37edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,164 edits
→Notation: add Sagittal notation
m →Notation: improve description
 
(88 intermediate revisions by 16 users not shown)
Line 6: Line 6:
}}
}}
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|37}}
{{ED intro}}


== Theory ==
== Theory ==
37edo is the 10th no-3 zeta 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.
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.
=== Odd harmonics ===
{{Harmonics in equal|37}}
=== 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.
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]].
=== Odd harmonics ===
{{Harmonics in equal|37}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Line 24: Line 27:


[[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.
[[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 ===
=== Dual fifths ===
Line 61: Line 54:


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).
=== Miscellaneous properties ===
37edo has the sharpest fifth of any edo that can possibly be [[diamond monotone]] in the [[15-odd-limit]]. The sharpest mapping of [[7/4]] where [[9/8]] is mapped no wider than [[8/7]] is 30\37, and the sharpest possible mapping of [[15/8]] where diamond monotone is achieveable is 34\37, where [[15/14]] is equated with [[14/13]][[~]][[13/12]] to half of [[7/6]]. Here [[5/4]] is mapped to 12\37, and [[10/9]] is mapped to 5\37. Equating both [[11/10]] and [[12/11]] with 10/9 makes the mappings for 9/8, 10/9, 11/10, and 12/11 add up to [[3/2]]. If the fifth was any sharper, then [[7/4]] and [[15/8]] would have to be flatter. Then 5/4 would have to be flatter, and therefore 10/9 as well, and at least one of 11/10 and 12/11 would have to be mapped wider than 10/9 for 9/8, 10/9, 11/10, and 12/11 to add up to 3/2. 37edo is, in fact, diamond monotone in the 15-odd-limit (see [[Monotonicity limits of small EDOs]]). Therefore, 22\37 is the sharpest fifth where 15-odd-limit diamond monotone is possible. The flattest fifth where 15-odd-limit diamond monotone is possible is [[19edo#Miscellaneous properties|11\19]].


