Ryan ASCII notation

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A system of using ASCII characters to notate any fractional frequency in Just Intonation (JI).
Also a system for naming octave equivalence classes in JI.
Created by the musician and theorist David Ryan

* Preprint: http://arxiv.org/pdf/1508.07739

**Abstract:**
Musical notation systems provide ways of recording which notes musicians should play at which times. One essential parameter described is the frequency. For twelve-note tuning systems the frequency can be described using letters A to G with sharp or flat symbols. For Just Intonation tuning systems these symbols are insufficient. This paper provides a system for describing any frequency which is a rational number multiplied by a suitable base frequency. Explicit notation is given for low prime numbers, and an algorithm for higher primes described. Also the differences between exact note names and note names up to octave equivalence are discussed.

**Key features:**
Can be inputted by computer keyboard alone (ASCII characters)
Can freely transpose keys in JI - done by multiplying notations - any two notations can be easily multiplied
Simple notations exist for 3-limit, 5-limit, 7-limit JI notes
Look-up table for providing ASCII notation for higher primes (11/8, 109/100, etc)
Algorithm for deriving these notations
Very compact notation for octave equivalence classes
Good for describing all the notes on a 5-limit or 7-limit tone lattice

**Challenges:**
For some notations it might not be clear whether exact frequencies or octave equivalence classes are being referenced.
- The suggested solution is to always include an item of octave information when the exact note is being described. E.g. use `0C for 1/1, not C
For a series of notes in ascending order of frequency, the octave numbering is not sequential.
- An example: the scale (1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1) has notations (`0C `1D' `0E' `0F `1G `0A' `1B' `1C) - the octave numbers are not sequential.

**Notation examples:**

__//2-limit://__
//Octave equivalence class://
C = {...1/4, 1/2, 1/1, 2/1, 4/1...}
//Individual notes://
`0C = 1/1
`C = `1C = 2/1 (definition of octave modifier using ` character to prefix)
,C = ,1C = 1/2 (definition of octave modifier using , character to prefix)
``C = `2C = 4/1
,,,C = ,3C = 1/8

__//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...}
//Individual notes://
`0F = 4/3 (definitions of the 7 note names here)
`0C = 1/1
`0G = 3/4
`0D = 9/16
`0A = 27/64
`0E = 81/256
`0B = 243/1024
`0Bb = 16/9 (definition of a flat b character)
`0F# = 729/4096 (definition of a sharp # character)
`0C# = 2187/16384 (this is equivalent to a sharp # character)
`0Cb = 16384/2187 (this is equivalent to a flat b character)

__//5-limit://__
//Octave equivalence classes://
E' = {...5/8, 5/4, 5/2, 5/1, 10/1...}
Ab. = {...1/10, 1/5, 2/5, 4/5, 8/5...}
//Individual notes://
`0E' = 5/4 (definition of ' modifier)
`0Ab. = 4/5 (definition of . modifier)
`0A' = 4/3
``E' = `2E' = 5/1
`B' = `1B' = 15/8

__//7-limit//__
//Octave equivalence classes://
Bb~7 = {...7/16, 7/8, 7/4, 7/2, 7/1, 14/1...}
D_7 = {...1/14, 1/7, 2/7, 4/7, 8/7, 16/7...}
//Individual notes://
`0Bb~7 = 7/8 (definition of ~7 modifier)
`0D_7 = 8/7 (definition of _7 modifier)
`0F~7 = 21/16
`Bb~7 = 7/4
`F~7 = 21/16
`0Eb~7 = 7/6

__//Higher p-limits//__
`0F#~11 = 11/8 (definition of ~11 modifier)
`0Gb_11 = 8/11 (definition of _11 modifier)
``F#~11 = 11/2
`0B~11 = 11/6
`0Ab~13 = 13/16 (definition of ~13 modifier)
`0C#~17 = 17/16
`0Eb~19 = 19/16
`0F#~23 = 23/32
etc (separate definition for each prime) and octave equivalence classes can be found using the procedure above.

