Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 627984405 - 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:TallKite|TallKite]] and made on <tt>2018-03- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-25 04:06:37 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>627986315</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> | 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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==Naming very large intervals== | ==Naming very large intervals== | ||
So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by "W". Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M is less than 3 octaves, and can be P4, P5, P11, P12, WWP4 or WWP5. | So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, **widening** by an 8ve is indicated by "W". Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M is less than 3 octaves, and can be P4, P5, P11, P12, WWP4 or WWP5. | ||
==Secondary splits== | ==Secondary splits== | ||
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==Notating non-8ve and no-5ths pergens== | ==Notating non-8ve and no-5ths pergens== | ||
In Blackwood-like pergens, the 5th is present but not independent. In non-5th pergens, the 5th is not present, and the prime subgroup doesn't contain 3. | In Blackwood-like pergens, the 5th is present but not independent. In non-5th pergens, the 5th is not present, and the prime subgroup doesn't contain 3. | ||
In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains. | In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains. | ||
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Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table can be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3). | Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table can be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3). | ||
Pergen squares | ==Pergen squares== | ||
For (P8, P5), the pergen square has 4 notes: | Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all, but assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions. | ||
For (P8, P5), the pergen square has 4 notes, shown here with octave numbers: | |||
C2 -- G2 | C2 -- G2 | ||
| | | | | | ||
C1 -- G1 | C1 -- G1 | ||
Splitting the | Splitting the 8ve or the multigen adds notes to the square. For (P8/2, P5), there are 6 notes. The new notes fall halfway between each 8ve: | ||
C2 --- G2 | C2 --- G2 | ||
F#v1 | F#v1 C#v2 | ||
C1 --- G1 | C1 --- G1 | ||
The square | A pergen square contains all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and C#v2 bisects it. G#v2 bisects the G2-A2 M2. Every "downed" note bisects an 8ve, a M2, a Wm7, a WM9, and many other intervals. | ||
C2 --- G2 --- D3 --- A3 | |||
F#v1 C#v2 G#v2 D#v3 | |||
C1 --- G1 --- D2 --- A2 | |||
Splitting the 5th adds notes to the horizontal edges of the square. Each note also bisects a P11, e.g. G1-C3. | |||
C3 Ev3 G3 | |||
|| | |||
C2 Ev2 G2 | |||
|| | |||
C1 Ev1 G1 | |||
From G1 to C2 is a 4th, so splitting the 4th adds A^1 halfway between them, in the center of the square. A^1 also bisects the P12 from C1 to G2. | |||
Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen. | |||
[[image:pergen squares.png]] | |||
A similar chart could be made for all rank-3 pergens, using pergen cubes. | A similar chart could be made for all rank-3 pergens, using pergen cubes. | ||
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See the screenshots in the next section for examples of which pergens are supported by a specific edo. | See the screenshots in the next section for examples of which pergens are supported by a specific edo. | ||
__**EDO-pair | __**EDO-pair names**__ | ||
Just as a pair of edos and a prime subgroup can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo & N'-edo, m = GCD (N,N'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16. | Just as a pair of edos and a prime subgroup can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo & N'-edo, m = GCD (N,N'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16. | ||
For each edo, find the nearest **edomapping** ( | For each edo, find the nearest **edomapping** (patent val) for the 2.3 subgroup. If the edo has a "b" wart, e.g. 13b-edo, use the second-nearest edomapping. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If d = m, -m or 0, the generator is the 5th, and the pergen is simply (P8/m, P5). | ||
For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4. | For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4. | ||
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To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24. 11/9 also works, it yields 243/242. | To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24. 11/9 also works, it yields 243/242. | ||
If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a Blackwood-like pergen. | If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a Blackwood-like pergen. | ||
The closer two edos are in the scale tree, the simpler the pergen they make. Examples:: | The closer two edos are in the scale tree, the simpler the pergen they make. Examples:: | ||