== Intervals ==
== Intervals ==
 
Inconsistent intervals are in ''italics''.
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! Degrees
! #
! Cents
! Cents
! Approximate Ratios<br>of 2.5.7.11.13.27 subgroup
! Approximate ratios<br>of 2.27.5.7.11.13 subgroup
! Additional Ratios of 3<br>with a sharp 3/2
! Additional ratios of 3<br>with a sharp 3/2
! Additional Ratios of 3<br>with a flat 3/2
! Additional ratios of 3<br>with a flat 3/2
! Additional Ratios of 9<br>with 194.59¢ 9/8
! Additional ratios of 9<br>with 194.59 ¢ 9/8
|-
|-
| 0
| 0
| 0.00
| 0.0
| 1/1
| 1/1
|
|
Line 81: Line 77:
|-
|-
| 1
| 1
| 32.43
| 32.4
| [[55/54]], [[56/55]]
| [[55/54]], [[56/55]]
|
|
Line 88: Line 84:
|-
|-
| 2
| 2
| 64.86
| 64.9
| [[27/26]], [[28/27]]
| [[27/26]], [[28/27]]
|
|
Line 95: Line 91:
|-
|-
| 3
| 3
| 97.30
| 97.3
| [[128/121]], [[55/52]]
| [[128/121]], [[55/52]]
| [[16/15]]
| [[16/15]]
Line 102: Line 98:
|-
|-
| 4
| 4
| 129.73
| 129.7
| [[14/13]]
| [[14/13]]
| [[13/12]], [[15/14]]
| [[13/12]], [[15/14]]
| [[12/11]]
| ''[[12/11]]''
|
|
|-
|-
| 5
| 5
| 162.16
| 162.2
| [[11/10]]
| [[11/10]]
| [[10/9]], [[12/11]]
| ''[[10/9]]'', [[12/11]]
| [[13/12]]
| ''[[13/12]]''
|
|
|-
|-
| 6
| 6
| 194.59
| 194.6
| [[28/25]]
| [[28/25]]
|
|
Line 123: Line 119:
|-
|-
| 7
| 7
| 227.03
| 227.0
| [[8/7]]
| [[8/7]]
| [[9/8]]
| ''[[9/8]]''
|
|
|
|
|-
|-
| 8
| 8
| 259.46
| 259.5
|
|
| [[7/6]], [[15/13]]
| [[7/6]], [[15/13]]
Line 137: Line 133:
|-
|-
| 9
| 9
| 291.89
| 291.9
| [[13/11]], [[32/27]]
| [[13/11]], [[32/27]]
|
|
| [[6/5]], [[7/6]]
| ''[[6/5]]'', ''[[7/6]]''
|
|
|-
|-
| 10
| 10
| 324.32
| 324.3
|
|
| [[6/5]], [[11/9]]
| [[6/5]], ''[[11/9]]''
|
|
|
|
|-
|-
| 11
| 11
| 356.76
| 356.8
| [[16/13]], [[27/22]]
| [[16/13]], [[27/22]]
|
|
Line 158: Line 154:
|-
|-
| 12
| 12
| 389.19
| 389.2
| [[5/4]]
| [[5/4]]
|
|
Line 165: Line 161:
|-
|-
| 13
| 13
| 421.62
| 421.6
| [[14/11]], [[32/25]]
| [[14/11]], [[32/25]]
|
|
Line 172: Line 168:
|-
|-
| 14
| 14
| 454.05
| 454.1
| [[13/10]]
| [[13/10]]
| [[9/7]]
| ''[[9/7]]''
|
|
|
|
|-
|-
| 15
| 15
| 486.49
| 486.5
|
|
| [[4/3]]
| [[4/3]]
Line 186: Line 182:
|-
|-
| 16
| 16
| 518.92
| 518.9
| [[27/20]]
| [[27/20]]
|
|
| [[4/3]]
| ''[[4/3]]''
|
|
|-
|-
| 17
| 17
| 551.35
| 551.4
| [[11/8]]
| [[11/8]]
| [[15/11]]
| [[15/11]]
Line 200: Line 196:
|-
|-
| 18
| 18
| 583.78
| 583.8
| [[7/5]]
| [[7/5]]
| [[18/13]]
| ''[[18/13]]''
|
|
|
|
|-
|-
| 19
| 19
| 616.22
| 616.2
| [[10/7]]
| [[10/7]]
| [[13/9]]
| ''[[13/9]]''
|
|
|
|
|-
|-
| 20
| 20
| 648.65
| 648.6
| [[16/11]]
| [[16/11]]
| [[22/15]]
| [[22/15]]
Line 221: Line 217:
|-
|-
| 21
| 21
| 681.08
| 681.1
| [[40/27]]
| [[40/27]]
|
|
| [[3/2]]
| ''[[3/2]]''
|
|
|-
|-
| 22
| 22
| 713.51
| 713.5
|
|
| [[3/2]]
| [[3/2]]
Line 235: Line 231:
|-
|-
| 23
| 23
| 745.95
| 745.9
| [[20/13]]
| [[20/13]]
| [[14/9]]
| ''[[14/9]]''
|
|
|
|
|-
|-
| 24
| 24
| 778.38
| 778.4
| [[11/7]], [[25/16]]
| [[11/7]], [[25/16]]
|
|
Line 249: Line 245:
|-
|-
| 25
| 25
| 810.81
| 810.8
| [[8/5]]
| [[8/5]]
|
|
Line 256: Line 252:
|-
|-
| 26
| 26
| 843.24
| 843.2
| [[13/8]], [[44/27]]
| [[13/8]], [[44/27]]
|
|
Line 263: Line 259:
|-
|-
| 27
| 27
| 875.68
| 875.7
|
|
| [[5/3]], [[18/11]]
| [[5/3]], ''[[18/11]]''
|
|
|
|
|-
|-
| 28
| 28
| 908.11
| 908.1
| [[22/13]], [[27/16]]
| [[22/13]], [[27/16]]
|
|
| [[5/3]], [[12/7]]
| ''[[5/3]], [[12/7]]''
|
|
|-
|-
| 29
| 29
| 940.54
| 940.5
|
|
| [[12/7]], [[26/15]]
| [[12/7]], [[26/15]]
Line 284: Line 280:
|-
|-
| 30
| 30
| 972.97
| 973.0
| [[7/4]]
| [[7/4]]
| [[16/9]]
| ''[[16/9]]''
|
|
|
|
|-
|-
| 31
| 31
| 1005.41
| 1005.4
| [[25/14]]
| [[25/14]]
|
|
Line 298: Line 294:
|-
|-
| 32
| 32
| 1037.84
| 1037.8
| [[20/11]]
| [[20/11]]
| [[9/5]], [[11/6]]
| ''[[9/5]]'', [[11/6]]
|
|
|
|
|-
|-
| 33
| 33
| 1070.27
| 1070.3
| [[13/7]]
| [[13/7]]
| [[24/13]], [[28/15]]
| [[24/13]], [[28/15]]
| [[11/6]]
| ''[[11/6]]''
|
|
|-
|-
| 34
| 34
| 1102.70
| 1102.7
| [[121/64]], [[104/55]]
| [[121/64]], [[104/55]]
| [[15/8]]
| [[15/8]]
Line 319: Line 315:
|-
|-
| 35
| 35
| 1135.14
| 1135.1
| [[27/14]], [[52/27]]
| [[27/14]], [[52/27]]
|
|
Line 326: Line 322:
|-
|-
| 36
| 36
| 1167.57
| 1167.6
|
|
|
|
Line 333: Line 329:
|-
|-
| 37
| 37
| 1200.00
| 1200.0
| [[2/1]]
| [[2/1]]
|
|
Line 340: Line 336:
|}
|}


