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| Augmented 1sn = 6\37 = 195¢
| Augmented 1sn = 6\37 = 195¢
}}
}}
'''37edo''' is a scale derived from dividing the octave into 37 equal steps. It is the 12th [[prime_numbers|prime]] edo, following [[31edo]] and coming before [[41edo]].
'''37EDO''' is a scale derived from dividing the octave into 37 equal steps. It is the 12th [[prime_numbers|prime]] EDO, following [[31edo|31EDO]] and coming before [[41edo|41EDO]].


== Theory ==
== Theory ==
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|}
|}


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" will be. Using its alternative flat fifth, it tempers out 16875/16384, making it a [[Negri|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" will be. Using its alternative flat fifth, it tempers out 16875/16384, making it a [[Negri|negri]] tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth ([[gorgo]]/[[laconic]]).


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 nonatonic MOS, which in 37-edo 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 nonatonic 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.


=== Subgroups ===
=== Subgroups ===
37edo offers close approximations to [[Overtone series|harmonics]] 5, 7, 11, and 13 [and a usable approximation of 9 as well].
37EDO offers close approximations to [[Overtone series|harmonics]] 5, 7, 11, and 13 [and a usable approximation of 9 as well].


12\37 = 389.2 cents
12\37 = 389.2 cents
Line 73: Line 73:
26\37 = 843.2 cents
26\37 = 843.2 cents


[6\37edo = 194.6 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 111et. 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 111et, 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 74et.
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.


=== The Two Fifths ===
=== The Two Fifths ===
The just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo:
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 flat fifth is 21\37 = 681.1 cents (37b val)
Line 98: Line 98:
If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of [[The_Biosphere|Biome]] temperament.
If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of [[The_Biosphere|Biome]] temperament.


Interestingly, the "major thirds" of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in 37edo.
Interestingly, 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).
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).


== Intervals ==
== Intervals ==
Line 122: Line 122:
| 1
| 1
| 32.43
| 32.43
|
| [[55/54]], [[56/55]]
|
|
|
|
Line 137: Line 137:
| 97.30
| 97.30
| [[55/52]]
| [[55/52]]
|
| [[16/15]]
|
|
|
|
Line 144: Line 144:
| 129.73
| 129.73
| [[14/13]]
| [[14/13]]
| [[13/12]]
| [[13/12]], [[15/14]]
| [[12/11]]
| [[12/11]]
|
|
Line 172: Line 172:
| 259.46
| 259.46
|
|
| [[7/6]]
| [[7/6]], [[15/13]]
|
|
|
|
Line 235: Line 235:
| 551.35
| 551.35
| [[11/8]]
| [[11/8]]
|
| [[15/11]]
|
|
| [[18/13]]
| [[18/13]]
Line 256: Line 256:
| 648.65
| 648.65
| [[16/11]]
| [[16/11]]
|
| [[22/15]]
|
|
| [[13/9]]
| [[13/9]]
Line 319: Line 319:
| 940.54
| 940.54
|
|
| [[12/7]]
| [[12/7]], [[26/15]]
|
|
|
|
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| 1070.27
| 1070.27
| [[13/7]]
| [[13/7]]
| [[24/13]]
| [[24/13]], [[28/15]]
| [[11/6]]
| [[11/6]]
|
|
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| 1102.70
| 1102.70
| [[104/55]]
| [[104/55]]
|
| [[15/8]]
|
|
|
|
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=== Temperament measures ===
=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 37et.  
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 37EDO.  
{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="2" |
! colspan="2" |
Line 671: Line 671:
|}
|}


* 37et is most prominent in the no-3 11-, 13-, 17-, 19- and 23-limit subgroups. The next ET that does better in these subgroups is 109, 581, 103, 124 and 93, respectively.  
* 37EDO is most prominent in the no-3 11-, 13-, 17-, 19- and 23-limit subgroups. The next EDO that does better in these subgroups is 109, 581, 103, 124 and 93, respectively.  


