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**Imported revision 577099497 - Original comment: **
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**Imported revision 577099845 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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>2016-03-10 05:57:23 UTC</tt>.<br>
: This revision was by author [[User:daveryan23|daveryan23]] and made on <tt>2016-03-10 06:03:45 UTC</tt>.<br>
: The original revision id was <tt>577099497</tt>.<br>
: The original revision id was <tt>577099845</tt>.<br>
: The revision comment was: <tt></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>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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**Abstract:**
**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
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:**
**Key features:**
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**Notation examples:**
**Notation examples:**


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


//3-limit://
__//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)
`0F = 4/3 (definitions of the 7 note names here)
`0C = 1/1
`0C = 1/1
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`0Cb = 16384/2187 (this is equivalent to a flat b character)
`0Cb = 16384/2187 (this is equivalent to a flat b character)


Octave equivalence classes are obtained from multiplying a note by all the powers of 2 (in class C), e.g.:
__//5-limit://__
`0G = 3/4 has octave equivalence class G = {...3/16, 3/8, 3/4, 3/2, 3/1, 6/1, 12/1...}
//Octave equivalence classes://
`0A = 27/64 has octave equivalence class A = {...27/128, 27/64, 27/32, 27/16, 27/8, 27/4, 27/2, 27/1, 54/1...}
E' = {...5/8, 5/4, 5/2, 5/1, 10/1...}
Bb = {...1/18, 1/9, 2/9, 4/9, 8/9, 16/9, 32/9, 64/9...}
Ab. = {...1/10, 1/5, 2/5, 4/5, 8/5...}
 
//Individual notes://
//5-limit://
`0E' = 5/4 (definition of ' modifier)
`0E' = 5/4 (definition of ' modifier)
`0Ab. = 4/5 (definition of . modifier)
`0Ab. = 4/5 (definition of . modifier)
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`B' = `1B' = 15/8
`B' = `1B' = 15/8


Octave equivalence classes:
__//7-limit//__
E' = {...5/8, 5/4, 5/2, 5/1, 10/1...}
//Octave equivalence classes://
Ab. = {...1/10, 1/5, 2/5, 4/5, 8/5...}
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...}
//7-limit//
//Individual notes://
`0Bb~7 = 7/8 (definition of ~7 modifier)
`0Bb~7 = 7/8 (definition of ~7 modifier)
`0D_7 = 8/7 (definition of _7 modifier)
`0D_7 = 8/7 (definition of _7 modifier)
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`0Eb~7 = 7/6
`0Eb~7 = 7/6


Octave equivalence classes:
__//Higher p-limits//__
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...}
 