**Calculation examples**
The golden rule in this notation is: to derive notation for a more complicated fraction, break it down into simpler fractions; fractions with notation already known. In particular, separate out the fractions for each higher prime.

55/32 = (11/8)*(5/4) = `0F#~11 * `0E' = `0A#'~11
7/5 = (7/8)*(4/5)*(2/1) = `0Bb~7 * `0Ab. * `1C = `1Bb.~7 * `0Ab = `1Gb.~7
30/1 = (2/1)*(3/1)*(5/1) = (32/1)*(3/4)*(5/4) = `````C * `0G * `0E' = `````G' * `0E = `````B' (Notice how modifiers ` for octave and ' for prime 5 can be moved about freely)
19/13 = (19/16) * (16/13) = `0Eb~19 * (`0Ab~13)^-1 = `0Eb~19 * `0E_13 = `0G~19_13

Calculating using octave equivalence classes is easier, since you don't have to keep track of the powers of two:
55/32 class = 5 class * 11 class = E' * F#~11 = A#'~11
7/5 class = 7 class * (1/5) class = Bb~7 * Ab. = Gb.~7

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

Original HTML content:

<html><head><title>Ryan ASCII notation</title></head><body>A system of using ASCII characters to notate any fractional frequency in Just Intonation (JI).<br />
Also a system for naming octave equivalence classes in JI.<br />
Created by the musician and theorist David Ryan<br />
<br />
<ul><li>Preprint: <!-- ws:start:WikiTextUrlRule:113:http://arxiv.org/pdf/1508.07739 --><a class="wiki_link_ext" href="http://arxiv.org/pdf/1508.07739" rel="nofollow">http://arxiv.org/pdf/1508.07739</a><!-- ws:end:WikiTextUrlRule:113 --></li></ul><br />
<strong>Abstract:</strong><br />
Musical notation systems provide ways of recording which notes musicians should play at which times. One essential parameter described is the frequency. For twelve-note tuning systems the frequency can be described using letters A to G with sharp or flat symbols. For Just Intonation tuning systems these symbols are insufficient. This paper provides a system for describing any frequency which is a rational number multiplied by a suitable base frequency. Explicit notation is given for low prime numbers, and an algorithm for higher primes described. Also the differences between exact note names and note names up to octave equivalence are discussed.<br />
<br />
<strong>Key features:</strong><br />
Can be inputted by computer keyboard alone (ASCII characters)<br />
Can freely transpose keys in JI - done by multiplying notations - any two notations can be easily multiplied<br />
Simple notations exist for 3-limit, 5-limit, 7-limit JI notes<br />
Look-up table for providing ASCII notation for higher primes (11/8, 109/100, etc)<br />
Algorithm for deriving these notations<br />
Very compact notation for octave equivalence classes<br />
Good for describing all the notes on a 5-limit or 7-limit tone lattice<br />
<br />
<strong>Challenges:</strong><br />
For some notations it might not be clear whether exact frequencies or octave equivalence classes are being referenced.<br />
- The suggested solution is to always include an item of octave information when the exact note is being described. E.g. use `0C for 1/1, not C<br />
For a series of notes in ascending order of frequency, the octave numbering is not sequential.<br />
- An example: the scale (1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1) has notations (`0C `1D' `0E' `0F `1G `0A' `1B' `1C) - the octave numbers are not sequential.<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 />
`0C = 1/1<br />
