User:TromboneBoi9

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 13-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 (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:

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.

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 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.