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<span style="display: block; text-align: right;">[[:de:37edo|Deutsch]]</span>
{{interwiki
 
| de = 37-EDO
'''37edo''' is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. It is the 12th [[prime_numbers|prime]] edo, following [[31edo]] and coming before [[41edo]].
| en = 37edo
| es =
| ja =
}}
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
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 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.  
 
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.


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


12\37 = 389.2 cents
=== Odd harmonics ===
{{Harmonics in equal|37}}


30\37 = 973.0 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]]).


17\37 = 551.4 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.


26\37 = 843.2 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.


[6\37edo = 194.6 cents]
=== Subsets and supersets ===
37edo is the 12th [[prime edo]], following [[31edo]] and coming before [[41edo]].  


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.
[[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.


=== The Two Fifths ===
=== Dual 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 42: 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.
 
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]].


Interestingly, the "major thirds" of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in 37edo.
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 67: Line 77:
|-
|-
| 1
| 1
| 32.43
| 32.4
|
| [[55/54]], [[56/55]]
|
|
|
|
Line 74: Line 84:
|-
|-
| 2
| 2
| 64.86
| 64.9
| 28/27, 27/26
| [[27/26]], [[28/27]]
|
|
|
|
Line 81: 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 151: 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 242: 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 312: Line 322:
|-
|-
| 36
| 36
| 1167.57
| 1167.6
|
|
|
|
Line 319: Line 329:
|-
|-
| 37
| 37
| 1200.00
| 1200.0
| 2/1
| [[2/1]]
|
|
|
|
Line 326: Line 336:
|}
|}


== Just approximation ==
=== Proposed interval names and solfèges ===
 
{| class="wikitable center-all right-2 left-3 mw-collapsible mw-collapsed"
=== Selected just intervals ===
|+ style="white-space: nowrap;" | Table of proposed interval names and solfèges
{| class="wikitable center-all"
|-
! colspan="2" |
! #
! prime 2
! Cents
! prime 3
! colspan="3" | [[Kite's ups and downs notation|Ups and downs notation]]
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
! prime 23
|-
|-
! rowspan="2" |Error
| 0
! absolute (¢)
| 0.0
| 0.0
| +11.56
| Perfect 1sn
| +2.88
| P1
| +4.15
| D
| +0.03
| +2.72
| -7.66
| -5.62
| -12.06
|-
|-
! [[Relative error|relative]] (%)
| 1
| 0.0
| 32.4
| +35.6
| Minor 2nd
| +8.9
| m2
| +12.8
| Eb
| +0.1
|-
| +8.4
| 2
| -23.6
| 64.9
| -17.3
| Upminor 2nd
| -37.2
| ^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
|}
|}


=== Temperament measures ===
== Notation ==
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 37et.  
=== Stein–Zimmermann–Gould notation ===
{| class="wikitable center-all"
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
! colspan="2" |
{{Sharpness-sharp6-szg}}
! 3-limit
 
! 5-limit
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
! 7-limit
{{Sharpness-sharp6-qt-szg}}
! 11-limit
 
! 13-limit
=== Kite's ups and downs notation ===
! no-3 11-limit
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.
! no-3 13-limit
{{Sharpness-sharp6a}}
! no-3 17-limit
 
! no-3 19-limit
Half-sharps and half-flats can be used to avoid triple arrows:
! no-3 23-limit
{{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
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.5.7
| -3.65
| 3136/3125, 4194304/4117715
| -2.85
| {{Mapping| 37 86 104 }}
| -2.50
| −0.905
| -2.00
| 0.647
| -1.79
| 2.00
| -0.681
| -0.692
| -0.265
| -0.0386
| +0.299
|-
|-
! rowspan="2" |Error
| 2.5.7.11
! [[TE error|absolute]] (¢)
| 176/175, 1375/1372, 65536/65219
| 3.64
| {{Mapping| 37 86 104 128 }}
| 3.18
| −0.681
| 2.82
| 2.71
| 2.52
| 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]]

Latest revision as of 14:18, 12 May 2026

← 36edo 37edo 38edo →
Prime factorization 37 (prime)
Step size 32.4324 ¢ 
Fifth 22\37 (713.514 ¢)
Semitones (A1:m2) 6:1 (194.6 ¢ : 32.43 ¢)
Dual sharp fifth 22\37 (713.514 ¢)
Dual flat fifth 21\37 (681.081 ¢)
Dual major 2nd 6\37 (194.595 ¢)
Consistency limit 7
Distinct consistency limit 7

37 equal divisions of the octave (abbreviated 37edo or 37ed2), also called 37-tone equal temperament (37tet) or 37 equal temperament (37et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 37 equal parts of about 32.4 ¢ each. Each step represents a frequency ratio of 21/37, or the 37th root of 2.

Theory

37edo has very accurate approximations of harmonics 5, 7, 11 and 13, making it a good choice for a no-threes approach. Harmonic 11 is particularly accurate, being only 0.03 cents sharp. A usable approximation of 9 is available at 6\37 (194.6 cents) as well, and the no-3 no-15 no-21 23-odd-limit is represented consistently.

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 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 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 antidiatonic scale.

