@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|37}}
+{{ED intro}}
 == Theory ==
-edo is the 10th [[No-threes subgroup temperaments|no-3]] zeta [[Zeta peak edo|peak]] edo, containing very accurate approximations of harmonics [[5/1|5]], [[7/1|7]], [[11/1|11]] and [[13/1|13]]. It has an extremely accurate harmonic 11, being only 0.03 cents sharp.
+edo 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]]).
@@ Line 15: / Line 15: @@
 edo 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]].
+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 ===
@@ Line 61: / Line 61: @@
 edo 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"
 |-
@@ Line 341: / Line 343: @@
 == Notation ==
+=== Ups and downs notation ===
-=== Ups and Downs notation ===
+edo can be notated using [[ups and downs notation]]:
 {| class="wikitable center-all right-2 left-3"
@@ Line 348: / Line 350: @@
 ! Degrees
 ! Cents
-! colspan="3" | [[Ups and Downs Notation]]
+! colspan="3" | [[Ups and downs notation]]
 |-
 | 0
@@ Line 579: / Line 581: @@
 |}
-===Sagittal notation===
+edo 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====
+==== Evo and Revo flavors ====
 <imagemap>
 File:37-EDO_Sagittal.svg
@@ Line 592: / Line 611: @@
 </imagemap>
-====Alternative Evo flavor====
+==== Alternative Evo flavor ====
 <imagemap>
 File:37-EDO_Alternative_Evo_Sagittal.svg
@@ Line 603: / Line 621: @@
 </imagemap>
-====Evo-SZ flavor====
+==== Evo-SZ flavor ====
 <imagemap>
 File:37-EDO_Evo-SZ_Sagittal.svg
@@ Line 616: / Line 633: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
@@ Line 628: / Line 646: @@
 | {{monzo| 86 -37 }}
 | {{mapping| 37 86 }}
-| -0.619
+| −0.619
 | 0.619
 | 1.91
@@ Line 635: / Line 653: @@
 | 3136/3125, 4194304/4117715
 | {{mapping| 37 86 104 }}
-| -0.905
+| −0.905
 | 0.647
 | 2.00
@@ Line 642: / Line 660: @@
 | 176/175, 1375/1372, 65536/65219
 | {{mapping| 37 86 104 128 }}
-| -0.681
+| −0.681
 | 0.681
 | 2.10
@@ Line 649: / Line 667: @@
 | 176/175, 640/637, 847/845, 1375/1372
 | {{mapping| 37 86 104 128 137 }}
-| -0.692
+| −0.692
 | 0.610
 | 1.88
@@ Line 738: / Line 756: @@
 == Scales ==
 * [[MOS Scales of 37edo]]
-* [[Roulette6]]
+* [[Chromatic pairs#Roulette|Roulette scales]]
-* [[Roulette7]]
-* [[Roulette13]]
-* [[Roulette19]]
 * [[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 ==
@@ Line 749: / Line 896: @@
 * [https://www.youtube.com/watch?v=IULi2zSdatA ''Mindless vibe''] (2023)
-; [[User:Francium|Francium]]
+; [[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) – porcupine[15] in 37edo tuning
+* [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]]
@@ Line 762: / Line 914: @@
 ; [[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]]
@@ Line 769: / Line 924: @@
 * [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) – passion in 37edo tuning
+* [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)
@@ Line 787: / Line 942: @@
 ; [[Ray Perlner]]
-* [https://www.youtube.com/watch?v=8reCr2nDGbw ''Porcupine Lullaby''] (2020) – porcupine in 37edo tuning
+* [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) – porcupine[7] in 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) – porcupine[7] in 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 ==
-* [[Lumatone mapping for 37edo]]
+* [[User:Unque/37edo Composition Theory|Unque's approach]]
 == External links ==

37edo: Difference between revisions