== Notation ==
=== Proposed interval names and solfèges ===
 
{| class="wikitable center-all right-2 left-3 mw-collapsible mw-collapsed"
=== Ups and Downs notation ===
|+ style="white-space: nowrap;" | Table of proposed interval names and solfèges
 
{| class="wikitable center-all right-2 left-3"
|-
|-
! Degrees
! #
! Cents
! Cents
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Kite's ups and downs notation|Ups and downs notation]]
|-
|-
| 0
| 0
| 0.00
| 0.0
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 357: Line 351:
|-
|-
| 1
| 1
| 32.43
| 32.4
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 363: Line 357:
|-
|-
| 2
| 2
| 64.86
| 64.9
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 369: Line 363:
|-
|-
| 3
| 3
| 97.30
| 97.3
| Downmid 2nd
| Downmid 2nd
| v~2
| v~2
Line 375: Line 369:
|-
|-
| 4
| 4
| 129.73
| 129.7
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 381: Line 375:
|-
|-
| 5
| 5
| 162.16
| 162.2
| Upmid 2nd
| Upmid 2nd
| ^~2
| ^~2
Line 387: Line 381:
|-
|-
| 6
| 6
| 194.59
| 194.6
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 393: Line 387:
|-
|-
| 7
| 7
| 227.03
| 227.0
| Major 2nd
| Major 2nd
| M2
| M2
Line 399: Line 393:
|-
|-
| 8
| 8
| 259.46
| 259.5
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 405: Line 399:
|-
|-
| 9
| 9
| 291.89
| 291.9
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 411: Line 405:
|-
|-
| 10
| 10
| 324.32
| 324.3
| Downmid 3rd
| Downmid 3rd
| v~3
| v~3
Line 417: Line 411:
|-
|-
| 11
| 11
| 356.76
| 356.8
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 423: Line 417:
|-
|-
| 12
| 12
| 389.19
| 389.2
| Upmid 3rd
| Upmid 3rd
| ^~3
| ^~3
Line 429: Line 423:
|-
|-
| 13
| 13
| 421.62
| 421.6
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 435: Line 429:
|-
|-
| 14
| 14
| 454.05
| 454.1
| Major 3rd
| Major 3rd
| M3
| M3
Line 441: Line 435:
|-
|-
| 15
| 15
| 486.49
| 486.5
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 447: Line 441:
|-
|-
| 16
| 16
| 518.92
| 518.9
| Up 4th, Dim 5th
| Up 4th, dim 5th
| ^4, d5
| ^4, d5
| ^G, Ab
| ^G, Ab
|-
|-
| 17
| 17
| 551.35
| 551.4
| Downmid 4th, Updim 5th
| Downmid 4th, updim 5th
| v~4, ^d5
| v~4, ^d5
| ^^G, ^Ab
| ^^G, ^Ab
|-
|-
| 18
| 18
| 583.78
| 583.8
| Mid 4th, Downmid 5th
| Mid 4th, downmid 5th
| ~4, v~5
| ~4, v~5
| Gt, ^^Ab
| Gt, ^^Ab
|-
|-
| 19
| 19
| 616.22
| 616.2
| Mid 5th, Upmid 4th
| Mid 5th, upmid 4th
| ~5, ^~4
| ~5, ^~4
| Ad, vvG#
| Ad, vvG#
|-
|-
| 20
| 20
| 648.65
| 648.6
| Upmid 5th, Downaug 5th
| Upmid 5th, downaug 5th
| ^~5, vA4
| ^~5, vA4
| vvA, vG#
| vvA, vG#
|-
|-
| 21
| 21
| 681.08
| 681.1
| Down 5th, Aug 4th
| Down 5th, aug 4th
| v5, A4
| v5, A4
| vA, G#
| vA, G#
|-
|-
| 22
| 22
| 713.51
| 713.5
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 489: Line 483:
|-
|-
| 23
| 23
| 745.95
| 745.9
| Minor 6th
| Minor 6th
| m6
| m6
Line 495: Line 489:
|-
|-
| 24
| 24
| 778.38
| 778.4
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 501: Line 495:
|-
|-
| 25
| 25
| 810.81
| 810.8
| Downmid 6th
| Downmid 6th
| v~6
| v~6
Line 507: Line 501:
|-
|-
| 26
| 26
| 843.24
| 843.2
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 513: Line 507:
|-
|-
| 27
| 27
| 875.68
| 875.7
| Upmid 6th
| Upmid 6th
| ^~6
| ^~6
Line 519: Line 513:
|-
|-
| 28
| 28
| 908.11
| 908.1
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 525: Line 519:
|-
|-
| 29
| 29
| 940.54
| 940.5
| Major 6th
| Major 6th
| M6
| M6
Line 531: Line 525:
|-
|-
| 30
| 30
| 972.97
| 973.0
| Minor 7th
| Minor 7th
| m7
| m7
Line 537: Line 531:
|-
|-
| 31
| 31
| 1005.41
| 1005.4
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 543: Line 537:
|-
|-
| 32
| 32
| 1037.84
| 1037.8
| Downmid 7th
| Downmid 7th
| v~7
| v~7
Line 549: Line 543:
|-
|-
| 33
| 33
| 1070.27
| 1070.3
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 555: Line 549:
|-
|-
| 34
| 34
| 1102.70
| 1102.7
| Upmid 7th
| Upmid 7th
| ^~7
| ^~7
Line 561: Line 555:
|-
|-
| 35
| 35
| 1135.14
| 1135.1
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 567: Line 561:
|-
|-
| 36
| 36
| 1167.57
| 1167.6
| Major 7th
| Major 7th
| M7
| M7
Line 573: Line 567:
|-
|-
| 37
| 37
| 1200.00
| 1200.0
| Perfect 8ve
| Perfect 8ve
| P8
| P8
Line 579: Line 573:
|}
|}


