Syntonic–rastmic subchroma notation

Syntonic–rastmic subchroma notation is a notation scheme developed by Aura et al. that is an expansion to the neutral circle-of-fifths notation.

While the neutral circle-of-fifths notation models the 2.3 subgroup of just intonation, with the neutral intervals capable of roughly modeling the harmonic 11, the syntonic–rastmic subchroma notation accurately captures the characteristics of the 2.3.5.11 subgroup, and is fit for a wider variety of equal temperaments and multirank temperaments. As it tries to strike a balance between the number and semantic consistency of the accidentals, it has the following three basic building blocks of accidentals: the conventional accidentals, the syntonic and demisyntonic accidentals, and the rastmic and demirastmic accidentals, detailed below.

Accidentals

Conventional accidentals

As in neutral circle-of-fifths notation, the demisharp raises the pitch by half a chromatic semitone, and the demiflat lowers the pitch by the same amount.

Series of conventional accidentals

Name	Ratio	Monzo	Textual Representation
…	…	…	…
Double sharp	(2187/2048)2	[-22 14⟩	x
Sesquisharp	(2187/2048)3/2	[-33/2 21/2⟩	t#
Sharp	(2187/2048)1	[-11 7⟩	#
Demisharp	(2187/2048)1/2	[-11/2 7/2⟩	t
(None)	(2187/2048)0	[0⟩
Demiflat	(2187/2048)−1/2	[11/2 -7/2⟩	d
Flat	(2187/2048)−1	[11 -7⟩	b
Sesquiflat	(2187/2048)−3/2	[33/2 -21/2⟩	db
Double flat	(2187/2048)−2	[22 -14⟩	bb
…	…	…	…

Syntonic and demisyntonic accidentals

Syntonic accidentals

Demisyntonic accidentals (WIP)

The syntonic and demisyntonic accidentals model the harmonic 5. The synsharp raises the pitch by a syntonic comma. The synflat lowers the pitch by the same amount.

Series of syntonic accidentals

Name	Ratio	Monzo	Textual Representation
…	…	…	…
Synsharp	(81/80)1	[-4 4 -1⟩	↑
Demisynsharp	(81/80)1/2	[-2 2 -1/2⟩	^
(None)	(81/80)0	[0⟩
Demisynflat	(81/80)−1/2	[2 -2 1/2⟩	v
Synflat	(81/80)−1	[4 -4 1⟩	↓
…	…	…	…

Rastmic and demirastmic accidentals

Rastmic accidentals

Demirastmic accidentals

The rastmic and demirastmic accidentals model the harmonic 11. The demirasharp raises the pitch by half a rastma. The demiraflat lowers the pitch by the same amount.

Series of rastmic and demirastmic accidentals

Name	Ratio	Subgroup Monzo (2.3.5.11)	Textual Representation
…	…	…	…
Double rasharp	(243/242)2	[-2 10 0 -4⟩	//
Sesquirasharp	(243/242)3/2	[-3/2 15/2 0 -3⟩	>/
Rasharp	(243/242)1	[-1 5 0 -2⟩	/
Demirasharp	(243/242)1/2	[-1/2 5/2 0 -1⟩	>
(None)	(243/242)0	[0⟩
Demiraflat	(243/242)−1/2	[1/2 -5/2 0 1⟩	<
Raflat	(243/242)−1	[1 -5 0 2⟩	\
Sesquiraflat	(243/242)−3/2	[3/2 -15/2 0 3⟩	<\
Double raflat	(243/242)−2	[2 -10 0 4⟩	\\
…	…	…	…

Combined accidentals

Combined quartertone-demirastmic accidentals

The demisharp/demiflat and the demirasharp/demiraflat are rarely used alone since they are irrational. They are usually combined for the following accidentals. These are the most common quartertones.

Overview of accidentals

Name	Ratio	Subgroup Monzo (2.3.5.11)	Textual Representation
Tendodemisharp	729/704	[-6 6 0 -1⟩	t>
Artodemisharp	33/32	[-5 1 0 1⟩	t<
Tendodemiflat	32/33	[5 -1 0 -1⟩	d>
Artodemiflat	704/729	[6 -6 0 1⟩	d<

Natural accidental

The natural accidental cancels all pitch alterations.

Notation guide for common tunings

The syntonic–rastmic subchroma notation is closely related to the syntonic-rastmic equivalence continuum, in that the accidental settings are similar among tunings of a particular equivalence number. Due to enharmonic equivalences, however, each tuning system can take many accidental settings. Each case shown in the tables below is only one of them. Users should choose the combination in question according to the semantics.

Equal temperaments are shown if the corresponding edo has {1, 3, 9}-diamond consistency. Others can be notated as subsets or be extrapolated.

