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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
#REDIRECT[[Rational comma notation]]
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:daveryan23|daveryan23]] and made on <tt>2015-11-13 07:56:28 UTC</tt>.<br>
: The original revision id was <tt>566321253</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<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 of using ASCII characters to notate any fractional frequency in Just Intonation. 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.
 
**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:**
Octaves are not sequential - easier to understand octave equivalence classes than exact notes . ( Example: C = 1/1 F = 4/3 G = 3/4 but 3/2 = `G so 3/2 requires an octave modifier to describe.)
 
**Notation examples:**
 
//2-limit://
C = 1/1
`C = 2/1 (definition of octave modifier using ` character to prefix)
,C = 1/2 (definition of octave modifier using , character to prefix)
,,,C = 1/8
 
//3-limit://
F = 4/3 (definitions of the 7 note names here)
C = 1/1
G = 3/4
D = 9/16
A = 27/64
E = 81/256
B = 243/1024
Bb = 16/9 (definition of a flat b character)
F# = 729/4096 (definition of a sharp # character)
C# = 2187/16384 (this is equivalent to a sharp # character)
Cb = 16384/2187 (this is equivalent to a flat b character)
`G = 3/2
```D = 9/2
 
//5-limit://
E' = 5/4 (definition of ' modifier)
Ab. = 4/5 (definition of . modifier)
A' = 4/3
``E' = 5/1
`B' = 15/8
 
//7-limit//
Bb~7 = 7/8 (definition of ~7 modifier)
D_7 = 8/7 (definition of _7 modifier)
F~7 = 21/16
`Bb~7 = 7/4
`F~7 = 21/16
Eb~7 = 7/6
 
//Higher p-limits//
F#~11 = 11/8 (definition of ~11 modifier)
Gb_11 = 8/11 (definition of _11 modifier)
``F#~11 = 11/2
B~11 = 11/6
Ab~13 = 13/16
C#~17 = 17/16
Eb~19 = 19/16
F#~23 = 23/32
etc (separate definition for each prime)
 
**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) = F#~11 * E' = A#'~11
7/5 = (7/8)*(4/5)*(2/1) = Bb~7 * Ab. * `C = `Bb.~7 * Ab = `Gb.~7
30/1 = (2/1)*(3/1)*(5/1) = (32/1)*(3/4)*(5/4) = `````C * G * E' = `````G' * E = `````B' (Notice how modifiers ` for octave and ' for prime 5 can be moved about freely)
19/13 = (19/16) * (16/13) = Eb~19 * (Ab~13)^-1 = Eb~19 * E_13 = G~19_13
 
**Other links**
Some music created using this notation is available at:
* Dave Ryan's SoundCloud page: https://soundcloud.com/daveryan23</pre></div>
<h4>Original HTML content:</h4>
<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Ryan ASCII notation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A system of using ASCII characters to notate any fractional frequency in Just Intonation. Created by the musician and theorist David Ryan&lt;br /&gt;
&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Preprint: &lt;!-- ws:start:WikiTextUrlRule:88:http://arxiv.org/pdf/1508.07739 --&gt;&lt;a class="wiki_link_ext" href="http://arxiv.org/pdf/1508.07739" rel="nofollow"&gt;http://arxiv.org/pdf/1508.07739&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:88 --&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;strong&gt;Abstract:&lt;/strong&gt;&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Key features:&lt;/strong&gt;&lt;br /&gt;
Can be inputted by computer keyboard alone (ASCII characters)&lt;br /&gt;
Can freely transpose keys in JI - done by multiplying notations - any two notations can be easily multiplied&lt;br /&gt;
Simple notations exist for 3-limit, 5-limit, 7-limit JI notes&lt;br /&gt;
Look-up table for providing ASCII notation for higher primes (11/8, 109/100, etc)&lt;br /&gt;
