Hello! My name is Andrew and I like screwing around with xenharmony, especially EDOs, JI, and various notations.
Here's my website...if you could call it that.
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.
Keep in mind that I have been making regular changes to this page, so I could have made a lot of mistakes here.
Extended Ups and Downs
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: / (pitch up) and \ (pitch down). Single edosteps are still notated with arrows: ^ and v.
Here's a full example in 37-EDO where the syntonic comma is 2 steps large:
| 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 ) (pitch up) and ( (pitch down) instead of / and \.
For anti-diatonic systems, the meantone-favoring flat and sharp symbols are not recommended, but use harmonic notation should they be needed.
An example in 13b-EDO where the syntonic comma is -2 steps (technically):
| 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 ad-lib quarter tone notation. Though for many systems (10, 17, 24, 31), the syntonic comma notation makes it redundant; perhaps it could be of use in larger systems like 41, 48, or 72.
An example in 41-EDO with + for a quartertone sharp and d for a quartertone flat:
| Steps | Pythagorean Notation | Old Notation | New Notation | With Quartertones |
|---|---|---|---|---|
| 0 | D | D | D | D |
| 1 | Cx | ^D | /D | /D |
| 2 | Fbb | ^^D, vEb | \Eb | D+ |
| 3 | Eb | vD#, Eb | Eb | Eb |
| 4 | D# | D#, ^Eb | D# | D# |
| 5 | C#x | ^D#, vvE | /D# | Ed |
| 6 | Fb | vE | \E | \E |
| 7 | E | E | E | E |
| 8 | Dx | ^E | /E | Fd |
| 9 | Gbb | vF | \F | E+ |
| 10 | F | F | F | F |
| 11 | E# | ^F | /F | /F |
| 12 | Abbb | ^^F, vGb | \Gb | F+ |
| 13 | Gb | vF#, Gb | Gb | Gb |
| 14 | F# | F#, ^Gb | F# | F# |
| 15 | Ex | ^F#, vvG | /F# | Gd |
| 16 | Abb | vG | \G | \G |
| 17 | G | G | G | G |
| 18 | Fx | ^G | /G | /G |
| 19 | Bbbb | ^^G, vAb | \Ab | G+ |
| 20 | Ab | vG#, Ab | Ab | Ab |
| 21 | G# | G#, ^Ab | G# | G# |
| 22 | F#x | ^G#, vvA | /G# | Ad |
| 23 | Bbb | vA | \A | \A |
| 24 | A | A | A | A |
| 25 | Gx | ^A | /A | /A |
| 26 | Cbb | ^^A, vBb | \Bb | A+ |
| 27 | Bb | vA#, Bb | Bb | Bb |
| 28 | A# | A#, ^Bb | A# | A# |
| 29 | G#x | ^A#, vvB | /A# | Bd |
| 30 | Cb | vB | \B | \B |
| 31 | B | B | B | B |
| 32 | Ax | ^B | /B | Cd |
| 33 | Dbb | vC | \C | B+ |
| 34 | C | C | C | C |
| 35 | B# | ^C | /C | /C |
| 36 | Ebbb | ^^C, vDb | \Db | C+ |
| 37 | Db | vC#, Db | Db | Db |
| 38 | C# | C#, ^Db | C# | C# |
| 39 | Bx | C#^, vvD | /C# | Dd |
| 40 | Ebb | vD | \D | \D |
| 41 | D | D | D | D |
Scales n' Stuff
Website
I have some scales ready in Scala format on my website here.
"Drewnian" Scale System
I recently developed an eleven-tone scale system, similar to Western theory in construction but by no means similar in sound. I like to think it's what Wstern tonality would look like if the 7/4 was prioritized over the 5/4.
