User:Unque/37edo Composition Theory: Difference between revisions

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{{breadcrumb|37edo}}
'''Note: This page is currently under construction, and will be subject to major expansion in the near future. Come back soon!'''
'''Note: This page is currently under construction, and will be subject to major expansion in the near future. Come back soon!'''


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{| class="wikitable"
{| class="wikitable"
|+
|+
Intervals of 37edo
!Intervals
!Intervals
!Cents
!Cents
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|1\37
|1\37
|32.43
|32.43
|[[55/54]], [[56/55]]
|[[55/54]]+, [[56/55]]
|
|
|D♭
|D♭
Line 32: Line 35:
|2\37
|2\37
|64.86
|64.86
|[[28/27]]
|[[28/27]]+
|[[Sycamore]]/[[Octaphore#2.3.7 Unicorn|Unicorn]]
|[[Sycamore]]
|Bｷ = E𝄫
|
|
|Sycamore uses diatonic fifth, Unicorn uses antidiatonic fifth
|-
|-
|3\37
|3\37
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|[[17/16]]
|[[17/16]]
|[[Passion]]
|[[Passion]]
|Cｷ
|Cｷ = F𝄫
|
|
|-
|-
|4\37
|4\37
|129.73
|129.73
|[[14/13]], [[16/15]]
|[[14/13]]
|[[Negri]]
|[[28812/28561#Cubical|Cubical]]
|
|A𝄪 = Dd
|Patent val 7 technically doesn't count as Negri, but it's the same structure
| Cubical uses antidiatonic fifth
|-
|-
|5\37
|5\37
|162.16
|162.16
|[[11/10|10/9]], [[9/8]]-
|[[11/10|10/9]]+, [[9/8]]-
|[[Porcupine]]
|[[Porcupine]]
|B♯
|B♯
|9/8 using antidiatonic fifth
|10/9 using diatonic fifth, or 9/8 using antidiatonic fifth
|-
|-
|6\37
|6\37
|194.59
|194.59
|[[19/17]], 9/8
|10/9, [[19/17]], 9/8
|[[Didacus]]
|[[Didacus]]
|C♯
|C♯
|9/8 using dual fifths
|10/9 = 9/8 using dual fifths
|-
|-
|7\37
|7\37
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|8\37
|8\37
|259.46
|259.46
|[[15/13]]
|[[15/13]]+
|[[Barbados]]
|[[Barbados]]
|E♭
|E♭
|
|Barbados utilizes split-3 shenanigans
|-
|-
|9\37
|9\37
|291.89
|291.89
|[[13/11]], [[19/16]]
|[[13/11]], [[19/16]]
|
|[[847/845|Cuthbert]]
|
|F♭
|
|
|-
|-
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|[[77/64]]
|[[77/64]]
|[[Orgone]]
|[[Orgone]]
|Dｷ
|Dｷ = G𝄫
|Not a good 6/5, but 77/64 is rather complex; I just think of it as half of 16/11
|Probably better interpreted as 8/(sqrt11)
|-
|-
|11\37
|11\37
|356.76
|356.76
|[[11/9]], [[16/13]]
|[[11/9]]+, [[16/13]]
|[[Beatles]]
|[[Beatles]]
|
|B𝄪 = Ed
|Bisects the diatonic fifth
|Bisects the diatonic fifth
|-
|-
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|[[5/4]]
|[[5/4]]
|[[Würschmidt|Wuerschmidt]]
|[[Würschmidt|Wuerschmidt]]
|
|C𝄪 = Fd
|
|
|-
|-
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|14\37
|14\37
|454.05
|454.05
|[[9/7]], [[13/10]]
|[[9/7]]+, [[13/10]]
|[[Ammonite]]
|[[Ammonite]]
|E
|E
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|4/3-
|4/3-
|[[Undecimation]]
|[[Undecimation]]
|
|G♭
|Antidiatonic fourth
|Antidiatonic fourth
