 5, 7, 11, and 13 [and a usable approximation of 9 as well].
{{Harmonics in equal|37}}


12\37 = 389.2 cents
=== 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]]).


30\37 = 973.0 cents
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.


17\37 = 551.4 cents
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.


26\37 = 843.2 cents
=== Subsets and supersets ===
37edo is the 12th [[prime edo]], following [[31edo]] and coming before [[41edo]].  


[6\37edo = 194.6 cents]
[[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.


This means 37 is quite accurate on the 2.5.7.11.13 subgroup, where it shares the same tuning as 111et. In fact, on the larger [[k*N_subgroups|3*37 subgroup]] 2.27.5.7.11.13.51.57 subgroup not only shares the same tuning as 19-limit 111et, it 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 74et.
=== Dual fifths ===
 
=== The Two Fifths ===
The just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo:
The just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo:


Line 94: Line 47:
"major third" = 14\37 = 454.1 cents
"major third" = 14\37 = 454.1 cents


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|Biome]] temperament.
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.  


Interestingly, the "major thirds" of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in 37edo.
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]].
 
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.


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).
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).
=== Miscellaneous properties ===
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]].


== Intervals ==
== Intervals ==
 
Inconsistent intervals are in ''italics''.
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! Degrees
! #
! Cents
! Cents
! Approximate Ratios<br>of 2.5.7.11.13.27 subgroup
! Approximate ratios<br>of 2.27.5.7.11.13 subgroup
! Additional Ratios of 3<br>with a sharp 3/2
! Additional ratios of 3<br>with a sharp 3/2
! Additional Ratios of 3<br>with a flat 3/2
! Additional ratios of 3<br>with a flat 3/2
! Additional Ratios of 9<br>with 194.59¢ 9/8
! Additional ratios of 9<br>with 194.59 ¢ 9/8
|-
|-
| 0
| 0
| 0.00
| 0.0
| 1/1
| 1/1
|
|
Line 119: Line 77:
|-
|-
| 1
| 1
| 32.43
| 32.4
|
| [[55/54]], [[56/55]]
|
|
|
|
Line 126: Line 84:
|-
|-
| 2
| 2
| 64.86
| 64.9
| 28/27, 27/26
| [[27/26]], [[28/27]]
|
|
|
|
Line 133: Line 91:
|-
|-
| 3
| 3
| 97.30
| 97.3
|
| [[128/121]], [[55/52]]
|
| [[16/15]]
|
|
|
|
|-
|-
| 4
| 4
| 129.73
| 129.7
| 14/13
| [[14/13]]
| 13/12
| [[13/12]], [[15/14]]
| 12/11
| ''[[12/11]]''
|
|
|-
|-
| 5
| 5
| 162.16
| 162.2
| 11/10
| [[11/10]]
| 10/9, 12/11
| ''[[10/9]]'', [[12/11]]
| 13/12
| ''[[13/12]]''
