Rational comma notation

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A system using either plaintext (ASCII) or richtext to notate any frequency in Just Intonation (JI).
Also names the pitch classes, which are notes up to octave equivalence.
Developed in 2015-2017 by the musician and music theorist David Ryan
* Paper (pre-print) for defining all the prime commas by algorithm and the basic notation: http://arxiv.org/abs/1612.01860 - also compares between some different choices for algorithm.
* Paper (pre-print) for defining how the notation breaks down into components and enables key changes and transposition: http://arxiv.org/abs/1508.07739

**Abstract (for 2 papers above):**
In Just Intonation every rational frequency has a prime factorisation. This can be reconstructed from an approximate 3-limit component, and a series of microtonal 'prime comma' adjustments, one for each higher prime (5 and above) present in the original prime factorisation. Each prime comma is of the form [p] = 2^a 3^b p. This means that different prime commas don't interact, which is helpful for notation since the effect of each higher prime is separated out, which will aid mapping between notations and frequencies. Prime commas are assigned by algorithm. The algorithm performs a tradeoff between using low numbers in the comma fraction, and keeping the comma as small as possible. All commas are microtonal, less than a semitone, the widest comma is thought to be [13]=26/27. Commas are only considered between a specific set of bounds for 3^b, since larger values of b become unmusical. Plaintext (ASCII) nnotations are defined of the form Lz[x/y] which have alternative forms in richtext. The notation breaks down into a set of four components: octave number, diatonic scale note, sharps/flats, prime commas or rational comma. This aids inversion, multiplication, division of notations, and enables transposition of music into any other key.

**Key features:**
Every rational-numbered frequency can be given a notation in this system of the form Lz[x/y] where L is 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

Note that this system (Dec 2016) 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!

**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.

**Other links**

Calculation examples are given in the second paper listed above.

Some music created using this notation is available at:
* Dave Ryan's SoundCloud page: http://soundcloud.com/daveryan23/tracks

Further work would be a computer / web stave where the commas could be used as accidentals, to enable anyone to make free-JI music online. If anyone wants to help with this project (as of Dec 2016), please contact David Ryan.

Original HTML content:

