Kite's thoughts on pergens: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-01 18:52:12 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-01 19:50:41 UTC</tt>.

: The original revision id was <tt>~~627091177~~</tt>.

: The original revision id was <tt>627092221</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Line 884:

The interval P8/2 has a "ratio" of the square root of 2, which equals 21/2, 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 ~~both (~~a,b) and n could be reduced by GCD (a,b). Since -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.

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.

Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.

Assume the pergen is a false double, and there's a comma C that splits both P8 and (a,b) appropriately. Can we prove r = 1? Let Q = the higher prime that C uses. Express P, G and C as monzos, i.e. in terms of 2, 3 and Q, by expanding the 2x2 pergen matrix to a 3x3 matrix A:

P = (1/m, 0, 0)

G = (a/n, b/n, 0)

C = (u, v, w)

The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80.

Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. To avoid fractions in the first 2 columns, A must be unimodular **//[I think, not positive]//**, and we have wb/mn = ±1, and w = ±mn/b. Inverting, we have:

2 = 2/1 = P8 = (m, 0, 0) · (P, G, C)

3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C)

Q = Q/1 = (±(av-bu)/n, -±v/m, ±b/mn) · (P, G, C)

For v/m to be an integer, v must equal km for some integer k. Likewise, av-bu must equal cn for some integer c. Thus bu = av - cn = akm - cn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb, where p and q are coprime. Substituting, we get bu = akprb - cqrb, and u = r(akp - cq). Furthermore, v = km = kprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r > 1, then C is a multiple of a simpler comma C'. //**[Not sure of this next part]**// Since C = r·C, P, G and C' must also form a basis for the 2.3.Q prime subgroup. Substituting

Assume the pergen is a true double, and r > 1. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.

Given a pergen (P8/m, (a,b)/n), how many parts is an arbitrary interval (a',b') split into?

Line 902:

Line 923:

If s is a multiple of n (happens when E is an A1) and s' is a multiple of n, let s = x·n and s' = y·n

GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')

Thus every such interval is split, e.g. half-5th splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones.

Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen.

~~Assume~~ the pergen ~~is a false double, and there's a comma C = (u,v,w) that splits both P8 and (a,b) appropriately~~. ~~Can we prove r = 1?~~

~~2.3.7 and (22,-5,-5) = P8/5~~

~~GCD (u,v,w) = 1~~

~~GCD (1 + ku, kv, kw) = m for some integer k, thus GCD (v,w) = m~~

~~GCD (a + k'u, b + k'v, k'w) = n for some integer k', thus |w| = LCM (m,n) = mn/r|b| = pqr|b|~~

~~Given an arbitrary x and y, GCD (x + k"u, y + k"v, k"w) >= r for some k~~

~~Assume GCD (x,y) = 1~~

To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts

Line 5,698:

Line 5,707:

finish proofs

link from: ups and downs page, Kite Giedraitis page, MOS scale names page,

<a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a>

<a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a>

<a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a>

<a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a>

<h3 id="toc22"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a>Notaion guide PDF</h3>

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.

<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>

<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>

(screenshot)

Line 5,714:

Line 5,723:

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. Written in Jesusonic, runs inside Reaper.

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a>

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a>

Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. Screenshots of the first 38 pergens:

Line 5,735:

Line 5,744:

The interval P8/2 has a &quot;ratio&quot; of the square root of 2, which equals 21/2, 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 ~~both (~~a,b) and n could be reduced by GCD (a,b). Since -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.

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.

Assume the pergen is a false double, and there's a comma C that splits both P8 and (a,b) appropriately. Can we prove r = 1? Let Q = the higher prime that C uses. Express P, G and C as monzos, i.e. in terms of 2, 3 and Q, by expanding the 2x2 pergen matrix to a 3x3 matrix A:

P = (1/m, 0, 0)

G = (a/n, b/n, 0)

C = (u, v, w)

The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80.

Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. To avoid fractions in the first 2 columns, A must be unimodular [I think, not positive], and we have wb/mn = ±1, and w = ±mn/b. Inverting, we have:

2 = 2/1 = P8 = (m, 0, 0) · (P, G, C)

3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C)

Q = Q/1 = (±(av-bu)/n, -±v/m, ±b/mn) · (P, G, C)

For v/m to be an integer, v must equal km for some integer k. Likewise, av-bu must equal cn for some integer c. Thus bu = av - cn = akm - cn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb, where p and q are coprime. Substituting, we get bu = akprb - cqrb, and u = r(akp - cq). Furthermore, v = km = kprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r &gt; 1, then C is a multiple of a simpler comma C'. [Not sure of this next part] Since C = r·C, P, G and C' must also form a basis for the 2.3.Q prime subgroup. Substituting

~~Let p = m/rb and q = n/rb~~, ~~where p~~ and ~~q are coprime integers, nonzero but possibly negative~~. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.