|
|
|-
|-
| 6
| 6
| 194.59
| 194.6
| [[28/25]]
|
|
|
|
|
| [[9/8]], [[10/9]]
| 9/8, 10/9
|-
|-
| 7
| 7
| 227.03
| 227.0
| 8/7
| [[8/7]]
| 9/8
| ''[[9/8]]''
|
|
|
|
|-
|-
| 8
| 8
| 259.46
| 259.5
|
|
| 7/6
| [[7/6]], [[15/13]]
|
|
|
|
|-
|-
| 9
| 9
| 291.89
| 291.9
| 13/11, 32/27
| [[13/11]], [[32/27]]
|
|
| 6/5, 7/6
| ''[[6/5]]'', ''[[7/6]]''
|
|
|-
|-
| 10
| 10
| 324.32
| 324.3
|
|
| 6/5, 11/9
| [[6/5]], ''[[11/9]]''
|
|
|
|
|-
|-
| 11
| 11
| 356.76
| 356.8
| 16/13, 27/22
| [[16/13]], [[27/22]]
|
|
|
|
| 11/9
| [[11/9]]
|-
|-
| 12
| 12
| 389.19
| 389.2
| 5/4
| [[5/4]]
|
|
|
|
Line 203: Line 161:
|-
|-
| 13
| 13
| 421.62
| 421.6
| 14/11
| [[14/11]], [[32/25]]
|
|
|
|
| 9/7
| [[9/7]]
|-
|-
| 14
| 14
| 454.05
| 454.1
| 13/10
| [[13/10]]
| 9/7
| ''[[9/7]]''
|
|
|
|
|-
|-
| 15
| 15
| 486.49
| 486.5
|
|
| 4/3
| [[4/3]]
|
|
|
|
|-
|-
| 16
| 16
| 518.92
| 518.9
| 27/20
| [[27/20]]
|
|
| 4/3
| ''[[4/3]]''
|
|
|-
|-
| 17
| 17
| 551.35
| 551.4
| [[11/8]]
| [[11/8]]
| [[15/11]]
|
|
|
| [[18/13]]
| 18/13
|-
|-
| 18
| 18
| 583.78
| 583.8
| 7/5
| [[7/5]]
| 18/13
| ''[[18/13]]''
|
|
|
|
|-
|-
| 19
| 19
| 616.22
| 616.2
| 10/7
| [[10/7]]
| 13/9
| ''[[13/9]]''
|
|
|
|
|-
|-
| 20
| 20
| 648.65
| 648.6
| [[16/11]]
| [[16/11]]
| [[22/15]]
|
|
|
| [[13/9]]
| 13/9
|-
|-
| 21
| 21
| 681.08
| 681.1
| 40/27
| [[40/27]]
|
|
| 3/2
| ''[[3/2]]''
|
|
|-
|-
| 22
| 22
| 713.51
| 713.5
|
|
| 3/2
| [[3/2]]
|
|
|
|
|-
|-
| 23
| 23
| 745.95
| 745.9
| 20/13
| [[20/13]]
| 14/9
| ''[[14/9]]''
|
|
|
|
|-
|-
| 24
| 24
| 778.38
| 778.4
| 11/7
| [[11/7]], [[25/16]]
|
|
|
|
| 14/9
| [[14/9]]
|-
|-
| 25
| 25
| 810.81
| 810.8
| 8/5
| [[8/5]]
|
|
|
|
Line 294: Line 252:
|-
|-
| 26
| 26
| 843.24
| 843.2
| 13/8, 44/27
| [[13/8]], [[44/27]]
|
|
|
|
| 18/11
| [[18/11]]
|-
|-
| 27
| 27
| 875.68
| 875.7
|
|
| 5/3, 18/11
| [[5/3]], ''[[18/11]]''
|
|
|
|
|-
|-
| 28
| 28
| 908.11
| 908.1
| 22/13, 27/16
| [[22/13]], [[27/16]]
|
|
| 5/3, 12/7
| ''[[5/3]], [[12/7]]''
|
|
|-
|-
| 29
| 29
| 940.54
| 940.5
|
|
| 12/7
| [[12/7]], [[26/15]]
|
|
|
|
|-
|-
| 30
| 30
| 972.97
| 973.0
| 7/4
| [[7/4]]
| 16/9
| ''[[16/9]]''
|
|
|
|
|-
|-
| 31
| 31
| 1005.41
| 1005.4
|
| [[25/14]]
|
|
|
|
| 16/9, 9/5
| [[16/9]], [[9/5]]
|-
|-
| 32
| 32
| 1037.84
| 1037.8
| 11/6
| [[20/11]]
| 9/5, 11/6
| ''[[9/5]]'', [[11/6]]
|
|
|
|
|-
|-
| 33
| 33
| 1070.27
| 1070.3
| 13/7
| [[13/7]]
| 24/13
| [[24/13]], [[28/15]]
| 11/6
| ''[[11/6]]''
|
|
|-
|-
| 34
| 34
| 1102.70
| 1102.7
|
| [[121/64]], [[104/55]]
|
| [[15/8]]
|
|
|
|
|-
|-
| 35
| 35
| 1135.14
| 1135.1
| 27/14, 52/27
| [[27/14]], [[52/27]]
|
|
|
|
Line 364: Line 322:
|-
|-
| 36
| 36
| 1167.57
| 1167.6
|
|
|
|
Line 371: Line 329:
|-
|-
| 37
| 37
| 1200.00
| 1200.0
| 2/1
| [[2/1]]
|
|
|
|
Line 378: Line 336:
|}
|}