Odd harmonics

Approximation of odd harmonics in 37edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +11.6 +2.9 +4.1 -9.3 +0.0 +2.7 +14.4 -7.7 -5.6 +15.7 -12.1
Relative (%) +35.6 +8.9 +12.8 -28.7 +0.1 +8.4 +44.5 -23.6 -17.3 +48.4 -37.2
Steps
(reduced) 		59
(22) 		86
(12) 		104
(30) 		117
(6) 		128
(17) 		137
(26) 		145
(34) 		151
(3) 		157
(9) 		163
(15) 		167
(19)

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 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

"minor third" = 10\37 = 324.3 cents

"major third" = 11\37 = 356.8 cents

22\37 generates an extreme superpythagorean scale: 7 7 1 7 7 7 1

"minor third" = 8\37 = 259.5 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 Oceanfront temperament.

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).

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 11\19.

Intervals

Inconsistent intervals are in italics.

# Cents Approximate ratios
of 2.27.5.7.11.13 subgroup 		Additional ratios of 3
with a sharp 3/2 		Additional ratios of 3
with a flat 3/2 		Additional ratios of 9
with 194.59 ¢ 9/8
0 0.0 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

Table of proposed interval names and solfèges
# Cents 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:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol
Flat symbol

If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
Flat symbol

Kite's ups and downs notation

37edo can also be notated using Kite's ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
         
             
Flat symbol          
             

Half-sharps and half-flats can be used to avoid triple arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
 
    
    
    
Flat symbol  
    
    
    

Ivan Wyschnegradsky's notation

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from 72edo can also be used:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 
  
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 

Sagittal notation

This notation uses the same sagittal sequence as edos 23b, 30, and 44.

Evo and Revo flavors

Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Alternative Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Approximation to JI

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 37edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 37edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/8, 16/11 0.033 0.1
13/10, 20/13 0.160 0.5
7/5, 10/7 1.272 3.9
13/7, 14/13 1.431 4.4
13/11, 22/13 2.682 8.3
13/8, 16/13 2.716 8.4
11/10, 20/11 2.842 8.8
5/4, 8/5 2.875 8.9
11/7, 14/11 4.114 12.7
7/4, 8/7 4.147 12.8
7/6, 12/7 7.411 22.9
5/3, 6/5 8.683 26.8
13/12, 24/13 8.843 27.3
9/8, 16/9 9.315 28.7
11/9, 18/11 9.349 28.8
15/14, 28/15 10.287 31.7
11/6, 12/11 11.525 35.5
3/2, 4/3 11.559 35.6
15/13, 26/15 11.718 36.1
13/9, 18/13 12.031 37.1
9/5, 10/9 12.191 37.6
9/7, 14/9 13.462 41.5
15/11, 22/15 14.401 44.4
15/8, 16/15 14.434 44.5
15-odd-limit intervals in 37edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/8, 16/11 0.033 0.1
13/10, 20/13 0.160 0.5
7/5, 10/7 1.272 3.9
13/7, 14/13 1.431 4.4
13/11, 22/13 2.682 8.3
13/8, 16/13 2.716 8.4
11/10, 20/11 2.842 8.8
5/4, 8/5 2.875 8.9
11/7, 14/11 4.114 12.7
7/4, 8/7 4.147 12.8
7/6, 12/7 7.411 22.9
5/3, 6/5 8.683 26.8
13/12, 24/13 8.843 27.3
15/14, 28/15 10.287 31.7
11/6, 12/11 11.525 35.5
3/2, 4/3 11.559 35.6
15/13, 26/15 11.718 36.1
15/11, 22/15 14.401 44.4
15/8, 16/15 14.434 44.5
9/7, 14/9 18.970 58.5
9/5, 10/9 20.242 62.4
13/9, 18/13 20.401 62.9
11/9, 18/11 23.084 71.2
9/8, 16/9 23.117 71.3

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.5 [86 -37⟩ [⟨37 86]] −0.619 0.619 1.91
2.5.7 3136/3125, 4194304/4117715 [⟨37 86 104]] −0.905 0.647 2.00
2.5.7.11 176/175, 1375/1372, 65536/65219 [⟨37 86 104 128]] −0.681 0.681 2.10
2.5.7.11.13 176/175, 640/637, 847/845, 1375/1372 [⟨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

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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Octave stretch or compression

37edo's primes 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, 86ed5, 104ed7, 133ed12 or 96ed6.

Scales

See also: MOS Scales of 37edo, Roulette scales

MOS scales

  • 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

Budjarn Lambeth's scales
Contains idiosyncratic terms.



  • 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)


Equally spaced scales

Every 8 steps of 37edo

Degrees Cents Approximate Ratios
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
Fretted instruments

Music

Modern renderings

Alessandro Marcello and Johann Sebastian Bach
Pietro Domenico Paradies

21st century

Beheld
Bryan Deister
Francium
groundfault
  • From Souvenirs of the Affliction (2025) – Bandcamp | YouTube
    • "The Life Unreachable"
    • "Not This Time"
Andrew Heathwaite
  • From Newbeams (2012)
    • "Shorn Brown"
    • "Jellybear"
Aaron Krister Johnson
JUMBLE
Fitzgerald Lee
Mandrake
Claudi Meneghin
Micronaive
Herman Miller
Joseph Monzo
Mundoworld
Ray Perlner
Phanomium
Togenom
Uncreative Name
Stephen Weigel
Xeno*n*

See also

External links

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