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==Supplemental materials*== | ==Supplemental materials*== | ||
the half-step glitch | the half-step glitch | ||
P8/5, P11/3 | |||
sixth-splits | |||
to do: | to do: | ||
finish proofs | finish proofs | ||
link from: ups and downs page, Kite Giedraitis page, MOS scale names page, | link from: ups and downs page, Kite Giedraitis page, MOS scale names page, | ||
http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments | http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments | ||
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This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens. | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens. | ||
http://www.tallkite.com/misc_files/pergens.pdf | http://www.tallkite.com/misc_files/pergens.pdf | ||
===pergenLister | Screenshots of the first 2 pages: | ||
[[image:pergens 1.png width="640" height="862"]] | |||
[[image:pergens 2.png width="640" height="674"]] | |||
=== === | |||
===alt-pergenLister=== | |||
Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair. Written in Jesusonic, runs inside Reaper. | Alt-pergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair. Written in Jesusonic, runs inside Reaper. | ||
http://www.tallkite.com/misc_files/alt-pergenLister.zip | http://www.tallkite.com/misc_files/alt-pergenLister.zip | ||
The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. | The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. | ||
Screenshots of the first | Screenshots of the first 60 pergens: | ||
[[image:alt-pergenLister.png width=" | [[image:alt-pergenLister 1.png width="800" height="427"]] | ||
[[image:xenharmonic/alt-pergenLister 2.png width="800" height="455"]] | |||
Pergens supported by 12edo: | |||
[[image:alt-pergenLister 12edo.png width="800" height="449"]] | |||
Pergens supported by 15edo. A red asterisk means partial support. | |||
[[image:alt-pergenLister 15edo.png width="800" height="493"]] | |||
Pergens supported by 19edo: | |||
[[image:alt-pergenLister 19edo.png width="800" height="459"]] | |||
Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up: | Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up: | ||
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The interval P8/2 has a "ratio" of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the **pergen matrix** [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping. | The interval P8/2 has a "ratio" of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the **pergen matrix** [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping. | ||
Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise a, b, and n could all be reduced by GCD (a,b), and the multigen could be simplified. Since GCD (a, b) = 1 and -am/b is an integer, it follows that m must be a multiple of |b| as well. Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|) = |b| · r. If r = 1, then GCD (m, n) = |b|, and vice versa, which is the proposed test for a false double. | Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise a, b, and n could all be reduced by GCD (a,b), and the multigen could be simplified. Since GCD (a, b) = 1 and -am/b is an integer, it follows that m must be a multiple of |b| as well. | ||
Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|) = |b| · r. If r = 1, then GCD (m, n) = |b|, and vice versa, which is the proposed test for a false double. | |||
If m = |b|, is the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m periods? | If m = |b|, is the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m periods? | ||
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==Miscellaneous Notes== | ==Miscellaneous Notes== | ||
__**Glossary**__ (to find the definition, use control-F to search for the bolded occurence of the word on this page) | |||
pergen | |||
split | |||
multigen | |||
ups and downs (the ^ and v symbols) | |||
higher prime (any prime > 3) | |||
color depth | |||
dependent/independent | |||
square mapping | |||
highs and lows (the / and \ symbols) | |||
enharmonic | |||
genchain | |||
perchain | |||
wide/widen (increased by an octave) | |||
single-split, double-split | |||
single-pair, double-pair (number of new accidentals in the notation) | |||
true double, false double | |||
explicitly false | |||
unreduced | |||
alternate vs. equivalent (generator or period) | |||
mapping comma | |||
keyspan | |||
stepspan | |||
gedra | |||
edomapping | |||
