Syntonic-rastmic subchroma notation

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The syntonic-rastmic subchroma notation is a notation scheme developed by Aura et al.[1] 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
Synflat (81/80)-1 [4 -4 1 ɔ

* "^" and "v" are acceptable variants of the syntonic accidentals in textual representation. Those are handy when input of non-ASCII characters are not available. Note: The textual representation forms of the syntonic and demisyntonic accidentals need to be checked over, while the graphical forms of the demisyntonic accidentals are a work in progress.

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, #ɔ, # Tetracot 27e, 34, 41, 48, 55c
6 +, tɔ, t, t+, #ɔ, # Larry 51ce, 58, 65, 72, 79
8 +, ++, tɔ, t, t+, #ɔɔ, #ɔ, # 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>, #ɔ, # Hitchcock restriction 39, 46, 53, 60e
7 +, ++, t<, t>, #ɔɔ, #ɔ, # 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>, #ɔ, #<, # Tetracot extension 68, 75, 82e
10 >, +, +>, t<, t, t>, #ɔ<, #ɔ, #<, # 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>, #ɔ, #\, # Sevond extension 56, 63, 70c
9 /, +, +/, t<, t>, #ɔ\, #ɔ, #\, # 7 & 87 80, 87, 94, 101c
11 /, +, +/, ++, t<, t>, #ɔɔ, #ɔ\, #ɔ, #\, # Twentcufo 104c, 111, 118, 125, 132e
13 /, +, +/, ++, ++/, t<, t>, #ɔɔ\, #ɔɔ, #ɔ\, #ɔ, #\, # 7 & 149 135c, 142, 149, 156e
Deuteromere tunings – order 2
Sharpness Accidentals Temperament Equal temperaments
18 >, /, +<, +, +>, +/, ++<, t<, t,
t>, #ɔɔ>, #ɔ\, #ɔ<, #ɔ, #ɔ>, #\, #<, #
Neutral 7 & 87 174, 181, 188c
20 >, /, +<, +, +>, +/, ++<, ++, t<, t,
t>, #ɔɔ, #ɔɔ>, #ɔ\, #ɔ<, #ɔ, #ɔ>, #\, #<, #
7 & 205 198, 205, 212, 219ce
22 >, /, +<, +, +>, +/, ++<, ++, ++>, t<,
t, t>, #ɔɔ<, #ɔɔ, #ɔɔ>, #ɔ\, #ɔ<, #ɔ, #ɔ>, #\, #<, #
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>, #ɔɔ/, #ɔ\, #ɔ, #ɔ/, #\, # 7 & 121 121, 128, 135, 142c
15 /, +\, +, +/, ++\, ++, t<, t>, #ɔɔ, #ɔɔ/, #ɔ\, #ɔ, #ɔ/, #\, # Trinity restriction 145, 152, 159, 166, 173c
17 /, +\, +, +/, ++\, ++, ++/, t<, t>, #ɔɔ\, #ɔɔ, #ɔɔ/, #ɔ\, #ɔ, #ɔ/, #\, # 7 & 183 169c, 176, 183, 190, 197e
19 /, +\, +, +/, ++\, ++, ++/, ++//, t<, t>, #ɔɔ\\, #ɔɔ\, #ɔɔ, #ɔɔ/, #ɔ\, #ɔ, #ɔ/, #\, # 7 & 207 200c, 207, 214
Tritomere tunings – order 2
Sharpness Accidentals Temperament Equal temperaments
28 >, /, />, +\, +<, +, +>, +/, +/>, ++\, ++<, ++, t<, t,
t>, #ɔɔ, #ɔɔ>, #ɔɔ/, #ɔ\<, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #\<, #\, #<, #
7 & 280 280, 287, 294c
30 >, /, />, +\, +<, +, +>, +/, +/>, ++\, ++<, ++, ++>, t<, t,
t>, #ɔɔ<, #ɔɔ, #ɔɔ>, #ɔɔ/, #ɔ\<, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #\<, #\, #<, #
Neutral trinity restriction 304, 311, 318, 325c
32 >, /, />, +\, +<, +, +>, +/, +/>, ++\, ++<, ++, ++>, ++/, t<, t,
t>, #ɔɔ\, #ɔɔ<, #ɔɔ, #ɔɔ>, #ɔɔ/, #ɔ\<, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #\<, #\, #<, #
7 & 335 321ce, 328, 335, 342, 349, 356ce
34 >, /, />, +\, +<, +, +>, +/, +/>, ++\, ++<, ++, ++>, ++/, ++/>, t<, t,
t>, #ɔɔ\<, #ɔɔ\, #ɔɔ<, #ɔɔ, #ɔɔ>, #ɔɔ/, #ɔ\<, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #\<, #\, #<, #
7 & 359 352c, 359, 366, 373
36 >, /, />, +\, +<, +, +>, +/, +/>, ++\, ++<, ++, ++>, ++/, ++/>, ++//, t<, t,
t>, #ɔɔ\\, #ɔɔ\<, #ɔɔ\, #ɔɔ<, #ɔɔ, #ɔɔ>, #ɔɔ/, #↓\<, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #\<, #\, #<, #
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>, #ɔ<, #ɔ, #ɔ>, #<, # 7 & 109 109, 116, 123ce
14 >, +<, +, +>, +/, t<, t, t>, #ɔ\, #ɔ<, #ɔ, #ɔ>, #<, # 7 & 140 133, 140, 147e
16 >, +<, +, +>, ++<, ++, t<, t, t>, #ɔɔ, #ɔɔ>, #ɔ<, #ɔ, #ɔ>, #<, # 7 & 157 157, 164, 171e