===Sagittal notation===
== Notation ==
This notation uses the same sagittal sequence as EDOs [[23edo#Sagittal notation|23b]], [[30edo#Sagittal notation|30]], and [[44edo#Sagittal notation|44]].
=== Stein–Zimmermann–Gould notation ===
[[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}}


Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
=== 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>
<imagemap>
File:37-EDO_Sagittal.svg
File:37-EDO_Sagittal.svg
Line 591: Line 605:
</imagemap>
</imagemap>


====Alternate Evo flavor====
==== Alternative Evo flavor ====
 
<imagemap>
<imagemap>
File:37-EDO_Alternative_Evo_Sagittal.svg
File:37-EDO_Alternative_Evo_Sagittal.svg
Line 602: Line 615:
</imagemap>
</imagemap>


====Evo-SZ flavor====
==== Evo-SZ flavor ====
 
<imagemap>
<imagemap>
File:37-EDO_Evo-SZ_Sagittal.svg
File:37-EDO_Evo-SZ_Sagittal.svg
Line 612: Line 624:
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 ==
{| 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]]
Line 625: Line 642:
|-
|-
| 2.5
| 2.5
| {{monzo| 86 -37 }}
| {{Monzo| 86 -37 }}
| {{mapping| 37 86 }}
| {{Mapping| 37 86 }}
| -0.619
| −0.619
| 0.619
| 0.619
| 1.91
| 1.91
Line 633: Line 650:
| 2.5.7
| 2.5.7
| 3136/3125, 4194304/4117715
| 3136/3125, 4194304/4117715
| {{mapping| 37 86 104 }}
| {{Mapping| 37 86 104 }}
| -0.905
| −0.905
| 0.647
| 0.647
| 2.00
| 2.00
Line 640: Line 657:
| 2.5.7.11
| 2.5.7.11
| 176/175, 1375/1372, 65536/65219
| 176/175, 1375/1372, 65536/65219
| {{mapping| 37 86 104 128 }}
| {{Mapping| 37 86 104 128 }}
| -0.681
| −0.681
| 0.681
| 0.681
| 2.10
| 2.10
Line 647: Line 664:
| 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
| {{mapping| 37 86 104 128 137 }}
| {{Mapping| 37 86 104 128 137 }}
| -0.692
| −0.692
| 0.610
| 0.610
| 1.88
| 1.88
Line 657: Line 674:
* [[List of 37et rank two temperaments by badness]]
* [[List of 37et rank two temperaments by badness]]