<html><head><title>Rational Comma Notation (RCN)</title></head><body>A system using either plaintext (ASCII) or richtext to notate any frequency in Just Intonation (JI).<br />
Also names the pitch classes, which are notes up to octave equivalence.<br />
Developed in 2015-2017 by the musician and music theorist David Ryan<br />
<ul><li>Paper (pre-print) for defining all the prime commas by algorithm and the basic notation: <!-- ws:start:WikiTextUrlRule:104:http://arxiv.org/abs/1612.01860 --><a class="wiki_link_ext" href="http://arxiv.org/abs/1612.01860" rel="nofollow">http://arxiv.org/abs/1612.01860</a><!-- ws:end:WikiTextUrlRule:104 --> - also compares between some different choices for algorithm.</li><li>Paper (pre-print) for defining how the notation breaks down into components and enables key changes and transposition: <!-- ws:start:WikiTextUrlRule:105:http://arxiv.org/abs/1508.07739 --><a class="wiki_link_ext" href="http://arxiv.org/abs/1508.07739" rel="nofollow">http://arxiv.org/abs/1508.07739</a><!-- ws:end:WikiTextUrlRule:105 --></li></ul><br />
<strong>Abstract (for 2 papers above):</strong><br />
In Just Intonation every rational frequency has a prime factorisation. This can be reconstructed from an approximate 3-limit component, and a series of microtonal 'prime comma' adjustments, one for each higher prime (5 and above) present in the original prime factorisation. Each prime comma is of the form [p] = 2^a 3^b p. This means that different prime commas don't interact, which is helpful for notation since the effect of each higher prime is separated out, which will aid mapping between notations and frequencies. Prime commas are assigned by algorithm. The algorithm performs a tradeoff between using low numbers in the comma fraction, and keeping the comma as small as possible. All commas are microtonal, less than a semitone, the widest comma is thought to be [13]=26/27. Commas are only considered between a specific set of bounds for 3^b, since larger values of b become unmusical. Plaintext (ASCII) nnotations are defined of the form Lz[x/y] which have alternative forms in richtext. The notation breaks down into a set of four components: octave number, diatonic scale note, sharps/flats, prime commas or rational comma. This aids inversion, multiplication, division of notations, and enables transposition of music into any other key.<br />
<br />
<strong>Key features:</strong><br />
Every rational-numbered frequency can be given a notation in this system of the form Lz[x/y] where L is note label, z is octave number, [x/y] is a rational comma comprising prime commas multiplied together.<br />
All pitch classes can be represented with a notation L[x/y]<br />
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<br />
3-limit notations correspond to notes on a standard musical stave, which may need one or more sharps or flats to fully describe<br />
The plaintext version of the notation can be inputted by computer keyboard alone (ASCII characters)<br />
Can transpose any JI music into any other key, by multiplying notations together. (In a computerised stave, this could be automated)<br />
Shorthand exists for 5-limit notations such as L' = L[5], L'' = L[25], L. = L[1/5], etc<br />
Shorthand exists for pitch classes, of form L~a_b which is the same pitch class as L[x/y]<br />
These shorthands allow the 7-limit tone lattice of pitch classes (3-dimensional) to be drawn quickly and accurately<br />
<br />
Note that this system (Dec 2016) 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!<br />
<br />
<strong>Notation examples:</strong><br />
<br />
<u><em>2-limit:</em></u><br />
<em>Octave equivalence class:</em><br />
C = {...1/4, 1/2, 1/1, 2/1, 4/1...}<br />
<em>Individual notes:</em><br />
C4 = 1/1<br />
C5 = 2/1<br />
C3 = 1/2<br />
C6 = 4/1<br />
C1 = 1/8<br />
C(-2) = 1/64<br />
<br />
<u><em>3-limit - Pythagorean - definitions of note names and sharps and flats are all here!</em></u><br />
<em>Octave equivalence classes:</em><br />
F = {...1/12, 1/6, 1/3, 2/3, 4/3, 8/3, 16/3...}<br />
G = {...3/16, 3/8, 3/4, 3/2, 3/1, 6/1, 12/1...}<br />
A = {...27/128, 27/64, 27/32, 27/16, 27/8, 27/4, 27/2, 27/1, 54/1...}<br />
Bb = {...1/18, 1/9, 2/9, 4/9, 8/9, 16/9, 32/9, 64/9...}<br />
<em>Basic Note Labels in Diatonic Scale (requiring no sharps or flats)</em><br />
C4 = 1/1<br />
D4 = 9/8<br />
E4 = 81/64<br />
F4 = 4/3<br />
G4 = 3/2<br />
A4 = 27/16<br />
B4 = 243/128<br />
<em>Individual notes:</em><br />
F5 = 8/3<br />
G5 = 3/1<br />
A8 = 27/1<br />
F2 = 1/3<br />
F#3 = 729/1024<br />
C#4 = 2187/2048 (equivalent to a sharp # character)<br />
Cb4 = 2048/2187 (equivalent to a flat b character)<br />
Larger number of sharps or flats continue indefinitely up or down the Pythagorean series of fifths.<br />
<br />
<u><em>5-limit:</em></u><br />
<em>Octave equivalence classes:</em><br />
E' = E[5] = {...5/8, 5/4, 5/2, 5/1, 10/1...}<br />
Ab. = Ab[1/5] = {...1/10, 1/5, 2/5, 4/5, 8/5...}<br />
<em>Individual notes:</em><br />
E'4 = E4[5] = 5/4<br />
Ab.4 = Ab4[1/5] = 4/5<br />
A'4 = A4[5] = 5/3<br />
Db.4 = Db4[1/5] = 16/15<br />
<br />
<u><em>7-limit</em></u><br />
<em>Octave equivalence classes:</em><br />
Bb~7 = Bb[7] = {...7/16, 7/8, 7/4, 7/2, 7/1, 14/1...}<br />
D_7 = D[1/7] = {...1/14, 1/7, 2/7, 4/7, 8/7, 16/7...}<br />
<em>Individual notes:</em><br />
Bb4[7] = 7/4<br />
D4[7] = 8/7<br />
F4[7] = 21/16<br />
Eb4[7] = 7/6<br />
<br />
<u><em>Higher p-limits</em></u><br />
F4[11] = 11/8<br />
A4[13] = 13/8<br />
C#4[17] = 17/16<br />
Eb4[19] = 19/16<br />
F#4[23] = 23/16<br />
Bb4[29] = 29/16<br />
C4[31] = 31/32 (which is itself a prime comma!)<br />
D4[37] = 37/32<br />
Notations can be derived for p/2^n for all higher p, using the prime comma algorithm.<br />
<br />
<strong>Other links</strong><br />
<br />
Calculation examples are given in the second paper listed above.<br />
<br />
Some music created using this notation is available at:<br />
<ul><li>Dave Ryan's SoundCloud page: <!-- ws:start:WikiTextUrlRule:106:http://soundcloud.com/daveryan23/tracks --><a class="wiki_link_ext" href="http://soundcloud.com/daveryan23/tracks" rel="nofollow">http://soundcloud.com/daveryan23/tracks</a><!-- ws:end:WikiTextUrlRule:106 --></li></ul><br />
Further work would be a computer / web stave where the commas could be used as accidentals, to enable anyone to make free-JI music online. If anyone wants to help with this project (as of Dec 2016), please contact David Ryan.</body></html>
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