Rational comma notation: Difference between revisions

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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 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.

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

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

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

__//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...}

Bb4[7] = 7/4

D4[7] = 8/7

F4[7] = 21/16

Eb4[7] = 7/6

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

Calculation examples are given in the second paper listed above.

Some music created using this notation is available at:

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.&lt;br /&gt;

&lt;strong&gt;Key features:&lt;/strong&gt;&lt;br /&gt;

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&lt;br /&gt;

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

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

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!&lt;br /&gt;

&lt;strong&gt;Notation examples:&lt;/strong&gt;&lt;br /&gt;

Eb4[7] = 7/6&lt;br /&gt;

F4[11] = 11/8&lt;br /&gt;