{| class="wikitable center-1"
{| class="wikitable center-1 center-2"
|-
|-
! Generator
! Generator*
! Cents*
! In patent val
! In patent val
! In 37b val
! In 37b val
|-
|-
| 1\37
| 1\37
| 32.4
|  
|  
|  
|  
|-
|-
| 2\37
| 2\37
| 64.9
| [[Sycamore]]
| [[Sycamore]]
|  
|  
|-
|-
| 3\37
| 3\37
| 97.3
| [[Passion]]
| [[Passion]]
|  
|  
|-
|-
| 4\37
| 4\37
| 129.7
| [[Twothirdtonic]]
| [[Twothirdtonic]]
| [[Negri]]
| [[Negri]] (37bd, out-of-tune)
|-
|-
| 5\37
| 5\37
| 162.2
| [[Porcupine]] / [[porcupinefish]]
| [[Porcupine]] / [[porcupinefish]]
|  
|  
|-
|-
| 6\37
| 6\37
| colspan="2" | [[Didacus]] / [[roulette]]
| 194.6
| [[Hemiwürschmidt]] / [[hemiwur]]
| [[Hemithirds]] (37b, out-of-tune)
|-
|-
| 7\37
| 7\37
| [[Shoe]] / [[semaja]]
| 227.0
| [[Shoe]] / [[laconic]] / [[gorgo]]
| [[Semaja]] / [[gorgik]]
| [[Gorgo]] (37b)
|-
|-
| 8\37
| 8\37
| 259.5
|  
|  
| [[Semaphore]] (37bd)
| [[Semaphore]] (37bd, out-of-tune)
|-
|-
| 9\37
| 9\37
| 291.9
| [[Quasitemp]]
|  
|  
| [[Gariberttet]]
|-
|-
| 10\37
| 10\37
|  
| 324.3
| [[Orgone]]
| [[Hyperkleismic]]
| [[Superkleismic]] (37bc, out-of-tune)
|-
|-
| 11\37
| 11\37
| 356.8
| [[Beatles]]
| [[Beatles]]
|  
|  
|-
|-
| 12\37
| 12\37
| 389.2
| [[Würschmidt]] (out-of-tune)
| [[Würschmidt]] (out-of-tune)
|  
|  
|-
|-
| 13\37
| 13\37
| [[Skwares]] (37dd)
| 421.6
| [[Skwares]] (37dd, out-of-tune)
|  
|  
|-
|-
| 14\37
| 14\37
| 454.1
| [[Ammonite]]
| [[Ammonite]]
|  
|  
|-
|-
| 15\37
| 15\37
| [[Ultrapyth]], [[oceanfront]]
| 486.5
| [[Ultrapyth]]
|  
|  
|-
|-
| 16\37
| 16\37
| 518.9
| [[Undecimation]]
| [[Undecimation]]
|  
| [[Shallowtone]] (37b)
|-
|-
| 17\37
| 17\37
| [[Freivald]], [[emka]], [[onzonic]]
| 551.4
| [[Freivald]], [[emka]]
|  
|  
|-
|-
| 18\37
| 18\37
|  
| 583.8
| [[Cotritone]]
|  
|  
|}
|}
<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 ==
37edo's [[prime]]s 3, 5, 7, 11 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]]. Some compressed-octave 37edo tunings (least to most compressed) include [[161zpi]], [[ed5|86ed5]], [[ed7|104ed7]], [[ed12|133ed12]] or [[ed6|96ed6]].


== Scales ==
== Scales ==
* [[MOS Scales of 37edo]]
''See also: [[MOS Scales of 37edo]], [[Chromatic pairs#Roulette|Roulette scales]]''
* [[Roulette6]]
 
* [[Roulette7]]
=== [[MOS scale]]s ===
* [[Roulette13]]
* [[Ammonite]][21]: 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1
* [[Roulette19]]
* [[Beatles]][7]: 4 7 4 7 4 7 4
* [[37ED4]]
* Beatles[10]: 4 3 4 4 3 4 4 4 3 4
* [[Square root of 13 over 10]]
* Beatles[17]: 3 1 3 1 3 3 1 3 1 3 1 3 3 1 3 1 3
* Ultrapyth[5] (quasi-[[equipentatonic]]): 7 8 7 8 7 (''recommended mode: 8 7 7 8 7'')
* Ultrapyth[7]: 7 1 7 7 7 1 7
* Ultrapyth[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'')
* 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'')
* 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[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[6]: 12 5 5 5 5 5
* Porcupine[7]: 5 5 5 7 5 5 5
* 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
* Twothirdtonic[7]: 13 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[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: [[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 ===
{| 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 ==
=== 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)


; [[User:Francium|Francium]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/TEzitpGJvt0 ''37edo''] (2023)
* [https://www.youtube.com/shorts/e7dLJTsS3PQ ''37edo''] (2025)
* [https://www.youtube.com/shorts/m9hmiH8zong ''37edo jam''] (2025)
* [https://www.youtube.com/shorts/mVRbcB2hoBU ''37edo prelude''] (2026)
* [https://www.youtube.com/shorts/Jt6_r6r3lGY ''37edo improv''] (2026)
 
; [[Francium]]
* [https://www.youtube.com/watch?v=jpPjVouoq3E ''5 days in''] (2023)
* [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
 
; [[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) [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2002%20Shorn%20Brown.mp3 play]{{dead link}}
* From [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)
* [https://andrewheathwaite.bandcamp.com/track/jellybear "Jellybear"] from ''Newbeams'' (2012) [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2003%20Jellybear.mp3 play]{{dead link}}
** "Shorn Brown"
** "Jellybear"