|-
|-
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|[[11/8]]
|[[11/8]]
|[[Emka]]
|[[Emka]]
|Eｷ
|Eｷ = A𝄫
|
|
|-
|-
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|[[10/7]]
|[[10/7]]
|
|
|
|D𝄪 = Gd
|
|
|-
|-
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|[[20/13]], [[14/9]]
|[[20/13]], [[14/9]]
|Ammonite
|Ammonite
|
|A♭
|
|
|-
|-
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|[[11/7]]
|[[11/7]]
|Lambeth
|Lambeth
|
|B𝄫
|
|
|-
|-
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|[[8/5]]
|[[8/5]]
|Wuerschmidt
|Wuerschmidt
|Gｷ
|Gｷ = C𝄫
|
|
|-
|-
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|[[13/8]], [[18/11]]
|[[13/8]], [[18/11]]
|Beatles
|Beatles
|
|E𝄪 = Ad
|
|
|-
|-
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|[[128/77]]
|[[128/77]]
|Orgone
|Orgone
|
|F𝄪
|
|Probably better interpreted as (sqrt11)/2
|-
|-
|28\37
|28\37
|908.11
|908.11
|[[32/19]], [[22/13]]
|[[32/19]], [[22/13]]
|
|Cuthbert
|G♯
|G♯
|
|
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|[[26/15]]
|[[26/15]]
|Barbados
|Barbados
|
|A
|
|
|-
|-
|30\37
| 30\37
|972.97
|972.97
|[[7/4]], [[16/9]]+
|[[7/4]], [[16/9]]+
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|16/9 using diatonic fourths
|16/9 using diatonic fourths
|-
|-
|31\37
| 31\37
|1005.41
|1005.41
|16/9
|16/9, 10/9
|Didacus
|Didacus
|C♭
|C♭
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|32\37
|32\37
|1037.84
|1037.84
|16/9-, [[20/11|9/5]]
|16/9-, [[20/11|9/5]]+
|Porcupine
|Porcupine
|
|Aｷ = D𝄫
|16/9 using antidiatonic fourths
|16/9 using antidiatonic fourths
|-
|-
|33\37
|33\37
|1070.27
|1070.27
|[[15/8]], 13/7
|[[13/7]]
|Negri
|Cubical
|Bd
|Bd
|
|
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|[[32/17]]
|[[32/17]]
|Passion
|Passion
|Cd
|G𝄪 = Cd
|
|
|-
|-
|35\37
|35\37
|1135.14
|1135.14
|[[27/14]]
|[[27/14]]+
|Sycamore/Unicorn
|Sycamore
|
|A♯
|
|
|-
|-
|36\37
|36\37
|1167.57
|1167.57
|[[55/28]], [[108/55]]
|[[55/28]], [[55/27]]+
|
|
|B
|B
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|
|
|}
|}
===Notation===
The table above uses the Diatonic fifth as the basis for notation, following the standard Circle of Fifths with additional half-flats and half-sharps used to more concisely represent notes that would otherwise require triple or even quadruple flats and sharps.
[[Ups and downs notation|Ups and Downs notation]] is also helpful for short accidentals; in a Blackdye piece that uses the antidiatonic 3/2, it may be helpful to notate that fifth as vG rather than F♯ to better indicate that it is being used as a perfect fifth rather than an augmented fourth.
Additionally, [[Diamond MOS notation]] may be more useful than Circle of Fifths notation for describing structures that do not adhere closely to the Circle of Fifths, such as Orgone or Cubical.
[[Category:Approaches to tuning systems]]