`C = `1C = 2/1 (definition of octave modifier using ` character to prefix)<br />
,C = ,1C = 1/2 (definition of octave modifier using , character to prefix)<br />
``C = `2C = 4/1<br />
,,,C = ,3C = 1/8<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>Individual notes:</em><br />
`0F = 4/3 (definitions of the 7 note names here)<br />
`0C = 1/1<br />
`0G = 3/4<br />
`0D = 9/16<br />
`0A = 27/64<br />
`0E = 81/256<br />
`0B = 243/1024<br />
`0Bb = 16/9 (definition of a flat b character)<br />
`0F# = 729/4096 (definition of a sharp # character)<br />
`0C# = 2187/16384 (this is equivalent to a sharp # character)<br />
`0Cb = 16384/2187 (this is equivalent to a flat b character)<br />
<br />
<u><em>5-limit:</em></u><br />
<em>Octave equivalence classes:</em><br />
E' = {...5/8, 5/4, 5/2, 5/1, 10/1...}<br />
Ab. = {...1/10, 1/5, 2/5, 4/5, 8/5...}<br />
<em>Individual notes:</em><br />
`0E' = 5/4 (definition of ' modifier)<br />
`0Ab. = 4/5 (definition of . modifier)<br />
`0A' = 4/3<br />
``E' = `2E' = 5/1<br />
`B' = `1B' = 15/8<br />
<br />
<u><em>7-limit</em></u><br />
<em>Octave equivalence classes:</em><br />
Bb~7 = {...7/16, 7/8, 7/4, 7/2, 7/1, 14/1...}<br />
D_7 = {...1/14, 1/7, 2/7, 4/7, 8/7, 16/7...}<br />
<em>Individual notes:</em><br />
`0Bb~7 = 7/8 (definition of ~7 modifier)<br />
`0D_7 = 8/7 (definition of _7 modifier)<br />
`0F~7 = 21/16<br />
`Bb~7 = 7/4<br />
`F~7 = 21/16<br />
`0Eb~7 = 7/6<br />
<br />
<u><em>Higher p-limits</em></u><br />
`0F#~11 = 11/8 (definition of ~11 modifier)<br />
`0Gb_11 = 8/11 (definition of _11 modifier)<br />
``F#~11 = 11/2<br />
`0B~11 = 11/6<br />
`0Ab~13 = 13/16 (definition of ~13 modifier)<br />
`0C#~17 = 17/16<br />
`0Eb~19 = 19/16<br />
`0F#~23 = 23/32<br />
etc (separate definition for each prime) and octave equivalence classes can be found using the procedure above.<br />
<br />
<strong>Calculation examples</strong><br />
The golden rule in this notation is: to derive notation for a more complicated fraction, break it down into simpler fractions; fractions with notation already known. In particular, separate out the fractions for each higher prime.<br />
<br />
55/32 = (11/8)*(5/4) = `0F#~11 * `0E' = `0A#'~11<br />
7/5 = (7/8)*(4/5)*(2/1) = `0Bb~7 * `0Ab. * `1C = `1Bb.~7 * `0Ab = `1Gb.~7<br />
30/1 = (2/1)*(3/1)*(5/1) = (32/1)*(3/4)*(5/4) = <!-- ws:start:WikiTextRawRule:00:```` --><!-- ws:end:WikiTextRawRule:00 -->`C * `0G * `0E' = <!-- ws:start:WikiTextRawRule:01:```` --><!-- ws:end:WikiTextRawRule:01 -->`G' * `0E = <!-- ws:start:WikiTextRawRule:02:```` --><!-- ws:end:WikiTextRawRule:02 -->`B' (Notice how modifiers ` for octave and ' for prime 5 can be moved about freely)<br />
19/13 = (19/16) * (16/13) = `0Eb~19 * (`0Ab~13)^-1 = `0Eb~19 * `0E_13 = `0G~19_13<br />
<br />
Calculating using octave equivalence classes is easier, since you don't have to keep track of the powers of two:<br />
55/32 class = 5 class * 11 class = E' * F#~11 = A#'~11<br />
7/5 class = 7 class * (1/5) class = Bb~7 * Ab. = Gb.~7<br />
<br />
<strong>Other links</strong><br />
Some music created using this notation is available at:<br />
<ul><li>Dave Ryan's SoundCloud page: <!-- ws:start:WikiTextUrlRule:114:https://soundcloud.com/daveryan23 --><a class="wiki_link_ext" href="https://soundcloud.com/daveryan23" rel="nofollow">https://soundcloud.com/daveryan23</a><!-- ws:end:WikiTextUrlRule:114 --></li></ul></body></html>