37edo: Difference between revisions

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== Theory ==
== Theory ==
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.
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]]{{clarify}}. <!-- what is the metric being used? -->
=== 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 62: Line 55:
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 ===
=== Miscellaneous properties ===
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.
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 83: Line 77:
|-
|-
| 1
| 1
| 32.43
| 32.4
| [[55/54]], [[56/55]]
| [[55/54]], [[56/55]]
|
|
Line 90: Line 84:
|-
|-
| 2
| 2
| 64.86
| 64.9
| [[27/26]], [[28/27]]
| [[27/26]], [[28/27]]
|
|
Line 97: Line 91:
|-
|-
| 3
| 3
| 97.30
| 97.3
| [[128/121]], [[55/52]]
| [[128/121]], [[55/52]]
| [[16/15]]
| [[16/15]]
Line 104: 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 125: 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 139: 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 160: Line 154:
|-
|-
| 12
| 12
| 389.19
| 389.2
| [[5/4]]
| [[5/4]]
|
|
Line 167: Line 161:
|-
|-
| 13
| 13
| 421.62
| 421.6
| [[14/11]], [[32/25]]
| [[14/11]], [[32/25]]
|
|
Line 174: 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 188: 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 202: 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 223: 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 237: 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 251: Line 245:
|-
|-
| 25
| 25
| 810.81
| 810.8
| [[8/5]]
| [[8/5]]
|
|
Line 258: Line 252:
|-
|-
| 26
| 26
| 843.24
| 843.2
| [[13/8]], [[44/27]]
| [[13/8]], [[44/27]]
|
|
Line 265: 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 286: 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 300: 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 321: Line 315:
|-
|-
| 35
| 35
| 1135.14
| 1135.1
| [[27/14]], [[52/27]]
| [[27/14]], [[52/27]]
|
|
Line 328: Line 322:
|-
|-
| 36
| 36
| 1167.57
| 1167.6
|
|
|
|
Line 335: Line 329:
|-
|-
| 37
| 37
| 1200.00
| 1200.0
| [[2/1]]
| [[2/1]]
|
|
Line 342: Line 336:
|}
|}


== Notation ==
=== Proposed interval names and solfèges ===
=== Ups and downs notation ===
{| class="wikitable center-all right-2 left-3 mw-collapsible mw-collapsed"
37edo can be notated using [[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 359: Line 351:
|-
|-
| 1
| 1
| 32.43
| 32.4
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 365: Line 357:
|-
|-
| 2
| 2
| 64.86
| 64.9
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 371: Line 363:
|-
|-
| 3
| 3
| 97.30
| 97.3
| Downmid 2nd
| Downmid 2nd
| v~2
| v~2
Line 377: Line 369:
|-
|-
| 4
| 4
| 129.73
| 129.7
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 383: Line 375:
|-
|-
| 5
| 5
| 162.16
| 162.2
| Upmid 2nd
| Upmid 2nd
| ^~2
| ^~2
Line 389: Line 381:
|-
|-
| 6
| 6
| 194.59
| 194.6
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 395: Line 387:
|-
|-
| 7
| 7
| 227.03
| 227.0
| Major 2nd
| Major 2nd
| M2
| M2
Line 401: Line 393:
|-
|-
| 8
| 8
| 259.46
| 259.5
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 407: Line 399:
|-
|-
| 9
| 9
| 291.89
| 291.9
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 413: Line 405:
|-
|-
| 10
| 10
| 324.32
| 324.3
| Downmid 3rd
| Downmid 3rd
| v~3
| v~3
Line 419: Line 411:
|-
|-
| 11
| 11
| 356.76
| 356.8
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 425: Line 417:
|-
|-
| 12
| 12
| 389.19
| 389.2
| Upmid 3rd
| Upmid 3rd
| ^~3
| ^~3
Line 431: Line 423:
|-
|-
| 13
| 13
| 421.62
| 421.6
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 437: Line 429:
|-
|-
| 14
| 14
| 454.05
| 454.1
| Major 3rd
| Major 3rd
| M3
| M3
Line 443: Line 435:
|-
|-
| 15
| 15
| 486.49
| 486.5
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 449: 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 491: Line 483:
|-
|-
| 23
| 23
| 745.95
| 745.9
| Minor 6th
| Minor 6th
| m6
| m6
Line 497: Line 489:
|-
|-
| 24
| 24
| 778.38
| 778.4
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 503: Line 495:
|-
|-
| 25
| 25
| 810.81
| 810.8
| Downmid 6th
| Downmid 6th
| v~6
| v~6
Line 509: Line 501:
|-
|-
| 26
| 26
| 843.24
| 843.2
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 515: Line 507:
|-
|-
| 27
| 27
| 875.68
| 875.7
| Upmid 6th
| Upmid 6th
| ^~6
| ^~6
Line 521: Line 513:
|-
|-
| 28
| 28
| 908.11
| 908.1
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 527: Line 519:
|-
|-
| 29
| 29
| 940.54
| 940.5
| Major 6th
| Major 6th
| M6
| M6
Line 533: Line 525:
|-
|-
| 30
| 30
| 972.97
| 973.0
| Minor 7th
| Minor 7th
| m7
| m7
Line 539: Line 531:
|-
|-
| 31
| 31
| 1005.41
| 1005.4
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 545: Line 537:
|-
|-
| 32
| 32
| 1037.84
| 1037.8
| Downmid 7th
| Downmid 7th
| v~7
| v~7
Line 551: Line 543:
|-
|-
| 33
| 33
| 1070.27
| 1070.3
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 557: Line 549:
|-
|-
| 34
| 34
| 1102.70
| 1102.7
| Upmid 7th
| Upmid 7th
| ^~7
| ^~7
Line 563: Line 555:
|-
|-
| 35
| 35
| 1135.14
| 1135.1
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 569: Line 561:
|-
|-
| 36
| 36
| 1167.57
| 1167.6
| Major 7th
| Major 7th
| M7
| M7
Line 575: Line 567:
|-
|-
| 37
| 37
| 1200.00
| 1200.0
| Perfect 8ve
| Perfect 8ve
| P8
| P8
Line 581: Line 573:
|}
|}


37edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.  
== Notation ==
=== 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}}
{{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}}


=== Sagittal notation ===
=== 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]].
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 ====
Line 630: 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 ==
Line 644: 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
Line 652: 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
Line 659: 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
Line 666: 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
Line 676: 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]] / [[hemiwur]] / [[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 ==
== Octave stretch or compression ==
Line 760: Line 779:
''See also: [[MOS Scales of 37edo]], [[Chromatic pairs#Roulette|Roulette scales]]''
''See also: [[MOS Scales of 37edo]], [[Chromatic pairs#Roulette|Roulette scales]]''


* [[37ed4]]
=== [[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
* [[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
Line 780: Line 797:
* 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]]
* 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: [[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 ===
Line 919: Line 979:


== 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)


; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/TEzitpGJvt0 ''37edo''] (2023)
* [https://www.youtube.com/shorts/e7dLJTsS3PQ ''37edo''] (2025)
* [https://www.youtube.com/shorts/e7dLJTsS3PQ ''37edo''] (2025)
* [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/Jt6_r6r3lGY ''37edo improv''] (2026)


; [[Francium]]
; [[Francium]]
Line 930: Line 1,001:
* [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]]
Line 941: Line 1,018:
* [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)


Line 949: 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) – 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]]
Line 965: 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]]
Line 983: Line 1,058:


; [[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 ==

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

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