Hello! My name is Andrew and I like screwing around with xenharmony, especially notation.
[I've had to make some changes--I do not have a website anymore, but soon enough I will have a new one.]
I also exist on the XA Discord, currently under the alias Sir Semiflat.
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.
(Update: Progress has started! It will be named "A Solo Spectrum Clavier".)
Keep in mind that I have constantly been making changes to this page, so I could have made a lot of mistakes here.
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 Pythagorean names and arrows for edosteps in between (which can get unwieldy in larger EDOs, e.g. 72-EDO), I considered giving the syntonic comma a single symbol instead. That way you can emphasize the 5-limit in your compositions more easily, and it will stay that way when directly read in a different EDO.
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 much notation-wise for EDOs with a syntonic comma of 0 or 1 step, of course, but it could have an effect on even "sharper" systems like 37-EDO with a larger comma.
The syntonic comma can be represented by slashes: / and \. Single edosteps are still notated with arrows: ^ and v.
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# | \D# |
| 5 | Cx | vD# | \E |
| 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# | \F# |
| 13 | E# | vF# | \G |
| 14 | F# | F# | F# |
| 15 | G | G | G |
| 16 | Ab | Ab | Ab |
| 17 | Bbb | ^Ab | /G |
| 18 | Cbb | ^^Ab | /Ab |
| 19 | Ex | vvG# | \G# |
| 20 | Fx | vG# | \A |
| 21 | G# | G# | G# |
| 22 | A | A | A |
| 23 | Bb | Bb | Bb |
| 24 | Cb | ^Bb | /A |
| 25 | Dbb | ^^Bb | \Bb |
| 26 | F#x | vvA# | \A# |
| 27 | Gx | vA# | \B |
| 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# | \C# |
| 35 | B# | vC# | \D |
| 36 | C# | C# | C# |
| 37 | D | D | D |
For systems with a negative syntonic comma (most often in sub-meantone systems), use ) and ( instead of / and \.
Use harmonic notation for anti-diatonic systems.
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 |
For systems with describable quarter tones, you can optionally use quarter tone notation. Though for many systems (24, 31), the syntonic comma notation makes it redundant; perhaps it could be of use in larger systems like 41, 48, or 72.
Scales n' Stuff
Cumulus 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 |
Blues scale in 10-EDO
I kinda like the 3 1 1 1 2 2 scale in 10-EDO, it works alright as a Blues scale. I think the second degree (3\10) is a bit sharper than it should, in fact a lot of intervals are "stretched out" in comparison to the Blues scale in 12, but before I didn't have very many scales in 10 under my belt except for the equipentatonic scale.
I dunno, I stick to theory more often than I should; I use theoretical diatonic intervals/scales more often than intervals/scales that actually sound diatonic. Luckily I've been experimenting with 14-EDO recently, and I think it's good territory to fix that.
Enneatonic scale in "3-limit"
Supahstar Saga described a scale in 19-EDO in his Exploring 19-TET YouTube series called the Enneatonic scale:
Since the third harmonic is an even number of steps in 19 (30 steps), splitting evenly into two harmonic (subminor) sevenths. If you take a major pentatonic scale, put a harmonic seventh in between each fifth, and reduce the whole scale into an octave, you get a nine-tone scale somewhat similar to the double harmonic scale in 12.
My thought was, if you use pure 3-limit Just Intonation, you can split the third harmonic into two "ratios" of √3. What would that sound like?
| Degree | Ratio | Cents |
|---|---|---|
| 1 | 1/1 | 0.000 |
| 2 | 9/8 | 203.910 |
| 3 | 81/64 | 407.820 |
| 4 | 3√3/4 | 452.933 |
| 5 | 27√3/32 | 656.843 |
| 6 | 3/2 | 701.955 |
| 7 | 27/16 | 905.865 |
| 8 | √3/1 | 950.978 |
| 9 | 9√3/8 | 1154.888 |
| 10 | 2/1 | 1200.000 |
Irrational HEJI Extensions
I've heard phi is somewhat useful in xen areas, as well as other popular irrational numbers, so what would it look like if I extended HEJI (my go-to Just Intonation notation) to support these numbers like factors?
Commas
Golden Ratio
The ratio phi adds up to 833.0903 cents, a sharp minor sixth. The Pythagorean minor sixth is 128/81, about 792.1800 cents. This leaves a comma of (81*phi)/128, about 40.9103 cents. I dub this interval the Golden quartertone.
Pi
The ratio pi/2 adds up to 781.7954 cents, an okay minor sixth. The Pythagorean minor sixth is 128/81, about 792.1800 cents. This leaves a comma of 256/(81*pi), about 10.3846 cents. I dub this interval the Circular comma.
Euler's constant
The ratio e/2 adds up to 531.2340 cents, a pretty sharp fourth. The Pythagorean perfect fourth is, of course, 4/3, 498.0450 cents. This leaves a comma of (3*e)/8, about 33.1890 cents. I dub this interval the Eulerian comma.
Notation
For the golden quartertone, I plan to use the symbol Blackwood used in his microtonal notation, because it already resembles a phi symbol (ϕ). For pi, I designed a symbol similar to the 55-comma symbol in Sagittal, but the "arrowhead" is replaced with a circular cap, making the symbol resemble a J with an extra shaft.
I'm yet to design a symbol for e.