== Just approximation ==
=== 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]].


=== Temperament measures ===
==== Evo and Revo flavors ====
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 37et.
<imagemap>
{| class="wikitable center-all"
File:37-EDO_Sagittal.svg
! colspan="2" |
desc none
! 3-limit
rect 80 0 300 50 [[Sagittal_notation]]
! 5-limit
rect 300 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
! 7-limit
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
! 11-limit
default [[File:37-EDO_Sagittal.svg]]
! 13-limit
</imagemap>
! no-3 11-limit
 
! no-3 13-limit
==== Alternative Evo flavor ====
! no-3 17-limit
<imagemap>
! no-3 19-limit
File:37-EDO_Alternative_Evo_Sagittal.svg
! no-3 23-limit
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"
|-
|-
! colspan="2" |Octave stretch (¢)
! rowspan="2" | [[Subgroup]]
| -3.65
! rowspan="2" | [[Comma list]]
| -2.85
! rowspan="2" | [[Mapping]]
| -2.50
! rowspan="2" | Optimal<br>8ve stretch (¢)
| -2.00
! colspan="2" | Tuning error
| -1.79
| -0.681
| -0.692
| -0.265
| -0.0386
| +0.299
|-
|-
! rowspan="2" |Error
! [[TE error|Absolute]] (¢)
! [[TE error|absolute]] (¢)
! [[TE simple badness|Relative]] (%)
| 3.64
|-
| 3.18
| 2.5
| 2.82
| {{Monzo| 86 -37 }}
| 2.71
| {{Mapping| 37 86 }}
| 2.52
| −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
| 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
| 0.610
| 1.11
| 1.17
| 1.41
|-
! [[TE simple badness|relative]] (%)
| 11.24
| 9.82
| 8.70
| 8.37
| 7.78
| 2.10
| 1.88
| 1.88
| 3.41
| 3.59
| 4.35
|}
|}
* 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.


* 37et is most prominent in the no-3 11-, 13-, 17-, 19- and 23-limit subgroups. The next ET that does better in these subgroups is 109, 581, 103, 124 and 93, respectively.
=== Rank-2 temperaments ===
 
== Scales ==
 
* [[MOS_Scales_of_37edo|MOS Scales of 37edo]]
* [[roulette6]]
* [[roulette7]]
* [[roulette13]]
* [[roulette19]]
* [[Chromatic_pairs#Shoe|Shoe]]
* [[37ED4]]
* [[square_root_of_13_over_10|The Square Root of 13/10]]
 
== Linear temperaments ==
* [[List of 37et rank two temperaments by badness]]
* [[List of 37et rank two temperaments by badness]]


