Syntonic–rastmic subchroma notation

The 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 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 accidentals

Syntonic accidentals

The syntonic 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⟩	↑
(None)	(81/80)0	[0⟩
Synflat	(81/80)-1	[4 -4 1⟩	↓
…	…	…	…

* "^" and "v" are acceptable variants of textual representation. Those are handy when input of non-ASCII characters are not available.

Rastmic and demirastmic accidentals

Rastmic accidentals

Demirastmic accidentals (WIP)

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. Note: The graphical forms of demirastmic accidentals are work in progress.

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

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.

Connections to interval naming

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