Assume the pergen is a true double, and r &gt; 1. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.

Given a pergen (P8/m, (a,b)/n), how many parts is an arbitrary interval (a',b') split into?

Line 5,753:

Line 5,783:

If s is a multiple of n (happens when E is an A1) and s' is a multiple of n, let s = x·n and s' = y·n

GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')

Thus every such interval is split, e.g. half-5th splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones.~~ ~~

Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen.

~~ ~~

~~Assume~~ the pergen ~~is a false double, and there's a comma C = (u,v,w) that splits both P8 and (a,b) appropriately~~. ~~Can we prove r = 1? ~~

~~ ~~

~~2.3.7 and (22,-5,-5) = P8/5 ~~

~~GCD (u,v,w) = 1 ~~

~~GCD (1 + ku, kv, kw) = m for some integer k, thus GCD (v,w) = m ~~

~~GCD (a + k'u, b + k'v, k'w) = n for some integer k', thus |w| = LCM (m,n) = mn/r|b| = pqr|b| ~~

~~Given an arbitrary x and y, GCD (x + k&quot;u, y + k&quot;v, k&quot;w) &gt;= r for some k ~~

~~Assume GCD (x,y) = 1 ~~

To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2018-03-01 18:52:12 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-01 19:50:41 UTC</tt>.<br>
-: The original revision id was <tt>627091177</tt>.<br>
+: The original revision id was <tt>627092221</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>
@@ Line 884: / Line 884: @@
 The interval P8/2 has a "ratio" of the square root of 2, which equals 2&lt;span style="vertical-align: super;"&gt;1/2&lt;/span&gt;, 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 both (a,b) and n could be reduced by GCD (a,b). Since -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.
+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.
-Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.
+Assume the pergen is a false double, and there's a comma C that splits both P8 and (a,b) appropriately. Can we prove r = 1? Let Q = the higher prime that C uses. Express P, G and C as monzos, i.e. in terms of 2, 3 and Q, by expanding the 2x2 pergen matrix to a 3x3 matrix A:
+P = (1/m, 0, 0)
+G = (a/n, b/n, 0)
+C = (u, v, w)
+The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80.
+Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. To avoid fractions in the first 2 columns, A must be unimodular **//[I think, not positive]//**, and we have wb/mn = ±1, and w = ±mn/b. Inverting, we have:
+= 2/1 = P8 = (m, 0, 0) · (P, G, C)
+= 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C)
+Q = Q/1 = (±(av-bu)/n, -±v/m, ±b/mn) · (P, G, C)
+For v/m to be an integer, v must equal km for some integer k. Likewise, av-bu must equal cn for some integer c. Thus bu = av - cn = akm - cn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb, where p and q are coprime. Substituting, we get bu = akprb - cqrb, and u = r(akp - cq). Furthermore, v = km = kprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r &gt; 1, then C is a multiple of a simpler comma C'. //**[Not sure of this next part]**// Since C = r·C, P, G and C' must also form a basis for the 2.3.Q prime subgroup. Substituting
+Assume the pergen is a true double, and r &gt; 1. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.
 Given a pergen (P8/m, (a,b)/n), how many parts is an arbitrary interval (a',b') split into?
@@ Line 902: / Line 923: @@
 If s is a multiple of n (happens when E is an A1) and s' is a multiple of n, let s = x·n and s' = y·n
 GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')
-Thus every such interval is split, e.g. half-5th splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones.
+Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen.
-Assume the pergen is a false double, and there's a comma C = (u,v,w) that splits both P8 and (a,b) appropriately. Can we prove r = 1?
-.3.7 and (22,-5,-5) = P8/5
-GCD (u,v,w) = 1
-GCD (1 + ku, kv, kw) = m for some integer k, thus GCD (v,w) = m
-GCD (a + k'u, b + k'v, k'w) = n for some integer k', thus |w| = LCM (m,n) = mn/r|b| = pqr|b|
-Given an arbitrary x and y, GCD (x + k"u, y + k"v, k"w) &gt;= r for some k
-Assume GCD (x,y) = 1
 To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts
@@ Line 5,698: / Line 5,707: @@
 finish proofs&lt;br /&gt;
 link from: ups and downs page, Kite Giedraitis page, MOS scale names page,&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7111:http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments"&gt;http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7111 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7120:http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments"&gt;http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7120 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7112:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments"&gt;http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7112 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7121:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments"&gt;http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7121 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:101:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc22"&gt;&lt;a name="Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:101 --&gt;Notaion guide PDF&lt;/h3&gt;