__**Staff notation**__ | __**Staff notation**__ | ||
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<h4>Original HTML content:</h4> | <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"><html><head><title>pergen</title></head><body><!-- ws:start:WikiTextHeadingRule:60:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:60 --> </h1> | <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"><html><head><title>pergen</title></head><body><!-- ws:start:WikiTextHeadingRule:60:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:60 --> </h1> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:118:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:121 --><!-- ws:start:WikiTextTocRule:122: --><div style="margin-left: 1em;"><a href="#Applications">Applications</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:122 --><!-- ws:start:WikiTextTocRule:123: --><div style="margin-left: 2em;"><a href="#Applications-Tipping points">Tipping points</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:123 --><!-- ws:start:WikiTextTocRule:124: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:124 --><!-- ws:start:WikiTextTocRule:125: --><div style="margin-left: 2em;"><a href="#Further Discussion-Naming very large intervals">Naming very large intervals</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:125 --><!-- ws:start:WikiTextTocRule:126: --><div style="margin-left: 2em;"><a href="#Further Discussion-Secondary splits">Secondary splits</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:126 --><!-- ws:start:WikiTextTocRule:127: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:127 --><!-- ws:start:WikiTextTocRule:128: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:128 --><!-- ws:start:WikiTextTocRule:129: --><div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:129 --><!-- ws:start:WikiTextTocRule:130: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:130 --><!-- ws:start:WikiTextTocRule:131: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:131 --><!-- ws:start:WikiTextTocRule:132: --><div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:132 --><!-- ws:start:WikiTextTocRule:133: --><div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:133 --><!-- ws:start:WikiTextTocRule:134: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating unsplit pergens">Notating unsplit pergens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:134 --><!-- ws:start:WikiTextTocRule:135: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating rank-3 pergens">Notating rank-3 pergens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:135 --><!-- ws:start:WikiTextTocRule:136: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating Blackwood-like pergens">Notating Blackwood-like pergens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:136 --><!-- ws:start:WikiTextTocRule:137: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating non-8ve and no-5ths pergens">Notating non-8ve and no-5ths pergens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:137 --><!-- ws:start:WikiTextTocRule:138: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergen squares">Pergen squares</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:138 --><!-- ws:start:WikiTextTocRule:139: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating tunings with an arbitrary generator">Notating tunings with an arbitrary generator</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:139 --><!-- ws:start:WikiTextTocRule:140: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:140 --><!-- ws:start:WikiTextTocRule:141: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:141 --><!-- ws:start:WikiTextTocRule:142: --><div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials*">Supplemental materials*</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:142 --><!-- ws:start:WikiTextTocRule:143: --><div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-Notaion guide PDF">Notaion guide PDF</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:143 --><!-- ws:start:WikiTextTocRule:144: --><div style="margin-left: 3em;"><a href="#toc25"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:144 --><!-- ws:start:WikiTextTocRule:145: --><div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-alt-pergenLister">alt-pergenLister</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:145 --><!-- ws:start:WikiTextTocRule:146: --><div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:146 --><!-- ws:start:WikiTextTocRule:147: --><div style="margin-left: 2em;"><a href="#Further Discussion-Miscellaneous Notes">Miscellaneous Notes</a></div> | ||
<!-- ws:end:WikiTextTocRule:147 --><!-- ws:start:WikiTextTocRule:148: --></div> | |||
<!-- ws:end:WikiTextTocRule:148 --><!-- ws:start:WikiTextHeadingRule:62:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:62 --><u><strong>Definition</strong></u></h1> | |||
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<!-- ws:start:WikiTextHeadingRule:72:&lt;h2&gt; --><h2 id="toc6"><a name="Further Discussion-Naming very large intervals"></a><!-- ws:end:WikiTextHeadingRule:72 -->Naming very large intervals</h2> | <!-- ws:start:WikiTextHeadingRule:72:&lt;h2&gt; --><h2 id="toc6"><a name="Further Discussion-Naming very large intervals"></a><!-- ws:end:WikiTextHeadingRule:72 -->Naming very large intervals</h2> | ||
