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== Theory ==
== Theory ==
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.  
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.  


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.  
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.  
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 ===
=== Odd harmonics ===
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=== As a tuning of other temperaments ===
=== 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]]).
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]]).
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.
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.
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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.


=== Miscellaneous properties ===
=== Miscellaneous properties ===
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== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
37edo can be notated using [[Kite's ups and downs notation|ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.  
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}
 
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt-szg}}
 
=== Kite's ups and downs notation ===
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.  
{{Sharpness-sharp6a}}
{{Sharpness-sharp6a}}


Half-sharps and half-flats can be used to avoid triple arrows:
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
{{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 ===
=== Ivan Wyschnegradsky's notation ===
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
{{Sharpness-sharp6-iw}}
{{Sharpness-sharp6-iw}}


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default [[File:37-EDO_Evo-SZ_Sagittal.svg]]
default [[File:37-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
</imagemap>
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals|37}}


== Regular temperament properties ==
== Regular temperament properties ==
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| 7\37
| 7\37
| 227.0
| 227.0
| [[Semaja]]
| [[Semaja]] / [[gorgik]]
| [[Gorgo]] (37b)
| [[Gorgo]] (37b)
|-
|-
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|  
|  
|}
|}
<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct
<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 ==
== Octave stretch or compression ==
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''See also: [[MOS Scales of 37edo]], [[Chromatic pairs#Roulette|Roulette scales]]''
''See also: [[MOS Scales of 37edo]], [[Chromatic pairs#Roulette|Roulette scales]]''


* [[37ed4]]: 2 2 2...
=== [[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
* [[Ammonite]][21]: 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1
* Opalised ammonite{{idio}} (modmos of Ammonite[8]): 5 4 6 5 2 5 4 6
* [[User:BudjarnLambeth/Antechinus|Antechinus]] (''nonoctave period'')
* [[Beatles]][7]: 4 7 4 7 4 7 4
* [[Beatles]][7]: 4 7 4 7 4 7 4
* Beatles[10]: 4 3 4 4 3 4 4 4 3 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
* Beatles[17]: 3 1 3 1 3 3 1 3 1 3 1 3 3 1 3 1 3
* [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]] (''octave-reduced ver.: 5 3 13 9 7'')
* Ultrapyth[5] (quasi-[[equipentatonic]]): 7 8 7 8 7 (''recommended mode: 8 7 7 8 7'')
* Oceanfront[5] (quasi-[[equipentatonic]]): 7 8 7 8 7
* Ultrapyth[7]: 7 1 7 7 7 1 7
* Oceanfront[7]: 7 1 7 7 7 1 7
* Ultrapyth[12]: 1 6 1 6 1 6 1 1 6 1 6 1
* Oceanfront[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'')
* Oceanfront[17]: 1 5 1 1 1 5 1 1 5 1 1 5 1 1 1 5 1
* 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'')
* Oceanfront[22]: 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1 1 4 1 1
* Passion[9]: 13 3 3 3 3 3 3 3 3
* 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[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
* 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[5]: 5 17 5 5 5
* Porcupine[6]: 12 5 5 5 5 5
* Porcupine[6]: 12 5 5 5 5 5
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* Porcupine[15]: 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2
* 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
* Porcupine[22]: 2 1 2 2 1 2 2 1 2 2 1 2 2 2 1 2 2 1 2 2 1 2
* [[Square root of 13 over 10]]: 7 7 7...
* Twothirdtonic[7]: 13 4 4 4 4 4 4
* Twothirdtonic[7]: 13 4 4 4 4 4 4
* Twothirdtonic[8]: 9 4 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[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
* 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: [[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 ===
=== Every 8 steps of 37edo ===
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== Music ==
== Music ==
=== Modern renderings ===
; {{W|Alessandro Marcello}} and {{w|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=HTAobydvC20 ''Oboe Concerto in D minor'', BWV 974] (1715) – arranged for oboe & organ by [[Claudi Meneghin]] (2022)
; {{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]]
; [[Beheld]]
* [https://www.youtube.com/watch?v=IULi2zSdatA ''Mindless vibe''] (2023)
* [https://www.youtube.com/watch?v=IULi2zSdatA ''Mindless vibe''] (2023)
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* [https://www.youtube.com/shorts/m9hmiH8zong ''37edo jam''] (2025)
* [https://www.youtube.com/shorts/m9hmiH8zong ''37edo jam''] (2025)
* [https://www.youtube.com/shorts/mVRbcB2hoBU ''37edo prelude''] (2026)
* [https://www.youtube.com/shorts/mVRbcB2hoBU ''37edo prelude''] (2026)
* [https://www.youtube.com/shorts/Jt6_r6r3lGY ''37edo improv''] (2026)


