Xenharmonic Wiki

Rational comma notation: Difference between revisions

← Older edit
VisualWikitext
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
m Fredg999 moved page Rational Comma Notation (RCN) to Rational comma notation: WP:NCCAPS + remove acronym in title
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
 
(4 intermediate revisions by 3 users not shown)
Line 1: Line 1:
A system using either plaintext (ASCII) or richtext to notate any frequency in Just Intonation (JI).
'''Rational comma notation''' ('''RCN''') is a [[musical notation]] system using either plaintext (ASCII) or richtext to notate any frequency in [[just intonation]] (JI). It also names the pitch classes, which are notes up to [[octave equivalence]]. It was developed in 2015-2017 by musician and music theorist [[David Ryan]].


Also names the pitch classes, which are notes up to [[octave equivalence]].
RCN is a combination of:


Developed in 2015-2017 by the musician and music theorist [[David_Ryan|David Ryan]].
* 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]


<ul><li>Paper (pre-print) for defining all the prime commas by algorithm and the basic notation: [http://arxiv.org/abs/1612.01860 http://arxiv.org/abs/1612.01860] - also compares between some different choices for algorithm.</li><li>Paper (pre-print) for defining how the notation breaks down into components and enables key changes and transposition: [http://arxiv.org/abs/1508.07739 http://arxiv.org/abs/1508.07739]</li></ul>The two links above give the abstract pages on arXiv, and the PDFs are free to download from arXiv.
Together these are able to notate the whole of free JI using notations of the form E4[5], Eb3[1/5], etc.
 
<ul><li>[http://www.retuner.net ReTuner] musical instrument, incorporating live calculation of RCN note names on instrument keys</li></ul>
 
'''Rational Comma Notation (RCN)''' is a combination of:
 
<ul><li>3-limit notation using a modified version of Scientific Pitch Notation (SPN), e.g. C4, C5, F#3</li><li>Rational commas for describing higher prime information in a frequency as microtonal adjustments of a Pythagorean frequency, e.g. [5], [1/7], [77/65]</li></ul>Together these are able to notate the whole of free-JI using notations of the form E4[5], Eb3[1/5], etc.


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.
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:'''
== 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 a 3-limit 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 a 3-limit note label, z is octave number, [x/y] is a rational comma comprising prime commas multiplied together.
Line 21: Line 16:
All pitch classes can be represented with a notation L[x/y]
All pitch classes can be represented with a notation L[x/y]


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
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


3-limit notations correspond to notes on a standard musical stave, which may need one or more sharps or flats to fully describe
3-limit notations correspond to notes on a standard musical stave, which may need one or more sharps or flats to fully describe
Line 29: Line 24:
Can transpose any JI music into any other key, by multiplying notations together. (In a computerised stave, this could be automated)
Can transpose any JI music into any other key, by multiplying notations together. (In a computerised stave, this could be automated)


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


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]
Line 39: Line 34:
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!
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!


'''Prime and Rational Comma examples under DR algorithm:'''
== Prime and rational comma examples under DR algorithm ==
 
Prime:
<u>Prime</u>:
* [5] = 80/81
 
* [7] = 63/64
[5] = 80/81
* [11] = 33/32
 
* [13] = 26/27
[7] = 63/64
* [17] = 2176/2187
 
* [19] = 513/512
[11] = 33/32
 
[13] = 26/27
 
[17] = 2176/2187
 
[19] = 513/512
 
<u>Rational</u>:
 
[1/5] = 1/[5] = 81/80
 
[35] = [5]*[7] = 35/36
 
[5/13] = [5]/[13] = 40/39
 
'''Notation examples:'''
 
<u>''2-limit:''</u>
 
''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
 
<u>''3-limit - Pythagorean - definitions of note names and sharps and flats are all here!''</u>
 
''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...}
 
''Basic Note Labels in Diatonic Scale (requiring no sharps or flats)''
 
C4 = 1/1
 
D4 = 9/8


E4 = 81/64
Rational:
* [1/5] = 1/[5] = 81/80
* [35] = [5]*[7] = 35/36
* [5/13] = [5]/[13] = 40/39


F4 = 4/3
== Notation examples ==
=== 2-limit ===
Octave equivalence class:
* C = {...1/4, 1/2, 1/1, 2/1, 4/1...}


G4 = 3/2
Individual notes:
* C4 = 1/1
* C5 = 2/1
* C3 = 1/2
* C6 = 4/1
* C1 = 1/8
* C(-2) = 1/64


A4 = 27/16
=== 3-limit ===
Pythagorean - definitions of note names and sharps and flats are all here!