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So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by &quot;W&quot;. Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M is less than 3 octaves, and can be P4, P5, P11, P12, WWP4 or WWP5.<br /> | So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, <strong>widening</strong> by an 8ve is indicated by &quot;W&quot;. Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M is less than 3 octaves, and can be P4, P5, P11, P12, WWP4 or WWP5.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:74:&lt;h2&gt; --><h2 id="toc7"><a name="Further Discussion-Secondary splits"></a><!-- ws:end:WikiTextHeadingRule:74 -->Secondary splits</h2> | <!-- ws:start:WikiTextHeadingRule:74:&lt;h2&gt; --><h2 id="toc7"><a name="Further Discussion-Secondary splits"></a><!-- ws:end:WikiTextHeadingRule:74 -->Secondary splits</h2> | ||
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<!-- ws:start:WikiTextHeadingRule:96:&lt;h2&gt; --><h2 id="toc18"><a name="Further Discussion-Notating non-8ve and no-5ths pergens"></a><!-- ws:end:WikiTextHeadingRule:96 -->Notating non-8ve and no-5ths pergens</h2> | <!-- ws:start:WikiTextHeadingRule:96:&lt;h2&gt; --><h2 id="toc18"><a name="Further Discussion-Notating non-8ve and no-5ths pergens"></a><!-- ws:end:WikiTextHeadingRule:96 -->Notating non-8ve and no-5ths pergens</h2> | ||
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In Blackwood-like pergens, the 5th is present but not independent. In non-5th pergens, the 5th is not present, and the prime subgroup doesn't contain 3. <br /> | In Blackwood-like pergens, the 5th is present but not independent. In non-5th pergens, the 5th is not present, and the prime subgroup doesn't contain 3.<br /> | ||
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In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.<br /> | In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.<br /> | ||
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Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table can be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3).<br /> | Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table can be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3).<br /> | ||
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Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all, but | <!-- ws:start:WikiTextHeadingRule:98:&lt;h2&gt; --><h2 id="toc19"><a name="Further Discussion-Pergen squares"></a><!-- ws:end:WikiTextHeadingRule:98 -->Pergen squares</h2> | ||
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Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all, but assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.<br /> | |||
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For (P8, P5), the pergen square has 4 notes:<br /> | For (P8, P5), the pergen square has 4 notes, shown here with octave numbers:<br /> | ||
C2 -- G2<br /> | C2 -- G2<br /> | ||
| |<br /> | | |<br /> | ||
C1 -- G1<br /> | C1 -- G1<br /> | ||
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Splitting the | Splitting the 8ve or the multigen adds notes to the square. For (P8/2, P5), there are 6 notes. The new notes fall halfway between each 8ve:<br /> | ||
C2 --- G2<br /> | C2 --- G2<br /> | ||
F#v1 | F#v1 C#v2<br /> | ||
C1 --- G1<br /> | C1 --- G1<br /> | ||
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The square | A pergen square contains all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and C#v2 bisects it. G#v2 bisects the G2-A2 M2. Every &quot;downed&quot; note bisects an 8ve, a M2, a Wm7, a WM9, and many other intervals.<br /> | ||
<br /> | C2 --- G2 --- D3 --- A3<br /> | ||
F#v1 C#v2 G#v2 D#v3<br /> | |||
C1 --- G1 --- D2 --- A2<br /> | |||
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Splitting the 5th adds notes to the horizontal edges of the square. Each note also bisects a P11, e.g. G1-C3. <br /> | |||
C3 Ev3 G3<br /> | |||
||<br /> | |||
C2 Ev2 G2<br /> | |||
||<br /> | |||
C1 Ev1 G1<br /> | |||
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From G1 to C2 is a 4th, so splitting the 4th adds A^1 halfway between them, in the center of the square. A^1 also bisects the P12 from C1 to G2.<br /> | |||
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Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.<br /> | |||
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A similar chart could be made for all rank-3 pergens, using pergen cubes.<br /> | A similar chart could be made for all rank-3 pergens, using pergen cubes.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:100:&lt;h2&gt; --><h2 id="toc20"><a name="Further Discussion-Notating tunings with an arbitrary generator"></a><!-- ws:end:WikiTextHeadingRule:100 -->Notating tunings with an arbitrary generator</h2> | ||
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Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.<br /> | Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.<br /> | ||
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See also the <a class="wiki_link" href="/Map%20of%20rank-2%20temperaments">map of rank-2 temperaments</a>.<br /> | See also the <a class="wiki_link" href="/Map%20of%20rank-2%20temperaments">map of rank-2 temperaments</a>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:102:&lt;h2&gt; --><h2 id="toc21"><a name="Further Discussion-Pergens and MOS scales"></a><!-- ws:end:WikiTextHeadingRule:102 -->Pergens and MOS scales</h2> | ||
