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Rational comma notation: Difference between revisions

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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>2017-02-01 06:39:37 UTC</tt>.<br>
: This revision was by author [[User:daveryan23|daveryan23]] and made on <tt>2017-02-01 06:56:09 UTC</tt>.<br>
: The original revision id was <tt>605223729</tt>.<br>
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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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* 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 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
* Paper (pre-print) for defining how the notation breaks down into components and enables key changes and transposition: http://arxiv.org/abs/1508.07739
The two links above give the abstract pages on arXiv, and the PDFs are free to download from arXiv.


**Abstract (for 2 papers above):**
**Rational Comma Notation (RCN)** is a combination of:
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 notation using a modified version of Scientific Pitch Notation (SPN), e.g. C4, C5, F#3
* Rational commas for describing higher prime information in a frequency as microtonal adjustments of a Pythagorean frequency, e.g. [5], [1/7], [77/65]
Together these are able to notate the whole of free-JI using notations of the form E4[5], Eb3[1/5], etc.


**Key features:**
RCN as a general scheme comes in different versions. The version RCN_X (subscript) allocates commas [5], [7], [11], [13]... to primes 5, 7, 11, 13... using algorithm X. By using different algorithms, different versions of RCN emerge. The first paper linked to above contains an algorithm due to the author (RCN_DR) as well as a comparison to RCN from two other algorithms (RCN_SAG, RCN_KG). Different composers might prefer different algorithms since there are different criteria to optimise for prime commas. A consensus has not been reached yet as to the best algorithm. This need not hinder using RCN, since it is possible to translate between any two versions of RCN, or indeed between any two free-JI notations.
 
**Key features of RCN:**
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.
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]
All pitch classes can be represented with a notation L[x/y]
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Shorthand exists for pitch classes, of form L~a_b which is the same pitch class as L[x/y]
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
These shorthands allow the 7-limit tone lattice of pitch classes (3-dimensional) to be drawn quickly and accurately
Optional shorthand for Pythagorean comma (B#3=531441/524288) and its inverse, which can help improve 3-limit note names


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!
Note that this system (Jan 2017) 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:**
**Notation examples:**
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Also names the pitch classes, which are notes up to octave equivalence.&lt;br /&gt;
Also names the pitch classes, which are notes up to octave equivalence.&lt;br /&gt;
Developed in 2015-2017 by the musician and music theorist David Ryan&lt;br /&gt;
Developed in 2015-2017 by the musician and music theorist David Ryan&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Paper (pre-print) for defining all the prime commas by algorithm and the basic notation: &lt;!-- ws:start:WikiTextUrlRule:104:http://arxiv.org/abs/1612.01860 --&gt;&lt;a class="wiki_link_ext" href="http://arxiv.org/abs/1612.01860" rel="nofollow"&gt;http://arxiv.org/abs/1612.01860&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:104 --&gt; - also compares between some different choices for algorithm.&lt;/li&gt;&lt;li&gt;Paper (pre-print) for defining how the notation breaks down into components and enables key changes and transposition: &lt;!-- ws:start:WikiTextUrlRule:105:http://arxiv.org/abs/1508.07739 --&gt;&lt;a class="wiki_link_ext" href="http://arxiv.org/abs/1508.07739" rel="nofollow"&gt;http://arxiv.org/abs/1508.07739&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:105 --&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Paper (pre-print) for defining all the prime commas by algorithm and the basic notation: &lt;!-- ws:start:WikiTextUrlRule:114:http://arxiv.org/abs/1612.01860 --&gt;&lt;a class="wiki_link_ext" href="http://arxiv.org/abs/1612.01860" rel="nofollow"&gt;http://arxiv.org/abs/1612.01860&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:114 --&gt; - also compares between some different choices for algorithm.&lt;/li&gt;&lt;li&gt;Paper (pre-print) for defining how the notation breaks down into components and enables key changes and transposition: &lt;!-- ws:start:WikiTextUrlRule:115:http://arxiv.org/abs/1508.07739 --&gt;&lt;a class="wiki_link_ext" href="http://arxiv.org/abs/1508.07739" rel="nofollow"&gt;http://arxiv.org/abs/1508.07739&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:115 --&gt;&lt;/li&gt;&lt;/ul&gt;The two links above give the abstract pages on arXiv, and the PDFs are free to download from arXiv.&lt;br /&gt;
&lt;strong&gt;Abstract (for 2 papers above):&lt;/strong&gt;&lt;br /&gt;
&lt;br /&gt;
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;Rational Comma Notation (RCN)&lt;/strong&gt; is a combination of:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;3-limit notation using a modified version of Scientific Pitch Notation (SPN), e.g. C4, C5, F#3&lt;/li&gt;&lt;li&gt;Rational commas for describing higher prime information in a frequency as microtonal adjustments of a Pythagorean frequency, e.g. [5], [1/7], [77/65]&lt;/li&gt;&lt;/ul&gt;Together these are able to notate the whole of free-JI using notations of the form E4[5], Eb3[1/5], etc.&lt;br /&gt;
&lt;br /&gt;
RCN as a general scheme comes in different versions. The version RCN_X (subscript) allocates commas [5], [7], [11], [13]... to primes 5, 7, 11, 13... using algorithm X. By using different algorithms, different versions of RCN emerge. The first paper linked to above contains an algorithm due to the author (RCN_DR) as well as a comparison to RCN from two other algorithms (RCN_SAG, RCN_KG). Different composers might prefer different algorithms since there are different criteria to optimise for prime commas. A consensus has not been reached yet as to the best algorithm. This need not hinder using RCN, since it is possible to translate between any two versions of RCN, or indeed between any two free-JI notations.&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 of RCN:&lt;/strong&gt;&lt;br /&gt;
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.&lt;br /&gt;
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.&lt;br /&gt;
All pitch classes can be represented with a notation L[x/y]&lt;br /&gt;
All pitch classes can be represented with a notation L[x/y]&lt;br /&gt;
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Shorthand exists for pitch classes, of form L~a_b which is the same pitch class as L[x/y]&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;
These shorthands allow the 7-limit tone lattice of pitch classes (3-dimensional) to be drawn quickly and accurately&lt;br /&gt;
These shorthands allow the 7-limit tone lattice of pitch classes (3-dimensional) to be drawn quickly and accurately&lt;br /&gt;
Optional shorthand for Pythagorean comma (B#3=531441/524288) and its inverse, which can help improve 3-limit note names&lt;br /&gt;
&lt;br /&gt;
&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;
Note that this system (Jan 2017) 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;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Notation examples:&lt;/strong&gt;&lt;br /&gt;
&lt;strong&gt;Notation examples:&lt;/strong&gt;&lt;br /&gt;
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&lt;br /&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:106:http://soundcloud.com/daveryan23/tracks --&gt;&lt;a class="wiki_link_ext" href="http://soundcloud.com/daveryan23/tracks" rel="nofollow"&gt;http://soundcloud.com/daveryan23/tracks&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:106 --&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Dave Ryan's SoundCloud page: &lt;!-- ws:start:WikiTextUrlRule:116:http://soundcloud.com/daveryan23/tracks --&gt;&lt;a class="wiki_link_ext" href="http://soundcloud.com/daveryan23/tracks" rel="nofollow"&gt;http://soundcloud.com/daveryan23/tracks&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:116 --&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
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.&lt;/body&gt;&lt;/html&gt;</pre></div>
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.&lt;/body&gt;&lt;/html&gt;</pre></div>
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