== Scales ==
== Scales ==
* [[MOS_Scales_of_37edo|MOS Scales of 37edo]]
* [[MOS_Scales_of_37edo|MOS Scales of 37edo]]
* [[roulette6]]
* [[roulette6]]
Line 710: Line 709:
|-
|-
| 5\37
| 5\37
| [[Porcupine]]/[[The_Biosphere#Oceanfront-Oceanfront Children-Porcupinefish|porcupinefish]]
| [[Porcupine]]/[[The Biosphere #Oceanfront-Oceanfront Children-Porcupinefish|porcupinefish]]
|  
|  
|-
|-
| 6\37
| 6\37
| colspan="2" | [[Chromatic_pairs#Roulette|Roulette]]
| colspan="2" | [[Chromatic pairs #Roulette|Roulette]]
|-
|-
| 7\37
| 7\37
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| 9\37
| 9\37
|  
|  
| [[Chromatic_pairs#Gariberttet|Gariberttet]]
| [[Chromatic pairs #Gariberttet|Gariberttet]]
|-
|-
| 10\37
| 10\37
Line 741: Line 740:
|-
|-
| 13\37
| 13\37
| [[Meantone_family#Squares|Squares]]
| [[Meantone family #Squares|Squares]]
|  
|  
|-
|-
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|-
|-
| 15\37
| 15\37
| [[The_Biosphere#Oceanfront-Oceanfront Children-Ultrapyth|Ultrapyth]], '''not''' [[superpyth]]
| [[The Biosphere#Oceanfront-Oceanfront Children-Ultrapyth|Ultrapyth]], '''not''' [[superpyth]]
|  
|  
|-
|-
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== Music ==
== Music ==
* [http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3 Toccata Bianca 37edo] by [http://www.akjmusic.com/ Aaron Krister Johnson]
* [http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3 Toccata Bianca 37EDO] by [http://www.akjmusic.com/ Aaron Krister Johnson]
* [http://andrewheathwaite.bandcamp.com/track/shorn-brown Shorn Brown] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2002%20Shorn%20Brown.mp3 play] and [http://andrewheathwaite.bandcamp.com/track/jellybear Jellybear] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2003%20Jellybear.mp3 play] by [[Andrew Heathwaite]]
* [http://andrewheathwaite.bandcamp.com/track/shorn-brown Shorn Brown] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2002%20Shorn%20Brown.mp3 play] and [http://andrewheathwaite.bandcamp.com/track/jellybear Jellybear] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2003%20Jellybear.mp3 play] by [[Andrew Heathwaite]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Monzo/monzo_kog-sisters_2014-0405.mp3 The Kog Sisters] by [[Joe Monzo]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Monzo/monzo_kog-sisters_2014-0405.mp3 The Kog Sisters] by [[Joe Monzo]]
Line 772: Line 771:


== Links ==
== Links ==
* [http://tonalsoft.com/enc/number/37-edo/37edo.aspx 37edo at Tonalsoft] [[Category:37edo| ]] <!-- main article -->
* [http://tonalsoft.com/enc/number/37-edo/37edo.aspx 37edo at Tonalsoft]
[[Category:37edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Prime EDO]]
[[Category:Subgroup]]
[[Category:Subgroup]]

Revision as of 10:09, 21 August 2021

Deutsch

← 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

37EDO is a scale derived from dividing the octave into 37 equal steps. It is the 12th prime EDO, following 31EDO and coming before 41EDO.

Theory

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19 prime 23
Error absolute (¢) 0.0 +11.6 +2.9 +4.1 +0.0 +2.7 -7.7 -5.6 -12.1
relative (%) 0 +36 +9 +13 +0 +8 -24 -17 -37
nearest edomapping 37 22 12 30 17 26 3 9 19

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" 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).

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 nonatonic 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.

Subgroups

37EDO offers close approximations to 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 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.

The Two 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 Biome temperament.

Interestingly, 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).

Intervals

Degrees Cents Approximate Ratios
of 2.5.7.11.13.27 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 0.00 1/1
1 32.43 55/54, 56/55
2 64.86 27/26, 28/27
3 97.30 55/52 16/15
4 129.73 14/13 13/12, 15/14 12/11
5 162.16 11/10 10/9, 12/11 13/12
6 194.59 9/8, 10/9
7 227.03 8/7 9/8
8 259.46 7/6, 15/13
9 291.89 13/11, 32/27 6/5, 7/6
10 324.32 6/5, 11/9
11 356.76 16/13, 27/22 11/9
12 389.19 5/4
13 421.62 14/11 9/7
14 454.05 13/10 9/7
15 486.49 4/3
16 518.92 27/20 4/3
17 551.35 11/8 15/11 18/13
18 583.78 7/5 18/13
19 616.22 10/7 13/9
20 648.65 16/11 22/15 13/9
21 681.08 40/27 3/2
22 713.51 3/2
23 745.95 20/13 14/9
24 778.38 11/7 14/9
25 810.81 8/5
26 843.24 13/8, 44/27 18/11
27 875.68 5/3, 18/11
28 908.11 22/13, 27/16 5/3, 12/7
29 940.54 12/7, 26/15
30 972.97 7/4 16/9
31 1005.41 16/9, 9/5
32 1037.84 11/6 9/5, 11/6
33 1070.27 13/7 24/13, 28/15 11/6
34 1102.70 104/55 15/8
35 1135.14 27/14, 52/27
36 1167.57
37 1200.00 2/1