//Higher p-limits//
`0F#~11 = 11/8 (definition of ~11 modifier)
`0F#~11 = 11/8 (definition of ~11 modifier)
`0Gb_11 = 8/11 (definition of _11 modifier)
`0Gb_11 = 8/11 (definition of _11 modifier)
``F#~11 = 11/2
``F#~11 = 11/2
`0B~11 = 11/6
`0B~11 = 11/6
`0Ab~13 = 13/16
`0Ab~13 = 13/16 (definition of ~13 modifier)
`0C#~17 = 17/16
`0C#~17 = 17/16
`0Eb~19 = 19/16
`0Eb~19 = 19/16
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Calculating using octave equivalence classes is easier, since you don't have to keep track of the powers of two:
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
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
7/5 class = 7 class * (1/5) class = Bb~7 * Ab. = Gb.~7
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Created by the musician and theorist David Ryan&lt;br /&gt;
Created by the musician and theorist David Ryan&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Preprint: &lt;!-- ws:start:WikiTextUrlRule:111: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:111 --&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Preprint: &lt;!-- ws:start:WikiTextUrlRule:113: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:113 --&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;strong&gt;Abstract:&lt;/strong&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. Also the differences between exact note names and note names up to octave equivalence are discussed&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. Also the differences between exact note names and note names up to octave equivalence are discussed.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Key features:&lt;/strong&gt;&lt;br /&gt;
&lt;strong&gt;Key features:&lt;/strong&gt;&lt;br /&gt;
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&lt;strong&gt;Notation examples:&lt;/strong&gt;&lt;br /&gt;
&lt;strong&gt;Notation examples:&lt;/strong&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;em&gt;2-limit:&lt;/em&gt;&lt;br /&gt;
&lt;u&gt;&lt;em&gt;2-limit:&lt;/em&gt;&lt;/u&gt;&lt;br /&gt;
C = {...1/4, 1/2, 1/1, 2/1, 4/1...} is the octave equivalence class for powers of 2&lt;br /&gt;
&lt;em&gt;Octave equivalence class:&lt;/em&gt;&lt;br /&gt;
C = {...1/4, 1/2, 1/1, 2/1, 4/1...}&lt;br /&gt;
&lt;em&gt;Individual notes:&lt;/em&gt;&lt;br /&gt;
`0C = 1/1&lt;br /&gt;
`0C = 1/1&lt;br /&gt;
`C = `1C = 2/1 (definition of octave modifier using ` character to prefix)&lt;br /&gt;
`C = `1C = 2/1 (definition of octave modifier using ` character to prefix)&lt;br /&gt;
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,,,C = ,3C = 1/8&lt;br /&gt;
,,,C = ,3C = 1/8&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;em&gt;3-limit:&lt;/em&gt;&lt;br /&gt;
&lt;u&gt;&lt;em&gt;3-limit - Pythagorean - definitions of note names and sharps and flats are all here!&lt;/em&gt;&lt;/u&gt;&lt;br /&gt;
&lt;em&gt;Octave equivalence classes:&lt;/em&gt;&lt;br /&gt;
F = {...1/12, 1/6, 1/3, 2/3, 4/3, 8/3, 16/3...}&lt;br /&gt;
G = {...3/16, 3/8, 3/4, 3/2, 3/1, 6/1, 12/1...}&lt;br /&gt;
A = {...27/128, 27/64, 27/32, 27/16, 27/8, 27/4, 27/2, 27/1, 54/1...}&lt;br /&gt;
Bb = {...1/18, 1/9, 2/9, 4/9, 8/9, 16/9, 32/9, 64/9...}&lt;br /&gt;
&lt;em&gt;Individual notes:&lt;/em&gt;&lt;br /&gt;
`0F = 4/3 (definitions of the 7 note names here)&lt;br /&gt;
`0F = 4/3 (definitions of the 7 note names here)&lt;br /&gt;
`0C = 1/1&lt;br /&gt;
`0C = 1/1&lt;br /&gt;
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`0Cb = 16384/2187 (this is equivalent to a flat b character)&lt;br /&gt;
`0Cb = 16384/2187 (this is equivalent to a flat b character)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Octave equivalence classes are obtained from multiplying a note by all the powers of 2 (in class C), e.g.:&lt;br /&gt;
