User:Unque/37edo Composition Theory
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 | 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.