B4 = 243/128
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:''
Basic note labels in diatonic scale (requiring no sharps or flats):
* C4 = 1/1
* D4 = 9/8
* E4 = 81/64
* F4 = 4/3
* G4 = 3/2
* A4 = 27/16
* B4 = 243/128


F5 = 8/3
Individual notes:
 
* F5 = 8/3
G5 = 3/1
* G5 = 3/1
 
* A8 = 27/1
A8 = 27/1
* F2 = 1/3
 
* F#3 = 729/1024
F2 = 1/3
* C#4 = 2187/2048 (equivalent to a sharp # character)
 
* Cb4 = 2048/2187 (equivalent to a flat b character)
F#3 = 729/1024
 
C#4 = 2187/2048 (equivalent to a sharp # character)
 
Cb4 = 2048/2187 (equivalent to a flat b character)


Larger number of sharps or flats continue indefinitely up or down the Pythagorean series of fifths.
Larger number of sharps or flats continue indefinitely up or down the Pythagorean series of fifths.


<u>''5-limit:''</u>
=== 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...}


''Octave equivalence classes:''
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


E' = E[5] = {...5/8, 5/4, 5/2, 5/1, 10/1...}
=== 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...}


Ab. = Ab[1/5] = {...1/10, 1/5, 2/5, 4/5, 8/5...}
Individual notes:
* Bb4[7] = 7/4
* D4[7] = 8/7
* F4[7] = 21/16
* Eb4[7] = 7/6


''Individual notes:''
=== Higher ''p''-limits ===
* F4[11] = 11/8
* A4[13] = 13/8
* C#4[17] = 17/16
* Eb4[19] = 19/16
* F#4[23] = 23/16
* Bb4[29] = 29/16
* C4[31] = 31/32 (which is itself a prime comma!)
* D4[37] = 37/32


E'4 = E4[5] = 5/4
Notations can be derived for ''p''/2^''n'' for all higher ''p'', using the prime comma algorithm.


Ab.4 = Ab4[1/5] = 4/5
== Future work ==
In future work, it is hoped to develop a computer based / online free-JI scoring system where the comma numbers (5, 1/7, 65/77, etc) are annotated directly onto notes in order to retune them from Pythagorean notes. The score would be Pythagorean, and commas used as accidentals to be able to compose in free JI and enable anyone to make free-JI music online. If anyone wants to help with this project, please contact David Ryan.


A'4 = A4[5] = 5/3
== External links ==
* [http://arxiv.org/abs/1612.01860 Paper (pre-print) for defining all the prime commas by algorithm and the basic notation] (also compares between some different choices for algorithm)
* [http://arxiv.org/abs/1508.07739 Paper (pre-print) for defining how the notation breaks down into components and enables key changes and transposition] (also includes calculation examples)
* [http://www.retuner.net ReTuner] musical instrument, incorporating live calculation of RCN note names on instrument keys
* Some music created using this notation is available on: [http://soundcloud.com/daveryan23/tracks Dave Ryan's SoundCloud profile]


Db.4 = Db4[1/5] = 16/15
{{Navbox notation}}


<u>''7-limit''</u>
[[Category:Notation]]
 
''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...}
 
''Individual notes:''
 
Bb4[7] = 7/4
 
D4[7] = 8/7
 
F4[7] = 21/16
 
Eb4[7] = 7/6
 
<u>''Higher p-limits''</u>
 
F4[11] = 11/8
 
A4[13] = 13/8
 
C#4[17] = 17/16
 
Eb4[19] = 19/16
 
F#4[23] = 23/16
 
Bb4[29] = 29/16
 
C4[31] = 31/32 (which is itself a prime comma!)
 
D4[37] = 37/32
 
Notations can be derived for p/2^n for all higher p, using the prime comma algorithm.
 
'''Other links'''
 
Calculation examples are given in the second paper listed above.
 
Some music created using this notation is available at:
 
<ul><li>Dave Ryan's SoundCloud page: [http://soundcloud.com/daveryan23/tracks http://soundcloud.com/daveryan23/tracks]</li></ul>
 
In future work, it is hoped to develop a computer based / online free-JI scoring system where the comma numbers (5, 1/7, 65/77, etc) are annotated directly onto notes in order to retune them from Pythagorean notes. The score would be Pythagorean, and commas used as accidentals to be able to compose in free-JI and enable anyone to make free-JI music online. If anyone wants to help with this project, please contact David Ryan.
 
[[Category:ASCII]]
[[Category:Just intonation]]
[[Category:Just intonation]]
[[Category:Notation]]
[[Category:RCN]]
Retrieved from "https://en.xen.wiki/w/Rational_comma_notation"