37edo: Difference between revisions

| de = 37-EDO

| en = 37edo

| ja =

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.

* 17\37 = 551.4 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 ===

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.

=== No-3 approach ===

{| 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 ==

 

 

[[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]]:

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

This notation uses the same sagittal sequence as EDOs [[23edo#Second-best fifth notation|23b]], [[30edo#Sagittal notation|30]], and [[44edo#Sagittal notation|44]].

rect 80 0 300 50 [[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]]

rect 80 0 300 50 [[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]]

==== Evo-SZ flavor ====

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]]

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve stretch (¢)

! [[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

| {{mapping| 37 86 104 128 137 }}

 

 

| 4\37

| [[Twothirdtonic]]

| [[Negri]]

| [[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

|  

|  

 

 

 

 

 

== Music ==

 

 

* [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

 

 

* [http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3 ''Toccata Bianca 37EDO'']{{dead link}}

 

 

 

 

 

 

 

 

 

* [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

 

* [https://www.youtube.com/watch?v=2otxZqUrvHc ''Elevated Floors''] (2025)

* [https://www.youtube.com/watch?v=BbexOU-9700 ''cat jam 37''] (2025)

 

 

* [https://www.youtube.com/watch?v=rE9L56yZ1Kw ''Winter''] (2025)

; <nowiki>XENO*n*</nowiki>

* ''[https://www.youtube.com/watch?v=_m5u4VviMXw Galantean Drift]'' (2025)

* [[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]]