Notation

Degrees Cents Ups and Downs Notation
0 0.00 Perfect 1sn P1 D
1 32.43 Minor 2nd m2 Eb
2 64.86 Upminor 2nd ^m2 ^Eb
3 97.30 Downmid 2nd v~2 ^^Eb
4 129.73 Mid 2nd ~2 Ed
5 162.16 Upmid 2nd ^~2 vvE
6 194.59 Downmajor 2nd vM2 vE
7 227.03 Major 2nd M2 E
8 259.46 Minor 3rd m3 F
9 291.89 Upminor 3rd ^m3 ^F
10 324.32 Downmid 3rd v~3 ^^F
11 356.76 Mid 3rd ~3 Ft
12 389.19 Upmid 3rd ^~3 vvF#
13 421.62 Downmajor 3rd vM3 vF#
14 454.05 Major 3rd M3 F#
15 486.49 Perfect 4th P4 G
16 518.92 Up 4th, Dim 5th ^4, d5 ^G, Ab
17 551.35 Downmid 4th, Updim 5th v~4, ^d5 ^^G, ^Ab
18 583.78 Mid 4th, Downmid 5th ~4, v~5 Gt, ^^Ab
19 616.22 Mid 5th, Upmid 4th ~5, ^~4 Ad, vvG#
20 648.65 Upmid 5th, Downaug 5th ^~5, vA4 vvA, vG#
21 681.08 Down 5th, Aug 4th v5, A4 vA, G#
22 713.51 Perfect 5th P5 A
23 745.95 Minor 6th m6 Bb
24 778.38 Upminor 6th ^m6 ^Bb
25 810.81 Downmid 6th v~6 ^^Bb
26 843.24 Mid 6th ~6 Bd
27 875.68 Upmid 6th ^~6 vvB
28 908.11 Downmajor 6th vM6 vB
29 940.54 Major 6th M6 B
30 972.97 Minor 7th m7 C
31 1005.41 Upminor 7th ^m7 ^C
32 1037.84 Downmid 7th v~7 ^^C
33 1070.27 Mid 7th ~7 Ct
34 1102.70 Upmid 7th ^~7 vvC#
35 1135.14 Downmajor 7th vM7 vC#
36 1167.57 Major 7th M7 C#
37 1200.00 Perfect 8ve P8 D

Just approximation

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 37EDO.

3-limit 5-limit 7-limit 11-limit 13-limit no-3 11-limit no-3 13-limit no-3 17-limit no-3 19-limit no-3 23-limit
Octave stretch (¢) -3.65 -2.85 -2.50 -2.00 -1.79 -0.681 -0.692 -0.265 -0.0386 +0.299
Error absolute (¢) 3.64 3.18 2.82 2.71 2.52 0.681 0.610 1.11 1.17 1.41
relative (%) 11.24 9.82 8.70 8.37 7.78 2.10 1.88 3.41 3.59 4.35
  • 37EDO is most prominent in the no-3 11-, 13-, 17-, 19- and 23-limit subgroups. The next EDO that does better in these subgroups is 109, 581, 103, 124 and 93, respectively.

Scales

Linear temperaments

Generator "Sharp 3/2" temperaments "Flat 3/2" temperaments (37b val)
1\37
2\37 Sycamore
3\37 Passion
4\37 Twothirdtonic Negri
5\37 Porcupine/porcupinefish
6\37 Roulette
7\37 Semaja Gorgo/Laconic
8\37 Semiphore
9\37 Gariberttet
10\37 Orgone
11\37 Beatles
12\37 Würschmidt (out-of-tune)
13\37 Squares
14\37 Ammonite
15\37 Ultrapyth, not superpyth
16\37 Not mavila (this is "undecimation")
17\37 Emka
18\37

Music

Links

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