Neutral tunings

Neutral tunings

Sharpness	Accidentals	Temperament	Equal temperaments
2	t, #	Mohaha	10e, 17, 24, 31, 38
4	↑, t, #↓, #	Tetracot	27e, 34, 41, 48, 55c
6	↑, t↓, t, t↑, #↓, #	Larry	51ce, 58, 65, 72, 79
8	↑, ↑↑, t↓, t, t↑, #↓↓, #↓, #	7 & 89	82c, 89, 96

55 can be notated as a mohaha tuning. 75 can be notated as a tetracot tuning. 123 and 137 can be notated as larry tunings.

Protomere tunings

Protomere tunings—order 1

Sharpness	Accidentals	Temperament	Equal temperaments
3	t<, t>, #	Porcupine restriction	15, 22, 29, 36ce
5	↑, t<, t>, #↓, #	Hitchcock restriction	39, 46, 53, 60e
7	↑, ↑↑, t<, t>, #↓↓, #↓, #	Absurdity extension	63c, 70, 77, 84e

Protomere tunings—order 2

Sharpness	Accidentals	Temperament	Equal temperaments
6	>, t<, t, t>, #<, #	Neutral porcupine restriction	51, 58ce
8	>, ↑, t<, t, t>, #↓, #<, #	Tetracot extension	68, 75, 82e
10	>, ↑, ↑>, t<, t, t>, #↓<, #↓, #<, #	Neutral hitchcock restriction	92, 99, 106e

51 and 99 can be notated as either order 1 or order 2.

Deuteromere tunings

Deuteromere tunings—order 1

Sharpness	Accidentals	Temperament	Equal temperaments
7	/, ↑, t<, t>, #↓, #\, #	Sevond extension	56, 63, 70c
9	/, ↑, ↑/, t<, t>, #↓\, #↓, #\, #	7 & 87	80, 87, 94, 101c
11	/, ↑, ↑/, ↑↑, t<, t>, #↓↓, #↓\, #↓, #\, #	Twentcufo	104c, 111, 118, 125, 132e
13	/, ↑, ↑/, ↑↑, ↑↑/, t<, t>, #↓↓\, #↓↓, #↓\, #↓, #\, #	7 & 149	135c, 142, 149, 156e

Deuteromere tunings—order 2

Sharpness	Accidentals	Temperament	Equal temperaments
18	>, /, ↑<, ↑, ↑>, ↑/, ↑↑<, t<, t, t>, #↓↓>, #↓\, #↓<, #↓, #↓>, #\, #<, #	Neutral 7 & 87	174, 181, 188c
20	>, /, ↑<, ↑, ↑>, ↑/, ↑↑<, ↑↑, t<, t, t>, #↓↓, #↓↓>, #↓\, #↓<, #↓, #↓>, #\, #<, #	7 & 205	198, 205, 212, 219ce
22	>, /, ↑<, ↑, ↑>, ↑/, ↑↑<, ↑↑, ↑↑>, t<, t, t>, #↓↓<, #↓↓, #↓↓>, #↓\, #↓<, #↓, #↓>, #\, #<, #	Neutral twentcufo	222c, 229, 236, 243e

181, 229, and 243e can be notated as either order 1 or order 2.

Tritomere tunings

Tritomere tunings—order 1

Sharpness	Accidentals	Temperament	Equal temperaments
13	/, ↑\, ↑, ↑/, ↑↑\, t<, t>, #↓↓/, #↓\, #↓, #↓/, #\, #	7 & 121	121, 128, 135, 142c
15	/, ↑\, ↑, ↑/, ↑↑\, ↑↑, t<, t>, #↓↓, #↓↓/, #↓\, #↓, #↓/, #\, #	Trinity restriction	145, 152, 159, 166, 173c
17	/, ↑\, ↑, ↑/, ↑↑\, ↑↑, ↑↑/, t<, t>, #↓↓\, #↓↓, #↓↓/, #↓\, #↓, #↓/, #\, #	7 & 183	169c, 176, 183, 190, 197e
19	/, ↑\, ↑, ↑/, ↑↑\, ↑↑, ↑↑/, ↑↑//, t<, t>, #↓↓\\, #↓↓\, #↓↓, #↓↓/, #↓\, #↓, #↓/, #\, #	7 & 207	200c, 207, 214