; [[Francium]]
; [[Francium]]
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* [https://www.youtube.com/watch?v=ngxSiuVadls ''A Dark Era Arises''] (2023) – in Porcupine[15], 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
* [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]]
; [[Andrew Heathwaite]]
* [https://andrewheathwaite.bandcamp.com/track/shorn-brown "Shorn Brown"] from ''Newbeams'' (2012)  
* From [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)
* [https://andrewheathwaite.bandcamp.com/track/jellybear "Jellybear"] from ''Newbeams'' (2012)
** "Shorn Brown"
** "Jellybear"


; [[Aaron Krister Johnson]]
; [[Aaron Krister Johnson]]
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* [https://www.youtube.com/watch?v=taT1DClJ2KM ''Tyrian and Gold''] (2024)
* [https://www.youtube.com/watch?v=taT1DClJ2KM ''Tyrian and Gold''] (2024)


; [[User:Fitzgerald Lee|Fitzgerald Lee]]
; [[Fitzgerald Lee]]
* [https://www.youtube.com/watch?v=Nr0cUJcL4SU ''Bittersweet End''] (2025)
* [https://www.youtube.com/watch?v=Nr0cUJcL4SU ''Bittersweet End''] (2025)


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; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=7dU8eyGbt9I ''Deck The Halls''] (2022)
* [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=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=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]]
; [[Micronaive]]
* [https://youtu.be/TMVRYLvg_cA No.27.50] (2022)
* [https://www.youtube.com/watch?v=TMVRYLvg_cA No.27.50] (2022)


; [[Herman Miller]]
; [[Herman Miller]]
* ''[https://soundcloud.com/morphosyntax-1/luck-of-the-draw Luck of the Draw]'' (2023)
* [https://soundcloud.com/morphosyntax-1/luck-of-the-draw ''Luck of the Draw''] (2023)


; [[Joseph Monzo]]
; [[Joseph Monzo]]
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; [[Mundoworld]]
; [[Mundoworld]]
* ''Reckless Discredit'' (2021) [https://www.youtube.com/watch?v=ovgsjSoHOkg YouTube] · [https://mundoworld.bandcamp.com/track/reckless-discredit Bandcamp]
* ''Reckless Discredit'' (2021) – [https://www.youtube.com/watch?v=ovgsjSoHOkg YouTube] · [https://mundoworld.bandcamp.com/track/reckless-discredit Bandcamp]


; [[Ray Perlner]]
; [[Ray Perlner]]
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; [[Stephen Weigel]]
; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=71yBnSVBsJk ''Leap Day Cloo (37-TET)''] (2025)
* [https://www.youtube.com/watch?v=71yBnSVBsJk ''Leap Day Cloo''] (2025)


; <nowiki>XENO*n*</nowiki>
; [[Xeno*n*]]
* ''[https://www.youtube.com/watch?v=_m5u4VviMXw Galantean Drift]'' (2025)
* [https://www.youtube.com/watch?v=_m5u4VviMXw ''Galantean Drift''] (2025)


== See also ==
== See also ==
Retrieved from "https://en.xen.wiki/w/37edo"