37edo: Difference between revisions

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37 edo 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.
37 edo 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.
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! | Degrees of 37edo
! | Degrees of 37edo
! | Cents Value
! | Cents Value
!pions
!7mus
! | Approximate Ratios
! | Approximate Ratios


Line 76: Line 79:
|-
|-
| | 0
| | 0
| | 0.00
| colspan="3"| 0.00
| | 1/1
| | 1/1
| |  
| |  
Line 85: Line 88:
| | 1
| | 1
| | 32.43
| | 32.43
|34.38
|41.51 (29.83<sub>16</sub>)
| |  
| |  
| |  
| |  
Line 92: Line 97:
|-
|-
| | 2
| | 2
| | 64.86
| | 64.865
|68.76
|83.03 (53.07<sub>16</sub>)
| | 28/27, 27/26
| | 28/27, 27/26
| |  
| |  
Line 100: Line 107:
|-
|-
| | 3
| | 3
| | 97.30
| | 97.3
|103.135
|124.54 (7C.8B<sub>16</sub>)
| |  
| |  
| |  
| |  
Line 109: Line 118:
| | 4
| | 4
| | 129.73
| | 129.73
|137.51
|166.05 (A6.0E<sub>16</sub>)
| | 14/13
| | 14/13
| | 13/12
| | 13/12
Line 117: Line 128:
| | 5
| | 5
| | 162.16
| | 162.16
|171.89
|207.57 (CF.91<sub>16</sub>)
| | 11/10
| | 11/10
| | 12/11
| | 12/11
Line 124: Line 137:
|-
|-
| | 6
| | 6
| | 194.59
| | 194.595
|206.27
|249.08 (F9.14<sub>16</sub>)
| |  
| |  
| |  
| |  
Line 133: Line 148:
| | 7
| | 7
| | 227.03
| | 227.03
|240.65
|290.595 (122.98<sub>16</sub>)
| | 8/7
| | 8/7
| |  
| |  
Line 141: Line 158:
| | 8
| | 8
| | 259.46
| | 259.46
|275.03
|332.11 (14C.1C<sub>16</sub>)
| |  
| |  
| | 7/6
| | 7/6
Line 149: Line 168:
| | 9
| | 9
| | 291.89
| | 291.89
|309.405
|373.63 (175.9F<sub>16</sub>)
| | 13/11, 32/27
| | 13/11, 32/27
| |  
| |  
Line 157: Line 178:
| | 10
| | 10
| | 324.32
| | 324.32
|343.78
|415.135 (19F.23<sub>16</sub>)
| |  
| |  
| | 6/5
| | 6/5
Line 165: Line 188:
| | 11
| | 11
| | 356.76
| | 356.76
|378.16
|456.65 (1C8.A6<sub>16</sub>)
| | 16/13, 27/22
| | 16/13, 27/22
| |  
| |  
Line 173: Line 198:
| | 12
| | 12
| | 389.19
| | 389.19
|412.54
|498.16 (1F2.298<sub>16</sub>)
| | 5/4
| | 5/4
| |  
| |  
Line 181: Line 208:
| | 13
| | 13
| | 421.62
| | 421.62
|446.92
|539.68 (21B.AD<sub>16</sub>)
| | 14/11
| | 14/11
| |  
| |  
Line 189: Line 218:
| | 14
| | 14
| | 454.05
| | 454.05
|481.3
|581.19 (245.3<sub>16</sub>)
| | 13/10
| | 13/10
| |  
| |  
Line 197: Line 228:
| | 15
| | 15
| | 486.49
| | 486.49
|515.68
|622.7 (26E.B4<sub>16</sub>)
| |  
| |  
| | 4/3
| | 4/3
Line 205: Line 238:
| | 16
| | 16
| | 518.92
| | 518.92
|550.05
|664.22 (298.37<sub>16</sub>)
| | 27/20
| | 27/20
| |  
| |  
Line 213: Line 248:
| | 17
| | 17
| | 551.35
| | 551.35
|584.43
|705.73 (2C1.BB<sub>16</sub>)
| | 11/8
| | 11/8
| |  
| |  
Line 221: Line 258:
| | 18
| | 18
| | 583.78
| | 583.78
|618.81