Tritomere tunings—order 2

Sharpness	Accidentals	Temperament	Equal temperaments
28	>, /, />, ↑\, ↑<, ↑, ↑>, ↑/, ↑/>, ↑↑\, ↑↑<, ↑↑, t<, t, t>, #↓↓, #↓↓>, #↓↓/, #↓\<, #↓\, #↓<, #↓, #↓>, #↓/, #\<, #\, #<, #	7 & 280	280, 287, 294c
30	>, /, />, ↑\, ↑<, ↑, ↑>, ↑/, ↑/>, ↑↑\, ↑↑<, ↑↑, ↑↑>, t<, t, t>, #↓↓<, #↓↓, #↓↓>, #↓↓/, #↓\<, #↓\, #↓<, #↓, #↓>, #↓/, #\<, #\, #<, #	Neutral trinity restriction	304, 311, 318, 325c
32	>, /, />, ↑\, ↑<, ↑, ↑>, ↑/, ↑/>, ↑↑\, ↑↑<, ↑↑, ↑↑>, ↑↑/, t<, t, t>, #↓↓\, #↓↓<, #↓↓, #↓↓>, #↓↓/, #↓\<, #↓\, #↓<, #↓, #↓>, #↓/, #\<, #\, #<, #	7 & 335	321ce, 328, 335, 342, 349, 356ce
34	>, /, />, ↑\, ↑<, ↑, ↑>, ↑/, ↑/>, ↑↑\, ↑↑<, ↑↑, ↑↑>, ↑↑/, ↑↑/>, t<, t, t>, #↓↓\<, #↓↓\, #↓↓<, #↓↓, #↓↓>, #↓↓/, #↓\<, #↓\, #↓<, #↓, #↓>, #↓/, #\<, #\, #<, #	7 & 359	352c, 359, 366, 373
36	>, /, />, ↑\, ↑<, ↑, ↑>, ↑/, ↑/>, ↑↑\, ↑↑<, ↑↑, ↑↑>, ↑↑/, ↑↑/>, ↑↑//, t<, t, t>, #↓↓\\, #↓↓\<, #↓↓\, #↓↓<, #↓↓, #↓↓>, #↓↓/, #↓\<, #↓\, #↓<, #↓, #↓>, #↓/, #\<, #\, #<, #	7 & 390	383c, 390, 397

311, 325c, 359 and 373 can be notated as either order 1 or order 2.

Hemitritomere tunings

Hemitritomere tunings

Sharpness	Accidentals	Temperament	Equal temperaments
12	>, ↑<, ↑, ↑>, t<, t, t>, #↓<, #↓, #↓>, #<, #	7 & 109	109, 116, 123ce
14	>, ↑<, ↑, ↑>, ↑/, t<, t, t>, #↓\, #↓<, #↓, #↓>, #<, #	7 & 140	133, 140, 147e
16	>, ↑<, ↑, ↑>, ↑↑<, ↑↑, t<, t, t>, #↓↓, #↓↓>, #↓<, #↓, #↓>, #<, #	7 & 157	157, 164, 171e

Tetartomere tunings

Tetartomere tunings—order 1

Sharpness	Accidentals	Temperament	Equal temperaments
19	/, //, ↑\, ↑, ↑/, ↑//, ↑↑\, ↑↑, t<, t>, #↓↓, #↓↓/, #↓\\, #↓\, #↓, #↓/, #\\, #\, #	7 & 193	186e, 193, 200, 207c
21	/, //, ↑\, ↑, ↑/, ↑//, ↑↑\, ↑↑, ↑↑/, t<, t>, #↓↓\, #↓↓, #↓↓/, #↓\\, #↓\, #↓, #↓/, #\\, #\, #	Brahmagupta restriction	210e, 217, 224, 231, 238c
23	/, //, ↑\, ↑, ↑/, ↑//, ↑↑\, ↑↑, ↑↑/, ↑↑//, t<, t>, #↓↓\\, #↓↓\, #↓↓, #↓↓/, #↓\\, #↓\, #↓, #↓/, #\\, #\, #	7 & 241	234c, 241, 248, 255

Tetartomere tunings—order 2

Sharpness	Accidentals	Temperament	Equal temperaments
40	>, /, />, //, ↑\<, ↑\, ↑<, ↑, ↑>, ↑/, ↑/>, ↑//, ↑↑\<, ↑↑\, ↑↑<, ↑↑, ↑↑>, ↑↑/, t<, t, t>, #↓↓\, #↓↓<, #↓↓, #↓↓>, #↓↓/, #↓↓/>, #↓\\, #↓\<, #↓\, #↓<, #↓, #↓>, #↓/, #↓/>, #\\, #\<, #\, #<, #	7 & 417	410e, 417, 424, 431c
42	>, /, />, //, ↑\<, ↑\, ↑<, ↑, ↑>, ↑/, ↑/>, ↑//, ↑↑\<, ↑↑\, ↑↑<, ↑↑, ↑↑>, ↑↑/, ↑↑/>, t<, t, t>, #↓↓\<, #↓↓\, #↓↓<, #↓↓, #↓↓>, #↓↓/, #↓↓/>, #↓\\, #↓\<, #↓\, #↓<, #↓, #↓>, #↓/, #↓/>, #\\, #\<, #\, #<, #	Neutral brahmagupta restriction	434e, 441, 448, 455
44	>, /, />, //, ↑\<, ↑\, ↑<, ↑, ↑>, ↑/, ↑/>, ↑//, ↑↑\<, ↑↑\, ↑↑<, ↑↑, ↑↑>, ↑↑/, ↑↑/>, ↑↑//, t<, t, t>, #↓↓\\, #↓↓\<, #↓↓\, #↓↓<, #↓↓, #↓↓>, #↓↓/, #↓↓/>, #↓\\, #↓\<, #↓\, #↓<, #↓, #↓>, #↓/, #↓/>, #\\, #\<, #\, #<, #	7 & 472	458ce, 465, 472, 479