; [[Aaron Krister Johnson]]
; [[Aaron Krister Johnson]]
Line 761: Line 1,017:
; [[JUMBLE]]
; [[JUMBLE]]
* [https://www.youtube.com/watch?v=taT1DClJ2KM ''Tyrian and Gold''] (2024)
* [https://www.youtube.com/watch?v=taT1DClJ2KM ''Tyrian and Gold''] (2024)
; [[Fitzgerald Lee]]
* [https://www.youtube.com/watch?v=Nr0cUJcL4SU ''Bittersweet End''] (2025)


; [[Mandrake]]
; [[Mandrake]]
Line 767: Line 1,026:
; [[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) – passion in 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]]
Line 783: Line 1,040:


; [[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]]
* [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]]
; [[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]
* "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)
; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=71yBnSVBsJk ''Leap Day Cloo''] (2025)
; [[Xeno*n*]]
* [https://www.youtube.com/watch?v=_m5u4VviMXw ''Galantean Drift''] (2025)


== See also ==
== See also ==
* [[Lumatone mapping for 37edo]]
* [[User:Unque/37edo Composition Theory|Unque's approach]]


== External links ==
== External links ==

Latest revision as of 14:18, 12 May 2026

← 36edo 37edo 38edo →
Prime factorization 37 (prime)
Step size 32.4324 ¢ 
Fifth 22\37 (713.514 ¢)
Semitones (A1:m2) 6:1 (194.6 ¢ : 32.43 ¢)
Dual sharp fifth 22\37 (713.514 ¢)
Dual flat fifth 21\37 (681.081 ¢)
Dual major 2nd 6\37 (194.595 ¢)
Consistency limit 7
Distinct consistency limit 7

37 equal divisions of the octave (abbreviated 37edo or 37ed2), also called 37-tone equal temperament (37tet) or 37 equal temperament (37et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 37 equal parts of about 32.4 ¢ each. Each step represents a frequency ratio of 21/37, or the 37th root of 2.

Theory

37edo has very accurate approximations of harmonics 5, 7, 11 and 13, making it a good choice for a no-threes approach. Harmonic 11 is particularly accurate, being only 0.03 cents sharp. A usable approximation of 9 is available at 6\37 (194.6 cents) as well, and the no-3 no-15 no-21 23-odd-limit is represented consistently.

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 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 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 antidiatonic scale.

Odd harmonics

Approximation of odd harmonics in 37edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +11.6 +2.9 +4.1 -9.3 +0.0 +2.7 +14.4 -7.7 -5.6 +15.7 -12.1
Relative (%) +35.6 +8.9 +12.8 -28.7 +0.1 +8.4 +44.5 -23.6 -17.3 +48.4 -37.2
Steps
(reduced) 		59
(22) 		86
(12) 		104
(30) 		117
(6) 		128
(17) 		137
(26) 		145
(34) 		151
(3) 		157
(9) 		163
(15) 		167
(19)

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 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.

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.

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 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).

Miscellaneous properties

37edo has the sharpest fifth of any edo that can possibly be diamond monotone in the 15-odd-limit. The sharpest mapping of 7/4 where 9/8 is mapped no wider than 8/7 is 30\37, and the sharpest possible mapping of 15/8 where diamond monotone is achieveable is 34\37, where 15/14 is equated with 14/13~13/12 to half of 7/6. Here 5/4 is mapped to 12\37, and 10/9 is mapped to 5\37. Equating both 11/10 and 12/11 with 10/9 makes the mappings for 9/8, 10/9, 11/10, and 12/11 add up to 3/2. If the fifth was any sharper, then 7/4 and 15/8 would have to be flatter. Then 5/4 would have to be flatter, and therefore 10/9 as well, and at least one of 11/10 and 12/11 would have to be mapped wider than 10/9 for 9/8, 10/9, 11/10, and 12/11 to add up to 3/2. 37edo is, in fact, diamond monotone in the 15-odd-limit (see Monotonicity limits of small EDOs). Therefore, 22\37 is the sharpest fifth where 15-odd-limit diamond monotone is possible. The flattest fifth where 15-odd-limit diamond monotone is possible is 11\19.

Intervals

Inconsistent intervals are in italics.

