User:TromboneBoi9/Approaches to weird EDOs
THIS PAGE IS A WORK IN PROGRESS AND WILL DEVELOP AS MY THEORIES DEVELOP
Outside of free just intonation, most of my xenharmonic work is exclusively in EDOs, generally various EDOs smaller than 36edo (although I have used larger ones in the past).
I work almost exclusively in notation software, so it's important that EDOs I'm working with can be worked into traditional diatonic notation in some form. However, as one would know, this is only the case for an EDO if it's approximation of 3/2 is between 686¢ (5\7) and 720¢ (3\5), which is not always the case.
Here are some of my own theoretical and notational approaches to various EDOs that break this mold.
13edo
I consider 13edo to be one of two EDOs which "phases out" from 12edo, the other being 11edo. These EDOs have specific properties in relation to 12edo:
- Intervals in the second and seventh interval regions are relatively close to their 12edo equivalents; consequently, various intervals like 9/8, 10/9, 9/5, 16/9, etc. may find good approximations.
- Intervals around the third and sixth interval regions are somewhat detuned; consequently, various intervals like 5/4, 7/6, 13/8, 5/3, etc. may be approximated depending on the EDO.
- Intervals around the fourth and fifth interval regions are nearly a quartertone off from their 12edo equivalents; consequently, 11/8 and 16/11 are approximated relatively well.
Thus, while 11edo and 13edo may be lost causes for traditional approaches, there is still potential from the JI approximation perspective.
For instance, the 2\13 interval is about 184¢, which is a fine major second; it's two cents flat of 10/9 and can act as a 9/8 if need be. This makes it a reasonable "fundamental consonance," taking the place of the fifth in traditional theory.
Scales generated by the 2\13 will also feature 4\13–a flat but recognizable 5/4–and 6\13–a dead-on 11/8–. In this way, a basic 6L1s "archaeotonic" scale is produced.
Any mode can be used, but the symmetrical 3|3 "Holthathian" mode captures 13edo's best intervals:
| Interval | Cents | Ratios | Note name | Pseudo-diatonic name |
|---|---|---|---|---|
| 0\13 | 0.00 | 1/1 | C | Perfect unison |
| 2\13 | 184.62 | 10/9, 9/8 | D | Major second |
| 4\13 | 369.23 | 5/4, 16/13 | E | Major third |
| 6\13 | 553.85 | 11/8 | F | Major fourth |
| 7\13 | 646.15 | 16/11 | G♭ | Minor fifth |
| 9\13 | 830.77 | 13/8, 8/5 | A♭ | Minor sixth |
| 11\13 | 1015.38 | 16/9, 9/5 | B♭ | Minor seventh |
| 13\13 | 1200.00 | 2/1 | C | Perfect octave |
You can also see demonstrated in the table above a useful notation system based on 6L1s, specifically the 6|0 "Ryonian" mode. This notation scheme is identical to the traditional 12edo notation system, except there is an extra step between E and F; E♯ and F♭ become enharmonics.
A potential downside to the compositional and notational use of 6L1s as a tonal system is that its small steps are too sparse, making it sound too much like an equalized whole tone scale melodically. This is opposed to the much more popular 5L3s "oneirotonic" system which, while generated on the dissonant 8\13 "major fifth," is far more melodically captivating; it also happens to support 18edo, another problematic EDO. 5L3s, however, is octatonic rather than heptatonic, which sacrifices clarity in staff notation greatly.