Tetartomere tunings

Tetartomere tunings – order 1
Sharpness Accidentals Temperament Equal temperaments
19 /, //, +\, +, +/, +//, ++\, ++, t<, t>, #ɔɔ, #ɔɔ/, #ɔ\\, #ɔ\, #ɔ, #ɔ/, #\\, #\, # 7 & 193 186e, 193, 200, 207c
21 /, //, +\, +, +/, +//, ++\, ++, ++/, t<, t>, #ɔɔ\, #ɔɔ, #ɔɔ/, #ɔ\\, #ɔ\, #ɔ, #ɔ/, #\\, #\, # Brahmagupta restriction 210e, 217, 224, 231, 238c
23 /, //, +\, +, +/, +//, ++\, ++, ++/, ++//, t<, t>, #ɔɔ\\, #ɔɔ\, #ɔɔ, #ɔɔ/, #ɔ\\, #ɔ\, #ɔ, #ɔ/, #\\, #\, # 7 & 241 234c, 241, 248, 255
Tetartomere tunings – order 2
Sharpness Accidentals Temperament Equal temperaments
40 >, /, />, //, +\<, +\, +<, +, +>, +/, +/>, +//, ++\<, ++\, ++<, ++, ++>, ++/, t<, t,
t>, #ɔɔ\, #ɔɔ<, #ɔɔ, #ɔɔ>, #ɔɔ/, #ɔɔ/>, #ɔ\\, #ɔ\<, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #ɔ/>, #\\, #\<, #\, #<, #
7 & 417 410e, 417, 424, 431c
42 >, /, />, //, +\<, +\, +<, +, +>, +/, +/>, +//, ++\<, ++\, ++<, ++, ++>, ++/, ++/>, t<, t,
t>, #ɔɔ\<, #ɔɔ\, #ɔɔ<, #ɔɔ, #ɔɔ>, #ɔɔ/, #ɔɔ/>, #ɔ\\, #ɔ\<, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #ɔ/>, #\\, #\<, #\, #<, #
Neutral brahmagupta restriction 434e, 441, 448, 455
44 >, /, />, //, +\<, +\, +<, +, +>, +/, +/>, +//, ++\<, ++\, ++<, ++, ++>, ++/, ++/>, ++//, t<, t,
t>, #ɔɔ\\, #ɔɔ\<, #ɔɔ\, #ɔɔ<, #ɔɔ, #ɔɔ>, #ɔɔ/, #ɔɔ/>, #ɔ\\, #ɔ\<, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #ɔ/>, #\\, #\<, #\, #<, #
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>, #ɔɔ\, #ɔɔ, #ɔɔ/, #ɔɔ//, #ɔ\\, #ɔ\, #ɔ, #ɔ/, #ɔ//, #\\, #\, # 7 & 258 251e, 258, 265, 272c
27 /, //, +\\, +\, +, +/, +//, ++\\, ++\, ++, ++/, ++//, t<, t>, #ɔɔ\\, #ɔɔ\, #ɔɔ, #ɔɔ/, #ɔɔ//, #ɔ\\, #ɔ\, #ɔ, #ɔ/, #ɔ//, #\\, #\, # 7 & 289 275e, 282, 289, 296, 303c
Pemptomere tunings – order 2
Sharpness Accidentals Temperament Equal temperaments
52 >, /, />, //, //>, +\\, +\<, +\, +<, +, +>, +/, +/>, +//, +//>, ++\\, ++\<, ++\, ++<, ++, ++>, ++/, ++/>, ++//, t<, t,
t>, #ɔɔ\\, #ɔɔ\<, #ɔɔ\, #ɔɔ<, #ɔɔ, #ɔɔ>, #ɔɔ/, #ɔɔ/>, #ɔ\\<, #ɔ\\, #ɔ\<, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #ɔ/>, #\\<, #\\, #\<, #\, #<, #
7 & 554 547e, 554, 561c
54 >, /, />, //, //>, +\\, +\<, +\, +<, +, +>, +/, +/>, +//, +//>, ++\\, ++\<, ++\, ++<, ++, ++>, ++/, ++/>, ++//, ++//>, t<, t,
t>, #ɔɔ\\<, #ɔɔ\\, #ɔɔ\<, #ɔɔ\, #ɔɔ<, #ɔɔ, #ɔɔ>, #ɔɔ/, #ɔɔ/>, #ɔ\\<, #ɔ\\, #ɔ\<, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #ɔ/>, #\\<, #\\, #\<, #\, #<, #
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>, #ɔɔ, #ɔɔ>, #ɔɔ/, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #\, #<, # 7 & 246 239, 246, 253, 260c
26 >, /, +\, +<, +, +>, +/, ++\, ++<, ++, ++>, t<, t, t>, #ɔɔ<, #ɔɔ, #ɔɔ>, #ɔɔ/, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #\, #<, # 7 & 270 256c, 263, 270, 277, 284, 291ce
28 >, /, +\, +<, +, +>, +/, ++\, ++<, ++, ++>, ++/, t<, t, t>, #ɔɔ\, #ɔɔ<, #ɔɔ, #ɔɔ>, #ɔɔ/, #ɔ\, #ɔ<, #ɔ, #ɔ>, #ɔ/, #\, #<, # 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".

Notes

  1. Other contributors include Flora Canou and HEHEHE I AM A SUPAHSTAR SAGA.