Rational comma notation

Rational comma notation (RCN) is a musical notation system using either plaintext (ASCII) or richtext to notate any frequency in just intonation (JI). It also names the pitch classes, which are notes up to octave equivalence. It was developed in 2015-2017 by musician and music theorist David Ryan.

RCN is a combination of:

• 3-limit notation using a modified version of scientific pitch notation (SPN), e.g. C4, C5, F#3
• Rational commas for describing higher prime information in a frequency as microtonal adjustments of a Pythagorean frequency, e.g. [5], [1/7], [77/65]

Together these are able to notate the whole of free JI using notations of the form E4[5], Eb3[1/5], etc.

RCN as a general scheme comes in different versions. The version RCN_X (subscript) allocates commas [5], [7], [11], [13]... to primes 5, 7, 11, 13... using algorithm X. By using different algorithms, different versions of RCN emerge. The first paper linked to above contains an algorithm due to the author (RCN_DR) as well as a comparison to RCN from two other algorithms (RCN_SAG, RCN_KG). Different composers might prefer different algorithms since there are different criteria to optimise for prime commas. A consensus has not been reached yet as to the best algorithm. This need not hinder using RCN, since it is possible to translate between any two versions of RCN, or indeed between any two free-JI notations.

Key features of RCN

Every rational-numbered frequency can be given a notation in this system of the form Lz[x/y] where L is a 3-limit note label, z is octave number, [x/y] is a rational comma comprising prime commas multiplied together.

All pitch classes can be represented with a notation L[x/y]

3-limit frequencies have the simplest notations, and have no comma, so their form is Lz or L(-z) for negative octave number, similar to scientific pitch notation

3-limit notations correspond to notes on a standard musical stave, which may need one or more sharps or flats to fully describe

The plaintext version of the notation can be inputted by computer keyboard alone (ASCII characters)

Can transpose any JI music into any other key, by multiplying notations together. (In a computerised stave, this could be automated)

Shorthand exists for 5-limit notations such as L' = L[5], L = L[25], L. = L[1/5], etc

Shorthand exists for pitch classes, of form L~a_b which is the same pitch class as L[x/y]

These shorthands allow the 7-limit tone lattice of pitch classes (3-dimensional) to be drawn quickly and accurately

Optional shorthand for Pythagorean comma (B#3=531441/524288) and its inverse, which can help improve 3-limit note names

Note that this system (Jan 2017) has had the algorithm, the notation style, and the octave numbering amended from previous drafts (e.g. Sept 2015) due to feedback from relevant Facebook groups. It is recommended to use the style Lz[x/y] set out in the papers above, and not the style from previous drafts!

Prime and rational comma examples under DR algorithm

Prime:

• [5] = 80/81
• [7] = 63/64
• [11] = 33/32
• [13] = 26/27
• [17] = 2176/2187
• [19] = 513/512

Rational:

• [1/5] = 1/[5] = 81/80
• [35] = [5]*[7] = 35/36
• [5/13] = [5]/[13] = 40/39

Notation examples

2-limit

Octave equivalence class:

• C = {...1/4, 1/2, 1/1, 2/1, 4/1...}

Individual notes:

• C4 = 1/1
• C5 = 2/1
• C3 = 1/2
• C6 = 4/1
• C1 = 1/8
• C(-2) = 1/64

3-limit

Pythagorean - definitions of note names and sharps and flats are all here!

Octave equivalence classes:

• F = {...1/12, 1/6, 1/3, 2/3, 4/3, 8/3, 16/3...}
• G = {...3/16, 3/8, 3/4, 3/2, 3/1, 6/1, 12/1...}
• A = {...27/128, 27/64, 27/32, 27/16, 27/8, 27/4, 27/2, 27/1, 54/1...}
• Bb = {...1/18, 1/9, 2/9, 4/9, 8/9, 16/9, 32/9, 64/9...}

Basic note labels in diatonic scale (requiring no sharps or flats):

• C4 = 1/1
• D4 = 9/8
• E4 = 81/64
• F4 = 4/3
• G4 = 3/2
• A4 = 27/16
• B4 = 243/128

Individual notes:

• F5 = 8/3
• G5 = 3/1
• A8 = 27/1
• F2 = 1/3
• F#3 = 729/1024
• C#4 = 2187/2048 (equivalent to a sharp # character)
• Cb4 = 2048/2187 (equivalent to a flat b character)

Larger number of sharps or flats continue indefinitely up or down the Pythagorean series of fifths.

5-limit

Octave equivalence classes:

• E' = E[5] = {...5/8, 5/4, 5/2, 5/1, 10/1...}
• Ab. = Ab[1/5] = {...1/10, 1/5, 2/5, 4/5, 8/5...}

Individual notes:

• E'4 = E4[5] = 5/4
• Ab.4 = Ab4[1/5] = 4/5
• A'4 = A4[5] = 5/3
• Db.4 = Db4[1/5] = 16/15

7-limit

Octave equivalence classes:

• Bb~7 = Bb[7] = {...7/16, 7/8, 7/4, 7/2, 7/1, 14/1...}
• D_7 = D[1/7] = {...1/14, 1/7, 2/7, 4/7, 8/7, 16/7...}

Individual notes:

• Bb4[7] = 7/4
• D4[7] = 8/7
• F4[7] = 21/16
• Eb4[7] = 7/6

Higher p-limits

• F4[11] = 11/8
• A4[13] = 13/8
• C#4[17] = 17/16
• Eb4[19] = 19/16
• F#4[23] = 23/16
• Bb4[29] = 29/16
• C4[31] = 31/32 (which is itself a prime comma!)
• D4[37] = 37/32

Notations can be derived for p/2^n for all higher p, using the prime comma algorithm.

Future work

In future work, it is hoped to develop a computer based / online free-JI scoring system where the comma numbers (5, 1/7, 65/77, etc) are annotated directly onto notes in order to retune them from Pythagorean notes. The score would be Pythagorean, and commas used as accidentals to be able to compose in free JI and enable anyone to make free-JI music online. If anyone wants to help with this project, please contact David Ryan.