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.
An idea for notation I had
Something I noticed in regular EDO notation, using Pythagorean names, is that the major third in a lot of temperaments is no longer the closest the EDO has to 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), 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 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# | ^A# |
| 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 x and y 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, xC |
| 2 | Eb | E | Eb, xD |
| 3 | Fx | Ex, Fb | xE, yF |
| 4 | F# | F# | F#, yG |
| 5 | F | Gb | F, yA |
| 6 | G | G# | G, yB |
| 7 | A | Ab | A, xF |
| 8 | B | A# | B, xG |
| 9 | Bb | Bb | Bb, xA |
| 10 | Cx | B# | xB, yC |
| 11 | C# | C | C#, yD |
| 12 | C | Cx, Dbb | C, yE |
| 13 | D | D | D |
Cloudy scales
I don't know about you, but I love the seventh harmonic. These scales are named after the cloudy comma.
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 Holo-Alpha.
| 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 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!
| 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 is an 3L8s MOS with 9/7 as the generator and 2/1 as the period. It approximates all intervals of 11-EDO within 10 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 |