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{{interwiki
{{interwiki
| de = 37edo
| de = 37-EDO
| en =  
| en = 37edo
| es =  
| es =  
| ja =  
| ja =  
}}
}}
{{Infobox ET}}
{{Infobox ET}}
The '''37 equal divisions of the octave''' ('''37edo'''), or the '''37(-tone) equal temperament''' ('''37tet''', '''37et''') when viewed from a regular temperament perspective, is the tuning system derived from dividing the octave into 37 [[equal]] steps.
{{ED intro}}


== Theory ==
== Theory ==
{{Primes in edo|37}}
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.


As a no-3s tuning, it is the 10th zeta peak edo. 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]]).
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 has an extremely precise 11/8, being only 0.03 cents sharp.
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.


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 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? -->


37edo is the 12th [[prime EDO]], following [[31edo]] and coming before [[41edo]].
=== 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 ===
=== Subgroups ===
Line 48: Line 54:
"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]].


As a consequence of large errors produced by having fifths so far from just as generators, 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).
=== 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 ==
== Intervals ==
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
Line 88: Line 98:
| 3
| 3
| 97.30
| 97.30
| [[55/52]]
| [[128/121]], [[55/52]]
| [[16/15]]
| [[16/15]]
|
|
Line 305: Line 315:
| 34
| 34
| 1102.70
| 1102.70
| [[104/55]]
| [[121/64]], [[104/55]]
| [[15/8]]
| [[15/8]]
|
|
Line 333: Line 343:


== Notation ==
== Notation ==
=== Ups and downs notation ===
37edo can be notated using [[ups and downs notation]]:
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! Degrees
! Degrees
! Cents
! Cents
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]
|-
|-
| 0
| 0
Line 567: Line 580:
| D
| 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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
Line 581: Line 645:
| 2.5
| 2.5
| {{monzo| 86 -37 }}
| {{monzo| 86 -37 }}
| [{{val| 37 86 }}]
| {{mapping| 37 86 }}
| -0.619
| −0.619
| 0.619
| 0.619
| 1.91
| 1.91
Line 588: Line 652:
| 2.5.7
| 2.5.7
| 3136/3125, 4194304/4117715
| 3136/3125, 4194304/4117715
| [{{val| 37 86 104 }}]
| {{mapping| 37 86 104 }}
| -0.905
| −0.905
| 0.647
| 0.647
| 2.00
| 2.00
Line 595: Line 659:
| 2.5.7.11
| 2.5.7.11
| 176/175, 1375/1372, 65536/65219
| 176/175, 1375/1372, 65536/65219
| [{{val| 37 86 104 128 }}]
| {{mapping| 37 86 104 128 }}
| -0.681
| −0.681
| 0.681
| 0.681
| 2.10
| 2.10
Line 602: Line 666:
| 2.5.7.11.13
| 2.5.7.11.13
| 176/175, 640/637, 847/845, 1375/1372
| 176/175, 640/637, 847/845, 1375/1372
| [{{val| 37 86 104 128 137 }}]
| {{mapping| 37 86 104 128 137 }}
| -0.692
| −0.692
| 0.610
| 0.610
| 1.88
| 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.  
* 37et is most prominent in the no-3 11-, 13-, 17-, 19- and 23-limit subgroups. The next ETs doing better in these subgroups are 109, 581, 103, 124 and 93, respectively.  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 640: Line 703:
|-
|-
| 6\37
| 6\37
| colspan="2" | [[Didacus]] / [[Chromatic_pairs#Roulette|roulette]]
| colspan="2" | [[Didacus]] / [[roulette]]
|-
|-
| 7\37
| 7\37
| [[Chromatic_pairs#Shoe|Shoe]] / [[semaja]]
| [[Shoe]] / [[semaja]]
| [[Chromatic_pairs#Shoe|Shoe]] / [[laconic]] / [[gorgo]]
| [[Shoe]] / [[laconic]] / [[gorgo]]
|-
|-
| 8\37
| 8\37
Line 652: Line 715:
| 9\37
| 9\37
|  
|  
| [[Chromatic_pairs#Gariberttet|Gariberttet]]
| [[Gariberttet]]
|-
|-
| 10\37
| 10\37
Line 675: Line 738:
|-
|-
| 15\37
| 15\37
| [[The_Biosphere#Ultrapyth|Ultrapyth]]
| [[Ultrapyth]], [[oceanfront]]
|  
|  
|-
|-
Line 693: Line 756:
== Scales ==
== Scales ==
* [[MOS Scales of 37edo]]
* [[MOS Scales of 37edo]]
* [[Roulette6]]
* [[Chromatic pairs#Roulette|Roulette scales]]
* [[Roulette7]]
* [[Roulette13]]
* [[Roulette19]]
* [[37ED4]]
* [[37ED4]]
* [[Square root of 13 over 10]]
* [[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 ==
== Music ==
* [http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3 Toccata Bianca 37EDO] by [http://www.akjmusic.com/ Aaron Krister Johnson]
; [[Beheld]]
* [https://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 [https://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]]
* [https://www.youtube.com/watch?v=IULi2zSdatA ''Mindless vibe''] (2023)
* [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=8reCr2nDGbw Porcupine Lullaby] by [[Ray Perlner]]
; [[Bryan Deister]]
* [https://youtu.be/TMVRYLvg_cA No.27.50] by [https://www.youtube.com/@Micronaive Micronaive]
* [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]]


== Links ==
== External links ==
* [http://tonalsoft.com/enc/number/37-edo/37edo.aspx 37edo at Tonalsoft]
* [http://tonalsoft.com/enc/number/37-edo/37edo.aspx 37-edo / 37-et / 37-tone equal-temperament] on [[Tonalsoft Encyclopedia]]


[[Category:37edo| ]] <!-- main article -->
[[Category:Listen]]
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Prime EDO]]
[[Category:Subgroup]]
Retrieved from "https://en.xen.wiki/w/37edo"