# Cents Approximate ratios
of 2.27.5.7.11.13 subgroup 		Additional ratios of 3
with a sharp 3/2 		Additional ratios of 3
with a flat 3/2 		Additional ratios of 9
with 194.59 ¢ 9/8
0 0.0 1/1
1 32.4 55/54, 56/55
2 64.9 27/26, 28/27
3 97.3 128/121, 55/52 16/15
4 129.7 14/13 13/12, 15/14 12/11
5 162.2 11/10 10/9, 12/11 13/12
6 194.6 28/25 9/8, 10/9
7 227.0 8/7 9/8
8 259.5 7/6, 15/13
9 291.9 13/11, 32/27 6/5, 7/6
10 324.3 6/5, 11/9
11 356.8 16/13, 27/22 11/9
12 389.2 5/4
13 421.6 14/11, 32/25 9/7
14 454.1 13/10 9/7
15 486.5 4/3
16 518.9 27/20 4/3
17 551.4 11/8 15/11 18/13
18 583.8 7/5 18/13
19 616.2 10/7 13/9
20 648.6 16/11 22/15 13/9
21 681.1 40/27 3/2
22 713.5 3/2
23 745.9 20/13 14/9
24 778.4 11/7, 25/16 14/9
25 810.8 8/5
26 843.2 13/8, 44/27 18/11
27 875.7 5/3, 18/11
28 908.1 22/13, 27/16 5/3, 12/7
29 940.5 12/7, 26/15
30 973.0 7/4 16/9
31 1005.4 25/14 16/9, 9/5
32 1037.8 20/11 9/5, 11/6
33 1070.3 13/7 24/13, 28/15 11/6
34 1102.7 121/64, 104/55 15/8
35 1135.1 27/14, 52/27
36 1167.6
37 1200.0 2/1

Proposed interval names and solfèges

Table of proposed interval names and solfèges
# Cents Ups and downs notation
0 0.0 Perfect 1sn P1 D
1 32.4 Minor 2nd m2 Eb
2 64.9 Upminor 2nd ^m2 ^Eb
3 97.3 Downmid 2nd v~2 ^^Eb
4 129.7 Mid 2nd ~2 Ed
5 162.2 Upmid 2nd ^~2 vvE
6 194.6 Downmajor 2nd vM2 vE
7 227.0 Major 2nd M2 E
8 259.5 Minor 3rd m3 F
9 291.9 Upminor 3rd ^m3 ^F
10 324.3 Downmid 3rd v~3 ^^F
11 356.8 Mid 3rd ~3 Ft
12 389.2 Upmid 3rd ^~3 vvF#
13 421.6 Downmajor 3rd vM3 vF#
14 454.1 Major 3rd M3 F#
15 486.5 Perfect 4th P4 G
16 518.9 Up 4th, dim 5th ^4, d5 ^G, Ab
17 551.4 Downmid 4th, updim 5th v~4, ^d5 ^^G, ^Ab
18 583.8 Mid 4th, downmid 5th ~4, v~5 Gt, ^^Ab
19 616.2 Mid 5th, upmid 4th ~5, ^~4 Ad, vvG#
20 648.6 Upmid 5th, downaug 5th ^~5, vA4 vvA, vG#
21 681.1 Down 5th, aug 4th v5, A4 vA, G#
22 713.5 Perfect 5th P5 A
23 745.9 Minor 6th m6 Bb
24 778.4 Upminor 6th ^m6 ^Bb
25 810.8 Downmid 6th v~6 ^^Bb
26 843.2 Mid 6th ~6 Bd
27 875.7 Upmid 6th ^~6 vvB
28 908.1 Downmajor 6th vM6 vB
29 940.5 Major 6th M6 B
30 973.0 Minor 7th m7 C
31 1005.4 Upminor 7th ^m7 ^C
32 1037.8 Downmid 7th v~7 ^^C
33 1070.3 Mid 7th ~7 Ct
34 1102.7 Upmid 7th ^~7 vvC#
35 1135.1 Downmajor 7th vM7 vC#
36 1167.6 Major 7th M7 C#
37 1200.0 Perfect 8ve P8 D

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats combined with quartertone accidentals and arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol
Flat symbol

If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
Flat symbol

Kite's ups and downs notation

37edo can also be notated using Kite's ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
         
             
Flat symbol          
             

Half-sharps and half-flats can be used to avoid triple arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
 
    
    
    
Flat symbol  
    
    
    

Ivan Wyschnegradsky's notation

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

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 
  
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 

Sagittal notation

This notation uses the same sagittal sequence as edos 23b, 30, and 44.

Evo and Revo flavors

Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Alternative Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Approximation to JI