|747.24 (2EB.3E<sub>16</sub>)
| | 7/5
| | 7/5
| |  
| |  
Line 229: Line 268:
| | 19
| | 19
| | 616.22
| | 616.22
|653.19
|788.76 (314.C2<sub>16</sub>)
| | 10/7
| | 10/7
| |  
| |  
Line 237: Line 278:
| | 20
| | 20
| | 648.65
| | 648.65
|687.57
|830.27 (33E.45<sub>16</sub>)
| | 16/11
| | 16/11
| |  
| |  
Line 245: Line 288:
| | 21
| | 21
| | 681.08
| | 681.08
|721.95
|871.78 (367.C9<sub>16</sub>)
| | 40/27
| | 40/27
| |  
| |  
Line 253: Line 298:
| | 22
| | 22
| | 713.51
| | 713.51
|756.32
|913.3 (391.4C<sub>16</sub>)
| |  
| |  
| | 3/2
| | 3/2
Line 261: Line 308:
| | 23
| | 23
| | 745.95
| | 745.95
|790.7
|954.81 (3BA.D<sub>16</sub>)
| | 20/13
| | 20/13
| |  
| |  
Line 269: Line 318:
| | 24
| | 24
| | 778.38
| | 778.38
|825.08
|996.32 (3E4.53<sub>16</sub>)
| | 11/7
| | 11/7
| |  
| |  
Line 277: Line 328:
| | 25
| | 25
| | 810.81
| | 810.81
|859.46
|1037.84 (40D.D68<sub>16</sub>)
| | 8/5
| | 8/5
| |  
| |  
Line 285: Line 338:
| | 26
| | 26
| | 843.24
| | 843.24
|893.84
|1079.35 (437.56<sub>16</sub>)
| | 13/8, 44/27
| | 13/8, 44/27
| |  
| |  
Line 293: Line 348:
| | 27
| | 27
| | 875.68
| | 875.68
|928.22
|1120.865 (460.DE<sub>16</sub>)
| |  
| |  
| | 5/3
| | 5/3
Line 301: Line 358:
| | 28
| | 28
| | 908.11
| | 908.11
|962.595
|1162.38 (48A.61<sub>16</sub>)
| | 22/13, 27/16
| | 22/13, 27/16
| |  
| |  
Line 309: Line 368:
| | 29
| | 29
| | 940.54
| | 940.54
|996.97
|1203.89 (4B3.E4<sub>16</sub>)
| |  
| |  
| | 12/7
| | 12/7
Line 317: Line 378:
| | 30
| | 30
| | 972.97
| | 972.97
|1031.35
|1245.405 (4DD.68<sub>16</sub>)
| | 7/4
| | 7/4
| |  
| |  
Line 324: Line 387:
|-
|-
| | 31
| | 31
| | 1005.41
| | 1005.405
|1065.73
|1286.92 (506.EB<sub>16</sub>)
| |  
| |  
| |  
| |  
Line 333: Line 398:
| | 32
| | 32
| | 1037.84
| | 1037.84
| | 20/11
|1100.11
|1328.43 (530.6F<sub>16</sub>)
| | 11/6
| | 11/6
| | 24/13
| | 24/13
| |  
| |  
| | 9/5
| | 9/5
|
|-
|-
| | 33
| | 33
| | 1070.27
| | 1070.27
|1134.49
|1369.95 (559.F2<sub>16</sub>)
| | 13/7
| | 13/7
| | 24/13
| | 24/13
Line 348: Line 417:
|-
|-
| | 34
| | 34
| | 1102.70
| | 1102.7
|1168.865
|1411.46 (583.76<sub>16</sub>)
| |  
| |  
| |  
| |  
Line 356: Line 427:
|-
|-
| | 35
| | 35
| | 1135.14
| | 1135.135
|1203.24
|1452.97 (5AC.F9<sub>16</sub>)
| | 27/14, 52/27
| | 27/14, 52/27
| |  
| |  
Line 365: Line 438:
| | 36
| | 36
| | 1167.57
| | 1167.57
|1237.62
|1494.49 (5D6.7D<sub>16</sub>)
| |  
| |  
| |  
| |  
Line 370: Line 445:
| |  
| |  
| |  
| |  
|-
|3
|1200
|1272
|1536 (600<sub>16</sub>)
|2/
|
|
|
|
|}
|}


Revision as of 21:05, 1 April 2019

Deutsch

37edo is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. It is the 12th prime edo, following 31edo and coming before 41edo.