{| class="wikitable"
{| class="wikitable center-1 center-2"
|-
|-
! Generator
! Generator*
! "Sharp 3/2" temperaments
! Cents*
! "Flat 3/2" temperaments (37b val)
! In patent val
! In 37b val
|-
|-
| 1\37
| 1\37
| 32.4
|  
|  
|  
|  
|-
|-
| 2\37
| 2\37
| [[Sycamore_family|Sycamore]]
| 64.9
| [[Sycamore]]
|  
|  
|-
|-
| 3\37
| 3\37
| 97.3
| [[Passion]]
| [[Passion]]
|  
|  
|-
|-
| 4\37
| 4\37
| 129.7
| [[Twothirdtonic]]
| [[Twothirdtonic]]
| [[Negri]]
| [[Negri]] (37bd, out-of-tune)
|-
|-
| 5\37
| 5\37
| [[Porcupine]]/[[The_Biosphere#Oceanfront-Oceanfront Children-Porcupinefish|porcupinefish]]
| 162.2
| [[Porcupine]] / [[porcupinefish]]
|  
|  
|-
|-
| 6\37
| 6\37
| colspan="2" | [[Chromatic_pairs#Roulette|Roulette]]
| 194.6
| [[Hemiwürschmidt]] / [[hemiwur]]
| [[Hemithirds]] (37b, out-of-tune)
|-
|-
| 7\37
| 7\37
| [[Semaja]]
| 227.0
| [[Gorgo]]/[[Laconic]]
| [[Semaja]] / [[gorgik]]
| [[Gorgo]] (37b)
|-
|-
| 8\37
| 8\37
| 259.5
|  
|  
| [[Semiphore]]
| [[Semaphore]] (37bd, out-of-tune)
|-
|-
| 9\37
| 9\37
| 291.9
| [[Quasitemp]]
|  
|  
| [[Chromatic_pairs#Gariberttet|Gariberttet]]
|-
|-
| 10\37
| 10\37
|  
| 324.3
| [[Orgone]]
| [[Hyperkleismic]]
| [[Superkleismic]] (37bc, out-of-tune)
|-
|-
| 11\37
| 11\37
| 356.8
| [[Beatles]]
| [[Beatles]]
|  
|  
|-
|-
| 12\37
| 12\37
| 389.2
| [[Würschmidt]] (out-of-tune)
| [[Würschmidt]] (out-of-tune)
|  
|  
|-
|-
| 13\37
| 13\37
| [[Meantone_family#Squares|Squares]]
| 421.6
| [[Skwares]] (37dd, out-of-tune)
|  
|  
|-
|-
| 14\37
| 14\37
| 454.1
| [[Ammonite]]
| [[Ammonite]]
|  
|  
|-
|-
| 15\37
| 15\37
| [[The_Biosphere#Oceanfront-Oceanfront Children-Ultrapyth|Ultrapyth]], '''not''' [[superpyth]]
| 486.5
| [[Ultrapyth]]
|  
|  
|-
|-
| 16\37
| 16\37
|  
| 518.9
| '''Not''' [[mavila]] (this is "undecimation")
| [[Undecimation]]
| [[Shallowtone]] (37b)
|-
|-
| 17\37
| 17\37
| [[Hemimean_clan#Emka|Emka]]
| 551.4
| [[Freivald]], [[emka]]
|  
|  
|-
|-
| 18\37
| 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 ==
== Music ==
* [http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3 Toccata Bianca 37edo] by [http://www.akjmusic.com/ Aaron Krister Johnson]
=== Modern renderings ===
* [http://andrewheathwaite.bandcamp.com/track/shorn-brown Shorn Brown] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2002%20Shorn%20Brown.mp3 play] and [http://andrewheathwaite.bandcamp.com/track/jellybear Jellybear] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2003%20Jellybear.mp3 play] by [[Andrew Heathwaite]]
; {{W|Alessandro Marcello}} and {{w|Johann Sebastian Bach}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Monzo/monzo_kog-sisters_2014-0405.mp3 The Kog Sisters] by [[Joe Monzo]]
* [https://www.youtube.com/watch?v=HTAobydvC20 ''Oboe Concerto in D minor'', BWV 974] (1715) – arranged for oboe & organ by [[Claudi Meneghin]] (2022)
* [https://www.youtube.com/watch?v=8reCr2nDGbw Porcupine Lullaby] by [[Ray Perlner]]
 
; {{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]]


== Links ==
[[Category:Listen]]
* [http://tonalsoft.com/enc/number/37-edo/37edo.aspx 37edo at Tonalsoft]  [[Category:37edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Subgroup]]
Retrieved from "https://en.xen.wiki/w/37edo"