Algorithm for deriving these notations&lt;br /&gt;
Very compact notation for octave equivalence classes&lt;br /&gt;
Good for describing all the notes on a 5-limit or 7-limit tone lattice&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Challenges:&lt;/strong&gt;&lt;br /&gt;
Octaves are not sequential - easier to understand octave equivalence classes than exact notes . ( Example: C = 1/1 F = 4/3 G = 3/4 but 3/2 = `G so 3/2 requires an octave modifier to describe.)&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Notation examples:&lt;/strong&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;em&gt;2-limit:&lt;/em&gt;&lt;br /&gt;
C = 1/1&lt;br /&gt;
`C = 2/1 (definition of octave modifier using ` character to prefix)&lt;br /&gt;
,C = 1/2 (definition of octave modifier using , character to prefix)&lt;br /&gt;
,,,C = 1/8&lt;br /&gt;
&lt;br /&gt;
&lt;em&gt;3-limit:&lt;/em&gt;&lt;br /&gt;
F = 4/3 (definitions of the 7 note names here)&lt;br /&gt;
C = 1/1&lt;br /&gt;
G = 3/4&lt;br /&gt;
D = 9/16&lt;br /&gt;
A = 27/64&lt;br /&gt;
E = 81/256&lt;br /&gt;
B = 243/1024&lt;br /&gt;
Bb = 16/9 (definition of a flat b character)&lt;br /&gt;
F# = 729/4096 (definition of a sharp # character)&lt;br /&gt;
C# = 2187/16384 (this is equivalent to a sharp # character)&lt;br /&gt;
Cb = 16384/2187 (this is equivalent to a flat b character)&lt;br /&gt;
`G = 3/2&lt;br /&gt;
```D = 9/2&lt;br /&gt;
&lt;br /&gt;
&lt;em&gt;5-limit:&lt;/em&gt;&lt;br /&gt;
E' = 5/4 (definition of ' modifier)&lt;br /&gt;
Ab. = 4/5 (definition of . modifier)&lt;br /&gt;
A' = 4/3&lt;br /&gt;
``E' = 5/1&lt;br /&gt;
`B' = 15/8&lt;br /&gt;
&lt;br /&gt;
&lt;em&gt;7-limit&lt;/em&gt;&lt;br /&gt;
Bb~7 = 7/8 (definition of ~7 modifier)&lt;br /&gt;
D_7 = 8/7 (definition of _7 modifier)&lt;br /&gt;
F~7 = 21/16&lt;br /&gt;
`Bb~7 = 7/4&lt;br /&gt;
`F~7 = 21/16&lt;br /&gt;
Eb~7 = 7/6&lt;br /&gt;
&lt;br /&gt;
&lt;em&gt;Higher p-limits&lt;/em&gt;&lt;br /&gt;
F#~11 = 11/8 (definition of ~11 modifier)&lt;br /&gt;
Gb_11 = 8/11 (definition of _11 modifier)&lt;br /&gt;
``F#~11 = 11/2&lt;br /&gt;
B~11 = 11/6&lt;br /&gt;
Ab~13 = 13/16&lt;br /&gt;
C#~17 = 17/16&lt;br /&gt;
Eb~19 = 19/16&lt;br /&gt;
F#~23 = 23/32&lt;br /&gt;
etc (separate definition for each prime)&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Calculation examples&lt;/strong&gt;&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
55/32 = (11/8)*(5/4) = F#~11 * E' = A#'~11&lt;br /&gt;
7/5 = (7/8)*(4/5)*(2/1) = Bb~7 * Ab. * `C = `Bb.~7 * Ab = `Gb.~7&lt;br /&gt;
30/1 = (2/1)*(3/1)*(5/1) = (32/1)*(3/4)*(5/4) = &lt;!-- ws:start:WikiTextRawRule:00:```` --&gt;&lt;!-- ws:end:WikiTextRawRule:00 --&gt;`C * G * E' = &lt;!-- ws:start:WikiTextRawRule:01:```` --&gt;&lt;!-- ws:end:WikiTextRawRule:01 --&gt;`G' * E = &lt;!-- ws:start:WikiTextRawRule:02:```` --&gt;&lt;!-- ws:end:WikiTextRawRule:02 --&gt;`B' (Notice how modifiers ` for octave and ' for prime 5 can be moved about freely)&lt;br /&gt;
19/13 = (19/16) * (16/13) = Eb~19 * (Ab~13)^-1 = Eb~19 * E_13 = G~19_13&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Other links&lt;/strong&gt;&lt;br /&gt;
Some music created using this notation is available at:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Dave Ryan's SoundCloud page: &lt;!-- ws:start:WikiTextUrlRule:89:https://soundcloud.com/daveryan23 --&gt;&lt;a class="wiki_link_ext" href="https://soundcloud.com/daveryan23" rel="nofollow"&gt;https://soundcloud.com/daveryan23&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:89 --&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
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