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).

37 edo 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.


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\37edo = 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 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.

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 of 37edo Cents Value pions 7mus Approximate Ratios

of 2.5.7.11.13.27 subgroup

Ratios of 3 with

a sharp 3/2

Ratios of 3 with

a flat 3/2

Ratios of 9 with

194.59¢ 9/8

Ratios of 9 with

227.03¢ 9/8

(two sharp

3/2's)

0 0.00 1/1
1 32.43 34.38 41.51 (29.8316)
2 64.865 68.76 83.03 (53.0716) 28/27, 27/26
3 97.3 103.135 124.54 (7C.8B16)
4 129.73 137.51 166.05 (A6.0E16) 14/13 13/12 12/11
5 162.16 171.89 207.57 (CF.9116) 11/10 12/11 13/12 10/9
6 194.595 206.27 249.08 (F9.1416) 9/8, 10/9
7 227.03 240.65 290.595 (122.9816) 8/7 9/8
8 259.46 275.03 332.11 (14C.1C16) 7/6
9 291.89 309.405 373.63 (175.9F16) 13/11, 32/27 6/5, 7/6
10 324.32 343.78 415.135 (19F.2316) 6/5 11/9
11 356.76 378.16 456.65 (1C8.A616) 16/13, 27/22 11/9
12 389.19 412.54 498.16 (1F2.29816) 5/4
13 421.62 446.92 539.68 (21B.AD16) 14/11 9/7
14 454.05 481.3 581.19 (245.316) 13/10 9/7
15 486.49 515.68 622.7 (26E.B416) 4/3
16 518.92 550.05 664.22 (298.3716) 27/20 4/3
17 551.35 584.43 705.73 (2C1.BB16) 11/8 18/13
18 583.78 618.81 747.24 (2EB.3E16) 7/5 18/13
19 616.22 653.19 788.76 (314.C216) 10/7 13/9
20 648.65 687.57 830.27 (33E.4516) 16/11 13/9
21 681.08 721.95 871.78 (367.C916) 40/27 3/2
22 713.51 756.32 913.3 (391.4C16) 3/2
23 745.95 790.7 954.81 (3BA.D16) 20/13 14/9
24 778.38 825.08 996.32 (3E4.5316) 11/7 14/9
25 810.81 859.46 1037.84 (40D.D6816) 8/5
26 843.24 893.84 1079.35 (437.5616) 13/8, 44/27 18/11
27 875.68 928.22 1120.865 (460.DE16) 5/3 18/11
28 908.11 962.595 1162.38 (48A.6116) 22/13, 27/16 5/3, 12/7
29 940.54 996.97 1203.89 (4B3.E416) 12/7
30 972.97 1031.35 1245.405 (4DD.6816) 7/4 16/9
31 1005.405 1065.73 1286.92 (506.EB16) 16/9, 9/5
32 1037.84 1100.11 1328.43 (530.6F16) 11/6 24/13 9/5
33 1070.27 1134.49 1369.95 (559.F216) 13/7 24/13 11/6
34 1102.7 1168.865 1411.46 (583.7616)
35 1135.135 1203.24 1452.97 (5AC.F916) 27/14, 52/27
36 1167.57 1237.62 1494.49 (5D6.7D16)
3 1200 1272 1536 (60016) 2/

Scales

MOS Scales of 37edo

roulette6

roulette7

roulette13

roulette19

Shoe

37ED4

The Square Root of 13/10

Linear temperaments

List of 37et rank two temperaments by badness

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
10\37
11\37 Beatles
12\37 Würschmidt (out-of-tune)
13\37
14\37 Ammonite
15\37 Ultrapyth, not superpyth
16\37 Not mavila (this is "undecimation")
17\37 Emka
18\37

Music in 37edo

Toccata Bianca 37edo by Aaron Krister Johnson

Shorn Brown play and Jellybear play by Andrew Heathwaite

The Kog Sisters by Joe Monzo

Links

37edo at Tonalsoft

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