Hello! My name is Andrew and I like screwing around with xenharmony, especially notation.
Currently experimenting with anti-diatonic stuff like EDOs 9, 11, 13, 16, and 23.
At some point I plan to make a piece similar to Easley Blackwood's 12 Etudes or Aaron Andrew Hunt's Equal-Tempered Keyboard, an "album" experimenting with a range of different EDO systems.
I also exist on the XA Discord, currently under the alias Sir Semiflat.
An idea for notation I had
Something I noticed in regular EDO notation, relying on Pythagorean names with an extra layer of accidentals, is that the Pythagorean major third in a lot of EDO systems doesn't match the just major third 5/4. So, with some thought from Just Intonation notations, I came up with something that I think might be promising.
In essence, instead of solely relying on Pythaogrean names and arrows for edosteps in between (which can get unwieldy in larger EDOs, e.g. 72-EDO), I considered making the arrow represent the syntonic comma instead. If your EDO has a different pitch for the just major third and the Pythagorean major third, then of course, it has a syntonic comma that hasn't been tempered out. This won't change anything for EDOs with a syntonic comma less than or equal to one step, of course, but it could have an effect on even "sharper" systems like 37-EDO.
For single edosteps, we can instead use a sort of slash-like symbol Bosanquet used in his notation, and perhaps stack them on top of each other to use less horizontal space.
Here's a full example in 37-EDO:
| Steps | Pythagorean notation | Old notation | New notation |
|---|---|---|---|
| 0 | D | D | D |
| 1 | Eb | Eb | Eb |
| 2 | Fb | ^Eb | ^D |
| 3 | Gbb | ^^Eb | ^Eb |
| 4 | Bx | vvD# | vD# |
| 5 | Cx | vD# | vE |
| 6 | D# | D# | D# |
| 7 | E | E | E |
| 8 | F | F | F |
| 9 | Gb | Gb | Gb |
| 10 | Abb | ^Gb | ^F |
| 11 | Bbbb | ^^Gb | ^Gb |
| 12 | Dx | vvF# | vF# |
| 13 | E# | vF# | vG |
| 14 | F# | F# | F# |
| 15 | G | G | G |
| 16 | Ab | Ab | Ab |
| 17 | Bbb | ^Ab | ^G |
| 18 | Cbb | ^^Ab | ^Ab |
| 19 | Ex | vvG# | vG# |
| 20 | Fx | vG# | vA |
| 21 | G# | G# | G# |
| 22 | A | A | A |
| 23 | Bb | Bb | Bb |
| 24 | Cb | ^Bb | ^A |
| 25 | Dbb | ^^Bb | vBb |
| 26 | F#x | vvA# | vA# |
| 27 | Gx | vA# | vB |
| 28 | A# | A# | A# |
| 29 | B | B | B |
| 30 | C | C | C |
| 31 | Db | Db | Db |
| 32 | Ebb | ^Db | ^C |
| 33 | Fbb | ^^Db | ^Db |
| 34 | Ax | vvC# | vC# |
| 35 | B# | vC# | vD |
| 36 | C# | C# | C# |
| 37 | D | D | D |
And for anti-diatonic systems, use ( and ) instead of ^ and v, using harmonic notation.
An example in 13-EDO:
| Steps | Pythagorean/old notation | 26-EDO Subset | New notation |
|---|---|---|---|
| 0 | D | D | D |
| 1 | E | Dx, Ebb | E, (C |
| 2 | Eb | E | Eb, (D |
| 3 | Fx | Ex, Fb | (E, )F |
| 4 | F# | F# | F#, )G |
| 5 | F | Gb | F, )A |
| 6 | G | G# | G, )B |
| 7 | A | Ab | A, (F |
| 8 | B | A# | B, (G |
| 9 | Bb | Bb | Bb, (A |
| 10 | Cx | B# | (B, )C |
| 11 | C# | C | C#, )D |
| 12 | C | Cx, Dbb | C, )E |
| 13 | D | D | D |
I have also devised custom accidentals for quartertones in both diatonic and anti-diatonic systems, but the image uploading process is being weird so I'll have to figure that out at some point.
Cloudy scales
I don't know about you, but I love the seventh harmonic. These MOS scales are named after the cloudy comma, and use different 7-limit intervals for generators.
Cumulus Alpha
Cumulus Alpha is a 5L6s MOS with 7/4 as the generator and 2/1 as the period. This appears to approximate a subset of 26-EDO; it approximates the whole of 26-EDO when extended to a 5L21s MOS, which I dub Cumulus Alpha Holo.
That's right. We're comparing JI to EDOs instead of the other way around.
| Steps | Ratio | Cents | Approx. 26-EDO Degree |
|---|---|---|---|
| 0 | 1/1 | 0.000 | 0 |
| 1 | 16807/16384 | 43.130 | 1 |
| 2 | 8/7 | 231.174 | 5 |
| 3 | 2401/2048 | 275.304 | 6 |
| 4 | 64/49 | 462.348 | 10 |
| 5 | 343/256 | 506.478 | 11 |
| 6 | 512/343 | 693.522 | 15 |
| 7 | 49/32 | 737.652 | 16 |
| 8 | 4096/2401 | 924.696 | 20 |
| 9 | 7/4 | 968.826 | 21 |
| 10 | 32768/16807 | 1155.870 | 25 |
| 11 | 2/1 | 1200.000 | 26 |
Cumulus Beta
Cumulus Beta is an 4L5s MOS with 7/6 as the generator and 2/1 as the period. Amazingly, it approximates all intervals of 9-EDO within a cent, with a mean difference of about 0.409 cents.
| Steps | Ratio | Cents | 9-EDO Difference |
|---|---|---|---|
| 0 | 1/1 | 0.000 | 0.000 |
| 1 | 2592/2401 | 132.516 | -0.817 |
| 2 | 7/6 | 266.871 | 0.204 |
| 3 | 432/343 | 399.387 | -0.613 |
| 4 | 49/36 | 533.742 | 0.409 |
| 5 | 72/49 | 666.258 | -0.409 |
| 6 | 343/216 | 800.613 | 0.613 |
| 7 | 12/7 | 933.129 | -0.204 |
| 8 | 2401/1296 | 1067.484 | 0.817 |
| 9 | 7/4 | 1200.000 | 0.000 |
Cumulus Gamma
Cumulus Gamma is an 3L8s MOS with 9/7 as the generator and 2/1 as the period. It approximates all intervals of 11-EDO within 7 cents, with a mean difference of 3.199 cents.
| Steps | Ratio | Cents | 11-EDO Difference |
|---|---|---|---|
| 0 | 1/1 | 0.000 | 0.000 |
| 1 | 729/686 | 105.252 | 3.839 |
| 2 | 67228/59049 | 224.580 | -6.398 |
| 3 | 98/81 | 329.832 | -2.559 |
| 4 | 9/7 | 435.084 | 1.280 |
| 5 | 6561/4802 | 540.336 | 5.119 |
| 6 | 9604/6561 | 659.664 | -5.119 |
| 7 | 14/9 | 764.916 | -1.280 |
| 8 | 81/49 | 870.168 | 2.559 |
| 9 | 59049/33614 | 975.420 | 6.398 |
| 10 | 1372/729 | 1094.748 | -3.839 |
| 11 | 2/1 | 1200.000 | 0.000 |