Latest revision as of 18:16, 14 April 2025

< 37edo

Note: This page is currently under construction, and will be subject to major expansion in the near future. Come back soon!

If it wasn't clear before, I definitely have a "type" when it comes to selecting tuning systems. 37 Equal Divisions of the Octave is another 11-limit system with a sharp diatonic fifth and supports Porcupine temperament. Being 15 + 22, fans of 15edo and 22edo will likely be drawn to 37edo as a structural extension of the two; additionally, fans of split-prime systems may also be drawn to 37 due to its slightly ambiguous harmonic nature (see Intervals).

As always, this page will be full of personal touches that may not reflect an objective truth or even wide consensus about how to use 37edo; I encourage learning musicians to experiment with different ideas and develop styles that best suit their own needs, rather than to take my word (or anyone else's for that matter) at face value as a great truth of music.

Intervals

37edo is a rather interesting 19-limit system, but it does have some amount of harmonic ambiguity in the 3-limit. The two mappings for 3/2 are respectively a diatonic and antidiatonic generator; I will here use + to indicate JI interpretations that use the diatonic 3-limit, and - to indicate those that use the antidiatonic 3-limit.

Intervals of 37edo
Intervals Cents JI Intervals As a generator Notation Notes
0\37 0.00 1/1 C
1\37 32.43 55/54+, 56/55 D♭
2\37 64.86 28/27+ Sycamore Bｷ = E𝄫
3\37 97.30 17/16 Passion Cｷ = F𝄫
4\37 129.73 14/13 Cubical A𝄪 = Dd Cubical uses antidiatonic fifth
5\37 162.16 10/9+, 9/8- Porcupine B♯ 10/9 using diatonic fifth, or 9/8 using antidiatonic fifth
6\37 194.59 10/9, 19/17, 9/8 Didacus C♯ 10/9 = 9/8 using dual fifths
7\37 227.03 9/8+, 8/7 Gorgo/Shoe D 9/8 using diatonic fifth
8\37 259.46 15/13+ Barbados E♭ Barbados utilizes split-3 shenanigans
9\37 291.89 13/11, 19/16 Cuthbert F♭
10\37 324.32 77/64 Orgone Dｷ = G𝄫 Probably better interpreted as 8/(sqrt11)
11\37 356.76 11/9+, 16/13 Beatles B𝄪 = Ed Bisects the diatonic fifth
12\37 389.19 5/4 Wuerschmidt C𝄪 = Fd
13\37 421.62 14/11 Lambeth D♯
14\37 454.05 9/7+, 13/10 Ammonite E
15\37 486.49 4/3+ Ultrapyth F Diatonic fourth
16\37 518.92 4/3- Undecimation G♭ Antidiatonic fourth
17\37 551.35 11/8 Emka Eｷ = A𝄫
18\37 583.78 7/5 Fｷ
19\37 616.22 10/7 D𝄪 = Gd
20\37 648.65 16/11 Emka E♯
21\37 681.08 3/2- Undecimation F♯ Antidiatonic fifth
22\37 713.51 3/2+ Ultrapyth G Diatonic fifth
23\37 745.95 20/13, 14/9 Ammonite A♭
24\37 778.38 11/7 Lambeth B𝄫
25\37 810.81 8/5 Wuerschmidt Gｷ = C𝄫
26\37 843.24 13/8, 18/11 Beatles E𝄪 = Ad
27\37 875.68 128/77 Orgone F𝄪 Probably better interpreted as (sqrt11)/2
28\37 908.11 32/19, 22/13 Cuthbert G♯
29\37 940.54 26/15 Barbados A
30\37 972.97 7/4, 16/9+ Gorgo/Shoe B♭ 16/9 using diatonic fourths
31\37 1005.41 16/9, 10/9 Didacus C♭ 16/9 using dual fourths
32\37 1037.84 16/9-, 9/5+ Porcupine Aｷ = D𝄫 16/9 using antidiatonic fourths
33\37 1070.27 13/7 Cubical Bd
34\37 1102.70 32/17 Passion G𝄪 = Cd
35\37 1135.14 27/14+ Sycamore A♯
36\37 1167.57 55/28, 55/27+ B
37\37 1200.00 2/1 C

Notation

The table above uses the Diatonic fifth as the basis for notation, following the standard Circle of Fifths with additional half-flats and half-sharps used to more concisely represent notes that would otherwise require triple or even quadruple flats and sharps.

Ups and Downs notation is also helpful for short accidentals; in a Blackdye piece that uses the antidiatonic 3/2, it may be helpful to notate that fifth as vG rather than F♯ to better indicate that it is being used as a perfect fifth rather than an augmented fourth.

Additionally, Diamond MOS notation may be more useful than Circle of Fifths notation for describing structures that do not adhere closely to the Circle of Fifths, such as Orgone or Cubical.

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