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Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.<br /> | Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.<br /> | ||
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Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4.<br /> | Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:104:&lt;h2&gt; --><h2 id="toc22"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:104 -->Pergens and EDOs</h2> | ||
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Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.<br /> | Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.<br /> | ||
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See the screenshots in the next section for examples of which pergens are supported by a specific edo.<br /> | See the screenshots in the next section for examples of which pergens are supported by a specific edo.<br /> | ||
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<u><strong>EDO-pair | <u><strong>EDO-pair names</strong></u><br /> | ||
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Just as a pair of edos and a prime subgroup can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo &amp; N'-edo, m = GCD (N,N'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16.<br /> | Just as a pair of edos and a prime subgroup can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo &amp; N'-edo, m = GCD (N,N'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16.<br /> | ||
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For each edo, find the nearest <strong>edomapping</strong> ( | For each edo, find the nearest <strong>edomapping</strong> (patent val) for the 2.3 subgroup. If the edo has a &quot;b&quot; wart, e.g. 13b-edo, use the second-nearest edomapping. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If d = m, -m or 0, the generator is the 5th, and the pergen is simply (P8/m, P5).<br /> | ||
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For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.<br /> | For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.<br /> | ||
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To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24. 11/9 also works, it yields 243/242.<br /> | To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24. 11/9 also works, it yields 243/242.<br /> | ||
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If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a Blackwood-like pergen. <br /> | If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a Blackwood-like pergen.<br /> | ||
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The closer two edos are in the scale tree, the simpler the pergen they make. Examples::<br /> | The closer two edos are in the scale tree, the simpler the pergen they make. Examples::<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:106:&lt;h2&gt; --><h2 id="toc23"><a name="Further Discussion-Supplemental materials*"></a><!-- ws:end:WikiTextHeadingRule:106 -->Supplemental materials*</h2> | ||
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the half-step glitch<br /> | the half-step glitch<br /> | ||
P8/5, P11/3<br /> | |||
sixth-splits<br /> | |||
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to do:<br /> | to do:<br /> | ||
finish proofs<br /> | finish proofs<br /> | ||
link from: ups and downs page, Kite Giedraitis page, MOS scale names page,<br /> | link from: ups and downs page, Kite Giedraitis page, MOS scale names page,<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:8214:http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments --><a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a><!-- ws:end:WikiTextUrlRule:8214 --><br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:8215:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --><a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a><!-- ws:end:WikiTextUrlRule:8215 --><br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:108:&lt;h3&gt; --><h3 id="toc24"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a><!-- ws:end:WikiTextHeadingRule:108 -->Notaion guide PDF</h3> | ||
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This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.<br /> | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:8216:http://www.tallkite.com/misc_files/pergens.pdf --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><!-- ws:end:WikiTextUrlRule:8216 --><br /> | ||
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Screenshots of the first 2 pages:<br /> | |||
<!-- ws:start:WikiTextLocalImageRule:4924:&lt;img src=&quot;/file/view/pergens%201.png/627986243/640x862/pergens%201.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 862px; width: 640px;&quot; /&gt; --><img src="/file/view/pergens%201.png/627986243/640x862/pergens%201.png" alt="pergens 1.png" title="pergens 1.png" style="height: 862px; width: 640px;" /><!-- ws:end:WikiTextLocalImageRule:4924 --><br /> | |||
