View source for 37edo

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| en = 37edo
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== Theory ==
37edo has very accurate approximations of harmonics [[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.

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]]).

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.

In the no-3 [[13-odd-limit]], 37edo maintains the smallest relative error of any edo until [[851edo]], and the smallest absolute error until [[103edo]]{{clarify}}. <!-- what is the metric being used? -->

=== Odd harmonics ===
{{Harmonics in equal|37}}

=== 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.

=== Subgroups ===
37edo offers close approximations to [[Harmonic series|harmonics]] 5, 7, 11, and 13, and a usable approximation of 9 as well.
* 12\37 = 389.2 cents
* 30\37 = 973.0 cents
* 17\37 = 551.4 cents
* 26\37 = 843.2 cents
* [6\37 = 194.6 cents]

This means 37 is quite 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 subgroup not only shares the same tuning as 19-limit 111edo, 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 [[74edo]].

=== 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 [[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]].

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).

=== No-3 approach ===
If prime 3 is ignored, 37edo represents the no-3 23-odd-limit consistently, and is distinctly consistent within the no-3 16-integer-limit.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! Degrees
! Cents
! Approximate Ratios<br>of 2.5.7.11.13.27 subgroup
! Additional Ratios of 3<br>with a sharp 3/2
! Additional Ratios of 3<br>with a flat 3/2
! Additional Ratios of 9<br>with 194.59¢ 9/8
|-
| 0
| 0.00
| 1/1
|
|
|
|-
| 1
| 32.43
| [[55/54]], [[56/55]]
|
|
|
|-
| 2
| 64.86
| [[27/26]], [[28/27]]
|
|
|
|-
| 3
| 97.30
| [[128/121]], [[55/52]]
| [[16/15]]
|
|
|-
| 4
| 129.73
| [[14/13]]
| [[13/12]], [[15/14]]
| [[12/11]]
|
|-
| 5
| 162.16
| [[11/10]]
| [[10/9]], [[12/11]]
| [[13/12]]
|
|-
| 6
| 194.59
| [[28/25]]
|
|
| [[9/8]], [[10/9]]
|-
| 7
| 227.03
| [[8/7]]
| [[9/8]]
|
|
|-
| 8
| 259.46
|
| [[7/6]], [[15/13]]
|
|
|-
| 9
| 291.89
| [[13/11]], [[32/27]]
|
| [[6/5]], [[7/6]]
|
|-
| 10
| 324.32
|
| [[6/5]], [[11/9]]
|
|
|-
| 11
| 356.76
| [[16/13]], [[27/22]]
|
|
| [[11/9]]
|-
| 12
| 389.19
| [[5/4]]
|
|
|
|-
| 13
| 421.62
| [[14/11]], [[32/25]]
|
|
| [[9/7]]
|-
| 14
| 454.05
| [[13/10]]
| [[9/7]]
|
|
|-
| 15
| 486.49
|
| [[4/3]]
|
|
|-
| 16
| 518.92
| [[27/20]]
|
| [[4/3]]
|
|-
| 17
| 551.35
| [[11/8]]
| [[15/11]]
|
| [[18/13]]
|-
| 18
| 583.78
| [[7/5]]
| [[18/13]]
|
|
|-
| 19
| 616.22
| [[10/7]]
| [[13/9]]
|
|
|-
| 20
| 648.65
| [[16/11]]
| [[22/15]]
|
| [[13/9]]
|-
| 21
| 681.08
| [[40/27]]
|
| [[3/2]]
|
|-
| 22
| 713.51
|
| [[3/2]]
|
|
|-
| 23
| 745.95
| [[20/13]]
| [[14/9]]
|
|
|-
| 24
| 778.38
| [[11/7]], [[25/16]]
|
|
| [[14/9]]
|-
| 25
| 810.81
| [[8/5]]
|
|
|
|-
| 26
| 843.24
| [[13/8]], [[44/27]]
|
|
| [[18/11]]
|-
| 27
| 875.68
|
| [[5/3]], [[18/11]]
|
|
|-
| 28
| 908.11
| [[22/13]], [[27/16]]
|
| [[5/3]], [[12/7]]
|
|-
| 29
| 940.54
|
| [[12/7]], [[26/15]]
|
|
|-
| 30
| 972.97
| [[7/4]]
| [[16/9]]
|
|
|-
| 31
| 1005.41
| [[25/14]]
|
|
| [[16/9]], [[9/5]]
|-
| 32
| 1037.84
| [[20/11]]
| [[9/5]], [[11/6]]
|
|
|-
| 33
| 1070.27
| [[13/7]]
| [[24/13]], [[28/15]]
| [[11/6]]
|
|-
| 34
| 1102.70
| [[121/64]], [[104/55]]
| [[15/8]]
|
|
|-
| 35
| 1135.14
| [[27/14]], [[52/27]]
|
|
|
|-
| 36
| 1167.57
|
|
|
|
|-
| 37
| 1200.00
| [[2/1]]
|
|
|
|}

== Notation ==
=== Ups and downs notation ===
37edo can be notated using [[ups and downs notation]]:

{| class="wikitable center-all right-2 left-3"
|-
! Degrees
! Cents
! colspan="3" | [[Ups and downs notation]]
|-
| 0
| 0.00
| Perfect 1sn
| P1
| D
|-
| 1
| 32.43
| Minor 2nd
| m2
| Eb
|-
| 2
| 64.86
| Upminor 2nd
| ^m2
| ^Eb
|-
| 3
| 97.30
| Downmid 2nd
| v~2
| ^^Eb
|-
| 4
| 129.73
| Mid 2nd
| ~2
| Ed
|-
| 5
| 162.16
| Upmid 2nd
| ^~2
| vvE
|-
| 6
| 194.59
| Downmajor 2nd
| vM2
| vE
|-
| 7
| 227.03
| Major 2nd
| M2
| E
|-
| 8
| 259.46
| Minor 3rd
| m3
| F
|-
| 9
| 291.89
| Upminor 3rd
| ^m3
| ^F
|-
| 10
| 324.32
| Downmid 3rd
| v~3
| ^^F
|-
| 11
| 356.76
| Mid 3rd
| ~3
| Ft
|-
| 12
| 389.19
| Upmid 3rd
| ^~3
| vvF#
|-
| 13
| 421.62
| Downmajor 3rd
| vM3
| vF#
|-
| 14
| 454.05
| Major 3rd
| M3
| F#
|-
| 15
| 486.49
| Perfect 4th
| P4
| G
|-
| 16
| 518.92
| Up 4th, Dim 5th
| ^4, d5
| ^G, Ab
|-
| 17
| 551.35
| Downmid 4th, Updim 5th
| v~4, ^d5
| ^^G, ^Ab
|-
| 18
| 583.78
| Mid 4th, Downmid 5th
| ~4, v~5
| Gt, ^^Ab
|-
| 19
| 616.22
| Mid 5th, Upmid 4th
| ~5, ^~4
| Ad, vvG#
|-
| 20
| 648.65
| Upmid 5th, Downaug 5th
| ^~5, vA4
| vvA, vG#
|-
| 21
| 681.08
| Down 5th, Aug 4th
| v5, A4
| vA, G#
|-
| 22
| 713.51
| Perfect 5th
| P5
| A
|-
| 23
| 745.95
| Minor 6th
| m6
| Bb
|-
| 24
| 778.38
| Upminor 6th
| ^m6
| ^Bb
|-
| 25
| 810.81
| Downmid 6th
| v~6
| ^^Bb
|-
| 26
| 843.24
| Mid 6th
| ~6
| Bd
|-
| 27
| 875.68
| Upmid 6th
| ^~6
| vvB
|-
| 28
| 908.11
| Downmajor 6th
| vM6
| vB
|-
| 29
| 940.54
| Major 6th
| M6
| B
|-
| 30
| 972.97
| Minor 7th
| m7
| C
|-
| 31
| 1005.41
| Upminor 7th
| ^m7
| ^C
|-
| 32
| 1037.84
| Downmid 7th
| v~7
| ^^C
|-
| 33
| 1070.27
| Mid 7th
| ~7
| Ct
|-
| 34
| 1102.70
| Upmid 7th
| ^~7
| vvC#
|-
| 35
| 1135.14
| Downmajor 7th
| vM7
| vC#
|-
| 36
| 1167.57
| Major 7th
| M7
| C#
|-
| 37
| 1200.00
| Perfect 8ve
| P8
| D
|}

37edo can be notated with 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}}

[[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have sharps and flats with arrows borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp6}}

If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt}}

=== 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>

== 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"
|-
! Generator
! In patent val
! In 37b val
|-
| 1\37
| 
| 
|-
| 2\37
| [[Sycamore]]
| 
|-
| 3\37
| [[Passion]]
| 
|-
| 4\37
| [[Twothirdtonic]]
| [[Negri]]
|-
| 5\37
| [[Porcupine]] / [[porcupinefish]]
| 
|-
| 6\37
| colspan="2" | [[Didacus]] / [[roulette]]
|-
| 7\37
| [[Shoe]] / [[semaja]]
| [[Shoe]] / [[laconic]] / [[gorgo]]
|-
| 8\37
| 
| [[Semaphore]] (37bd)
|-
| 9\37
| 
| [[Gariberttet]]
|-
| 10\37
| 
| [[Orgone]]
|-
| 11\37
| [[Beatles]]
| 
|-
| 12\37
| [[Würschmidt]] (out-of-tune)
| 
|-
| 13\37
| [[Skwares]] (37dd)
| 
|-
| 14\37
| [[Ammonite]]
| 
|-
| 15\37
| [[Ultrapyth]], [[oceanfront]]
| 
|-
| 16\37
| [[Undecimation]]
| 
|-
| 17\37
| [[Freivald]], [[emka]], [[onzonic]]
| 
|-
| 18\37
| 
| 
|}

== Scales ==
* [[MOS Scales of 37edo]]
* [[Chromatic pairs#Roulette|Roulette scales]]
* [[37ED4]]
* [[Square root of 13 over 10]]

=== 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 ==
; [[Beheld]]
* [https://www.youtube.com/watch?v=IULi2zSdatA ''Mindless vibe''] (2023)

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/e7dLJTsS3PQ ''37edo''] (2025)
* [https://www.youtube.com/shorts/m9hmiH8zong ''37edo jam''] (2025)

; [[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

; [[Andrew Heathwaite]]
* [https://andrewheathwaite.bandcamp.com/track/shorn-brown "Shorn Brown"] from ''Newbeams'' (2012) 
* [https://andrewheathwaite.bandcamp.com/track/jellybear "Jellybear"] from ''Newbeams'' (2012) 

; [[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)

; [[User:Fitzgerald Lee|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=HTAobydvC20 Marcello - Bach: Adagio from BWV 974, arranged for Oboe & Organ, tuned into 37edo] (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)
* [https://www.youtube.com/watch?v=AJ2sa-fRqbE Paradies, Toccata, Arranged for Organ and Tuned into 37edo] (2023)

; [[Micronaive]]
* [https://youtu.be/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)

; <nowiki>XENO*n*</nowiki>
* ''[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]]