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 37edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 37edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/8, 16/11 0.033 0.1
13/10, 20/13 0.160 0.5
7/5, 10/7 1.272 3.9
13/7, 14/13 1.431 4.4
13/11, 22/13 2.682 8.3
13/8, 16/13 2.716 8.4
11/10, 20/11 2.842 8.8
5/4, 8/5 2.875 8.9
11/7, 14/11 4.114 12.7
7/4, 8/7 4.147 12.8
7/6, 12/7 7.411 22.9
5/3, 6/5 8.683 26.8
13/12, 24/13 8.843 27.3
9/8, 16/9 9.315 28.7
11/9, 18/11 9.349 28.8
15/14, 28/15 10.287 31.7
11/6, 12/11 11.525 35.5
3/2, 4/3 11.559 35.6
15/13, 26/15 11.718 36.1
13/9, 18/13 12.031 37.1
9/5, 10/9 12.191 37.6
9/7, 14/9 13.462 41.5
15/11, 22/15 14.401 44.4
15/8, 16/15 14.434 44.5
15-odd-limit intervals in 37edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/8, 16/11 0.033 0.1
13/10, 20/13 0.160 0.5
7/5, 10/7 1.272 3.9
13/7, 14/13 1.431 4.4
13/11, 22/13 2.682 8.3
13/8, 16/13 2.716 8.4
11/10, 20/11 2.842 8.8
5/4, 8/5 2.875 8.9
11/7, 14/11 4.114 12.7
7/4, 8/7 4.147 12.8
7/6, 12/7 7.411 22.9
5/3, 6/5 8.683 26.8
13/12, 24/13 8.843 27.3
15/14, 28/15 10.287 31.7
11/6, 12/11 11.525 35.5
3/2, 4/3 11.559 35.6
15/13, 26/15 11.718 36.1
15/11, 22/15 14.401 44.4
15/8, 16/15 14.434 44.5
9/7, 14/9 18.970 58.5
9/5, 10/9 20.242 62.4
13/9, 18/13 20.401 62.9
11/9, 18/11 23.084 71.2
9/8, 16/9 23.117 71.3

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.5 [86 -37⟩ [⟨37 86]] −0.619 0.619 1.91
2.5.7 3136/3125, 4194304/4117715 [⟨37 86 104]] −0.905 0.647 2.00
2.5.7.11 176/175, 1375/1372, 65536/65219 [⟨37 86 104 128]] −0.681 0.681 2.10
2.5.7.11.13 176/175, 640/637, 847/845, 1375/1372 [⟨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

Generator* Cents* In patent val In 37b val
1\37 32.4
2\37 64.9 Sycamore
3\37 97.3 Passion
4\37 129.7 Twothirdtonic Negri (37bd, out-of-tune)
5\37 162.2 Porcupine / porcupinefish
6\37 194.6 Hemiwürschmidt / hemiwur Hemithirds (37b, out-of-tune)
7\37 227.0 Semaja / gorgik Gorgo (37b)
8\37 259.5 Semaphore (37bd, out-of-tune)
9\37 291.9 Quasitemp
10\37 324.3 Hyperkleismic Superkleismic (37bc, out-of-tune)
11\37 356.8 Beatles
12\37 389.2 Würschmidt (out-of-tune)
13\37 421.6 Skwares (37dd, out-of-tune)
14\37 454.1 Ammonite
15\37 486.5 Ultrapyth
16\37 518.9 Undecimation Shallowtone (37b)
17\37 551.4 Freivald, emka
18\37 583.8 Cotritone

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Octave stretch or compression

37edo's primes 3, 5, 7, 11 and 13 are all tuned sharp, so it can benefit from octave shrinking. Some compressed-octave 37edo tunings (least to most compressed) include 161zpi, 86ed5, 104ed7, 133ed12 or 96ed6.

Scales

See also: MOS Scales of 37edo, Roulette scales

MOS scales

  • Ammonite[21]: 1 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 3 1
  • Beatles[7]: 4 7 4 7 4 7 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
  • Ultrapyth[5] (quasi-equipentatonic): 7 8 7 8 7 (recommended mode: 8 7 7 8 7)
  • Ultrapyth[7]: 7 1 7 7 7 1 7
  • Ultrapyth[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)
  • 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)
  • 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[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[6]: 12 5 5 5 5 5
  • Porcupine[7]: 5 5 5 7 5 5 5
  • 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
  • Twothirdtonic[7]: 13 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[19]: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1

Scales by individuals

Budjarn Lambeth's scales
Contains idiosyncratic terms.



  • 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)


Equally spaced scales

Every 8 steps of 37edo

Degrees Cents Approximate Ratios
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
Fretted instruments

Music

Modern renderings

Alessandro Marcello and Johann Sebastian Bach
Pietro Domenico Paradies

21st century

Beheld
Bryan Deister
Francium
groundfault
  • From Souvenirs of the Affliction (2025) – Bandcamp | YouTube
    • "The Life Unreachable"
    • "Not This Time"
Andrew Heathwaite
  • From Newbeams (2012)
    • "Shorn Brown"
    • "Jellybear"
Aaron Krister Johnson
JUMBLE
Fitzgerald Lee
Mandrake
Claudi Meneghin
Micronaive
Herman Miller
Joseph Monzo
Mundoworld
Ray Perlner
Phanomium
Togenom
Uncreative Name
Stephen Weigel
Xeno*n*

See also

External links

Retrieved from "https://en.xen.wiki/index.php?title=37edo&oldid=230133"