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<!-- ws:start:WikiTextHeadingRule:110:&lt;h3&gt; --><h3 id="toc25"><!-- ws:end:WikiTextHeadingRule:110 --> </h3> | |||
<!-- ws:start:WikiTextHeadingRule:112:&lt;h3&gt; --><h3 id="toc26"><a name="Further Discussion-Supplemental materials*-alt-pergenLister"></a><!-- ws:end:WikiTextHeadingRule:112 -->alt-pergenLister</h3> | |||
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Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair. Written in Jesusonic, runs inside Reaper.<br /> | Alt-pergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair. Written in Jesusonic, runs inside Reaper.<br /> | ||
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The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.<br /> | The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.<br /> | ||
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Screenshots of the first | Screenshots of the first 60 pergens:<br /> | ||
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Pergens supported by 12edo:<br /> | |||
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Pergens supported by 15edo. A red asterisk means partial support.<br /> | |||
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Pergens supported by 19edo:<br /> | |||
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Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | ||
| Line 6,589: | Line 6,660: | ||
The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.<br /> | The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:114:&lt;h2&gt; --><h2 id="toc27"><a name="Further Discussion-Various proofs (unfinished)"></a><!-- ws:end:WikiTextHeadingRule:114 -->Various proofs (unfinished)</h2> | ||
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The interval P8/2 has a &quot;ratio&quot; of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the <strong>pergen matrix</strong> [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.<br /> | The interval P8/2 has a &quot;ratio&quot; of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the <strong>pergen matrix</strong> [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.<br /> | ||
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Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise a, b, and n could all be reduced by GCD (a,b), and the multigen could be simplified. Since GCD (a, b) = 1 and -am/b is an integer, it follows that m must be a multiple of |b| as well. Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|) = |b| · r. If r = 1, then GCD (m, n) = |b|, and vice versa, which is the proposed test for a false double.<br /> | Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise a, b, and n could all be reduced by GCD (a,b), and the multigen could be simplified. Since GCD (a, b) = 1 and -am/b is an integer, it follows that m must be a multiple of |b| as well. <br /> | ||
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Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|) = |b| · r. If r = 1, then GCD (m, n) = |b|, and vice versa, which is the proposed test for a false double.<br /> | |||
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If m = |b|, is the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m periods?<br /> | If m = |b|, is the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m periods?<br /> | ||
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Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.<br /> | Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:116:&lt;h2&gt; --><h2 id="toc28"><a name="Further Discussion-Miscellaneous Notes"></a><!-- ws:end:WikiTextHeadingRule:116 -->Miscellaneous Notes</h2> | ||
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<u><strong>Glossary</strong></u> (to find the definition, use control-F to search for the bolded occurence of the word on this page)<br /> | |||
pergen<br /> | |||
split<br /> | |||
multigen<br /> | |||
ups and downs (the ^ and v symbols)<br /> | |||
higher prime (any prime &gt; 3)<br /> | |||
color depth<br /> | |||
dependent/independent<br /> | |||
square mapping<br /> | |||
highs and lows (the / and \ symbols)<br /> | |||
enharmonic<br /> | |||
genchain<br /> | |||
perchain<br /> | |||
wide/widen (increased by an octave)<br /> | |||
single-split, double-split<br /> | |||
single-pair, double-pair (number of new accidentals in the notation)<br /> | |||
true double, false double<br /> | |||
explicitly false<br /> | |||
unreduced<br /> | |||
alternate vs. equivalent (generator or period)<br /> | |||
mapping comma<br /> | |||
keyspan<br /> | |||
stepspan<br /> | |||
gedra<br /> | |||
edomapping<br /> | |||
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<u><strong>Staff notation</strong></u><br /> | <u><strong>Staff notation</strong></u><br /> | ||
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