Xenharmonic Wiki

37edo: Difference between revisions

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{{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 ===
====Evo and Revo flavors====
[[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 592: Line 605:
</imagemap>
</imagemap>


====Alternative Evo flavor====
==== Alternative Evo flavor ====
 
<imagemap>
<imagemap>
File:37-EDO_Alternative_Evo_Sagittal.svg
File:37-EDO_Alternative_Evo_Sagittal.svg
Line 603: 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 613: 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 626: 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 634: 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 641: 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 648: 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 658: 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 762: 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 768: 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 784: 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 ==
Retrieved from "https://en.xen.wiki/w/37edo"