&lt;u&gt;&lt;em&gt;5-limit:&lt;/em&gt;&lt;/u&gt;&lt;br /&gt;
`0G = 3/4 has octave equivalence class G = {...3/16, 3/8, 3/4, 3/2, 3/1, 6/1, 12/1...}&lt;br /&gt;
&lt;em&gt;Octave equivalence classes:&lt;/em&gt;&lt;br /&gt;
`0A = 27/64 has octave equivalence class A = {...27/128, 27/64, 27/32, 27/16, 27/8, 27/4, 27/2, 27/1, 54/1...}&lt;br /&gt;
E' = {...5/8, 5/4, 5/2, 5/1, 10/1...}&lt;br /&gt;
Bb = {...1/18, 1/9, 2/9, 4/9, 8/9, 16/9, 32/9, 64/9...}&lt;br /&gt;
Ab. = {...1/10, 1/5, 2/5, 4/5, 8/5...}&lt;br /&gt;
&lt;br /&gt;
&lt;em&gt;Individual notes:&lt;/em&gt;&lt;br /&gt;
&lt;em&gt;5-limit:&lt;/em&gt;&lt;br /&gt;
`0E' = 5/4 (definition of ' modifier)&lt;br /&gt;
`0E' = 5/4 (definition of ' modifier)&lt;br /&gt;
`0Ab. = 4/5 (definition of . modifier)&lt;br /&gt;
`0Ab. = 4/5 (definition of . modifier)&lt;br /&gt;
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`B' = `1B' = 15/8&lt;br /&gt;
`B' = `1B' = 15/8&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Octave equivalence classes:&lt;br /&gt;
&lt;u&gt;&lt;em&gt;7-limit&lt;/em&gt;&lt;/u&gt;&lt;br /&gt;
E' = {...5/8, 5/4, 5/2, 5/1, 10/1...}&lt;br /&gt;
&lt;em&gt;Octave equivalence classes:&lt;/em&gt;&lt;br /&gt;
Ab. = {...1/10, 1/5, 2/5, 4/5, 8/5...}&lt;br /&gt;
Bb~7 = {...7/16, 7/8, 7/4, 7/2, 7/1, 14/1...}&lt;br /&gt;
&lt;br /&gt;
D_7 = {...1/14, 1/7, 2/7, 4/7, 8/7, 16/7...}&lt;br /&gt;
&lt;em&gt;7-limit&lt;/em&gt;&lt;br /&gt;
&lt;em&gt;Individual notes:&lt;/em&gt;&lt;br /&gt;
`0Bb~7 = 7/8 (definition of ~7 modifier)&lt;br /&gt;
`0Bb~7 = 7/8 (definition of ~7 modifier)&lt;br /&gt;
`0D_7 = 8/7 (definition of _7 modifier)&lt;br /&gt;
`0D_7 = 8/7 (definition of _7 modifier)&lt;br /&gt;
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`0Eb~7 = 7/6&lt;br /&gt;
`0Eb~7 = 7/6&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Octave equivalence classes:&lt;br /&gt;
&lt;u&gt;&lt;em&gt;Higher p-limits&lt;/em&gt;&lt;/u&gt;&lt;br /&gt;
Bb~7 = {...7/16, 7/8, 7/4, 7/2, 7/1, 14/1...}&lt;br /&gt;
D_7 = {...1/14, 1/7, 2/7, 4/7, 8/7, 16/7...}&lt;br /&gt;
&lt;br /&gt;
&lt;em&gt;Higher p-limits&lt;/em&gt;&lt;br /&gt;
`0F#~11 = 11/8 (definition of ~11 modifier)&lt;br /&gt;
`0F#~11 = 11/8 (definition of ~11 modifier)&lt;br /&gt;
`0Gb_11 = 8/11 (definition of _11 modifier)&lt;br /&gt;
`0Gb_11 = 8/11 (definition of _11 modifier)&lt;br /&gt;
``F#~11 = 11/2&lt;br /&gt;
``F#~11 = 11/2&lt;br /&gt;
`0B~11 = 11/6&lt;br /&gt;
`0B~11 = 11/6&lt;br /&gt;
`0Ab~13 = 13/16&lt;br /&gt;
`0Ab~13 = 13/16 (definition of ~13 modifier)&lt;br /&gt;
`0C#~17 = 17/16&lt;br /&gt;
`0C#~17 = 17/16&lt;br /&gt;
`0Eb~19 = 19/16&lt;br /&gt;
`0Eb~19 = 19/16&lt;br /&gt;
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&lt;br /&gt;
&lt;br /&gt;
Calculating using octave equivalence classes is easier, since you don't have to keep track of the powers of two:&lt;br /&gt;
Calculating using octave equivalence classes is easier, since you don't have to keep track of the powers of two:&lt;br /&gt;
&lt;br /&gt;
55/32 class = 5 class * 11 class = E' * F#~11 = A#'~11&lt;br /&gt;
55/32 class = 5 class * 11 class = E' * F#~11 = A#'~11&lt;br /&gt;
7/5 class = 7 class * (1/5) class = Bb~7 * Ab. = Gb.~7&lt;br /&gt;
7/5 class = 7 class * (1/5) class = Bb~7 * Ab. = Gb.~7&lt;br /&gt;
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&lt;strong&gt;Other links&lt;/strong&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;
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:112: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:112 --&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
&lt;ul&gt;&lt;li&gt;Dave Ryan's SoundCloud page: &lt;!-- ws:start:WikiTextUrlRule:114: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:114 --&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
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