441 and 455 can be notated as either order 1 or order 2.

Pemptomere tunings

Pemptomere tunings—order 1

Sharpness	Accidentals	Temperament	Equal temperaments
25	/, //, ↑\\, ↑\, ↑, ↑/, ↑//, ↑↑\\, ↑↑\, ↑↑, ↑↑/, t<, t>, #↓↓\, #↓↓, #↓↓/, #↓↓//, #↓\\, #↓\, #↓, #↓/, #↓//, #\\, #\, #	7 & 258	251e, 258, 265, 272c
27	/, //, ↑\\, ↑\, ↑, ↑/, ↑//, ↑↑\\, ↑↑\, ↑↑, ↑↑/, ↑↑//, t<, t>, #↓↓\\, #↓↓\, #↓↓, #↓↓/, #↓↓//, #↓\\, #↓\, #↓, #↓/, #↓//, #\\, #\, #	7 & 289	275e, 282, 289, 296, 303c

Pemptomere tunings—order 2

Sharpness	Accidentals	Temperament	Equal temperaments
52	>, /, />, //, //>, ↑\\, ↑\<, ↑\, ↑<, ↑, ↑>, ↑/, ↑/>, ↑//, ↑//>, ↑↑\\, ↑↑\<, ↑↑\, ↑↑<, ↑↑, ↑↑>, ↑↑/, ↑↑/>, ↑↑//, t<, t, t>, #↓↓\\, #↓↓\<, #↓↓\, #↓↓<, #↓↓, #↓↓>, #↓↓/, #↓↓/>, #↓\\<, #↓\\, #↓\<, #↓\, #↓<, #↓, #↓>, #↓/, #↓/>, #\\<, #\\, #\<, #\, #<, #	7 & 554	547e, 554, 561c
54	>, /, />, //, //>, ↑\\, ↑\<, ↑\, ↑<, ↑, ↑>, ↑/, ↑/>, ↑//, ↑//>, ↑↑\\, ↑↑\<, ↑↑\, ↑↑<, ↑↑, ↑↑>, ↑↑/, ↑↑/>, ↑↑//, ↑↑//>, t<, t, t>, #↓↓\\<, #↓↓\\, #↓↓\<, #↓↓\, #↓↓<, #↓↓, #↓↓>, #↓↓/, #↓↓/>, #↓\\<, #↓\\, #↓\<, #↓\, #↓<, #↓, #↓>, #↓/, #↓/>, #\\<, #\\, #\<, #\, #<, #	Neutral 7 & 289	571e, 578, 585

571e and 585 can be notated as either order 1 or order 2.

Hemipemptomere tunings

Hemipemptomere tunings

Sharpness	Accidentals	Temperament	Equal temperaments
24	>, /, ↑\, ↑<, ↑, ↑>, ↑/, ↑↑\, ↑↑<, ↑↑, t<, t, t>, #↓↓, #↓↓>, #↓↓/, #↓\, #↓<, #↓, #↓>, #↓/, #\, #<, #	7 & 246	239, 246, 253, 260c
26	>, /, ↑\, ↑<, ↑, ↑>, ↑/, ↑↑\, ↑↑<, ↑↑, ↑↑>, t<, t, t>, #↓↓<, #↓↓, #↓↓>, #↓↓/, #↓\, #↓<, #↓, #↓>, #↓/, #\, #<, #	7 & 270	256c, 263, 270, 277, 284, 291ce
28	>, /, ↑\, ↑<, ↑, ↑>, ↑/, ↑↑\, ↑↑<, ↑↑, ↑↑>, ↑↑/, t<, t, t>, #↓↓\, #↓↓<, #↓↓, #↓↓>, #↓↓/, #↓\, #↓<, #↓, #↓>, #↓/, #\, #<, #	7 & 294	287c, 294, 301, 308e

Connections to interval naming

Courtesy of collaboration between Aura and Lillian Hearne, syntonic–rastmic subchroma notation also has connections to SKULO interval names in which raising by the rastma is represented by "R" and lowering by the rastma is represented by "r".