It involves two scales, each built with a seven-based tetrachord. One is built with a 8/7, 8/7, 49/48 tetrachord, generating the "major" intervals; and the other is built with a 49/48, 8/7, 8/7 tetrachord, generating the "minor" intervals. As a result, the scale has very large major seconds and very small minor seconds.
| Degree | Name | Ratio |
|---|---|---|
| 0 | P1 | 1/1 |
| 1 | m2 | 49/48 |
| 2 | M2 | 8/7 |
| 3 | m3 | 7/6 |
| 4 | M3 | 64/49 |
| 5 | P4 | 4/3 |
| 6 | P5 | 3/2 |
| 7 | m6 | 49/32 |
| 8 | M6 | 12/7 |
| 9 | m7 | 7/4 |
| 10 | M7 | 96/49 |
| 11 | P8 | 2/1 |
The intent with these weird superpyth-like constructions is to make 5-EDO-like scales with some extra small intervals for spice.
It's very easily possible to map this scale to a 12-tone keyboard, in fact it is by design a mimicry of Western tonality. However, there is no tritone. For this interval you can either use the eleventh harmonic 11/8, a septimal whole tone below the "minor sixth". or a septimal whole tone above the "major third".
Enneatonic scale in JI
Supahstar Saga described a scale in 19-EDO in his Exploring 19-TET YouTube series called the Enneatonic scale:
In 19-tone, 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, measuring around 950.978 cents. What would that sound like?
| Degree | Ratio | Decimal | Cents |
|---|---|---|---|
| 1 | 1/1 | 1.0000 | 0.000 |
| 2 | 9/8 | 1.1250 | 203.910 |
| 3 | 81/64 | 1.2656 | 407.820 |
| 4 | 3√3/4 | 1.2990 | 452.933 |
| 5 | 27√3/32 | 1.4614 | 656.843 |
| 6 | 3/2 | 1.5000 | 701.955 |
| 7 | 27/16 | 1.6875 | 905.865 |
| 8 | √3/1 | 1.7321 | 950.978 |
| 9 | 9√3/8 | 1.9486 | 1154.888 |
| 10 | 2/1 | 2.0000 | 1200.000 |
But of course Saga wasn't looking for a √3 interval, he meant to use the harmonic (subminor) seventh. The two intervals are rather close though: the seventh is about 17.848 cents sharper.
Using harmonic sevenths of 968.826 cents:
| Degree | Ratio | Decimal | Cents |
|---|---|---|---|
| 1 | 1/1 | 1.0000 | 0.000 |
| 2 | 9/8 | 1.1250 | 203.910 |
| 3 | 81/64 | 1.2656 | 407.820 |
| 4 | 21/16 | 1.3125 | 470.781 |
| 5 | 189/128 | 1.4766 | 674.691 |
| 6 | 3/2 | 1.5000 | 701.955 |
| 7 | 27/16 | 1.6875 | 905.865 |
| 8 | 7/4 | 1.7500 | 968.826 |
| 9 | 63/32 | 1.9486 | 1172.736 |
| 10 | 2/1 | 2.0000 | 1200.000 |
Using harmonic sevenths inverted around the third harmonic, or just supermajor sixths (12/7), of 933.129 cents:
| Degree | Ratio | Decimal | Cents |
|---|---|---|---|
| 1 | 1/1 | 1.0000 | 0.000 |
| 2 | 9/8 | 1.1250 | 203.910 |
| 3 | 81/64 | 1.2656 | 407.820 |
| 4 | 9/7 | 1.2857 | 435.084 |
| 5 | 81/56 | 1.4464 | 638.9941 |
| 6 | 3/2 | 1.5000 | 701.955 |
| 7 | 27/16 | 1.6875 | 905.865 |
| 8 | 12/7 | 1.7143 | 933.129 |
| 9 | 27/14 | 1.9286 | 1137.039 |
| 10 | 2/1 | 2.0000 | 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 81ϕ/128, about 40.9103 cents. I dub this interval the Golden quartertone.
Pi
The ratio π/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 256/81π, 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 3e/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.