   &lt;br /&gt;
 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.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7113:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7113 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7122:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7122 --&gt;&lt;br /&gt;
 (screenshot)&lt;br /&gt;
 &lt;br /&gt;
@@ Line 5,714: / Line 5,723: @@
   &lt;br /&gt;
 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. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7114:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7114 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7123:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7123 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. Screenshots of the first 38 pergens:&lt;br /&gt;
@@ Line 5,735: / Line 5,744: @@
 The interval P8/2 has a &amp;quot;ratio&amp;quot; of the square root of 2, which equals 2&lt;span style="vertical-align: super;"&gt;1/2&lt;/span&gt;, 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 &lt;strong&gt;pergen matrix&lt;/strong&gt; [(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.&lt;br /&gt;
 &lt;br /&gt;
-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 both (a,b) and n could be reduced by GCD (a,b). Since -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.&lt;br /&gt;
+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.&lt;br /&gt;
+&lt;br /&gt;
+Assume the pergen is a false double, and there's a comma C that splits both P8 and (a,b) appropriately. Can we prove r = 1? Let Q = the higher prime that C uses. Express P, G and C as monzos, i.e. in terms of 2, 3 and Q, by expanding the 2x2 pergen matrix to a 3x3 matrix A:&lt;br /&gt;
+&lt;br /&gt;
+P = (1/m, 0, 0)&lt;br /&gt;
+G = (a/n, b/n, 0)&lt;br /&gt;
+C = (u, v, w)&lt;br /&gt;
+&lt;br /&gt;
+The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80.&lt;br /&gt;
+&lt;br /&gt;
+Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. To avoid fractions in the first 2 columns, A must be unimodular &lt;strong&gt;&lt;em&gt;[I think, not positive]&lt;/em&gt;&lt;/strong&gt;, and we have wb/mn = ±1, and w = ±mn/b. Inverting, we have:&lt;br /&gt;
+&lt;br /&gt;
+= 2/1 = P8 = (m, 0, 0) · (P, G, C)&lt;br /&gt;
+&lt;br /&gt;
+= 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C)&lt;br /&gt;
+Q = Q/1 = (±(av-bu)/n, -±v/m, ±b/mn) · (P, G, C)&lt;br /&gt;
+&lt;br /&gt;
+For v/m to be an integer, v must equal km for some integer k. Likewise, av-bu must equal cn for some integer c. Thus bu = av - cn = akm - cn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb, where p and q are coprime. Substituting, we get bu = akprb - cqrb, and u = r(akp - cq). Furthermore, v = km = kprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r &amp;gt; 1, then C is a multiple of a simpler comma C'. &lt;em&gt;&lt;strong&gt;[Not sure of this next part]&lt;/strong&gt;&lt;/em&gt; Since C = r·C, P, G and C' must also form a basis for the 2.3.Q prime subgroup. Substituting&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
 &lt;br /&gt;
-Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.&lt;br /&gt;
+Assume the pergen is a true double, and r &amp;gt; 1. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.&lt;br /&gt;
 &lt;br /&gt;
 Given a pergen (P8/m, (a,b)/n), how many parts is an arbitrary interval (a',b') split into?&lt;br /&gt;
@@ Line 5,753: / Line 5,783: @@
 If s is a multiple of n (happens when E is an A1) and s' is a multiple of n, let s = x·n and s' = y·n&lt;br /&gt;
 GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')&lt;br /&gt;
-Thus every such interval is split, e.g. half-5th splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones.&lt;br /&gt;
+Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen.&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-Assume the pergen is a false double, and there's a comma C = (u,v,w) that splits both P8 and (a,b) appropriately. Can we prove r = 1?&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
-.3.7 and (22,-5,-5) = P8/5&lt;br /&gt;
-GCD (u,v,w) = 1&lt;br /&gt;
-GCD (1 + ku, kv, kw) = m for some integer k, thus GCD (v,w) = m&lt;br /&gt;
-GCD (a + k'u, b + k'v, k'w) = n for some integer k', thus |w| = LCM (m,n) = mn/r|b| = pqr|b|&lt;br /&gt;
-Given an arbitrary x and y, GCD (x + k&amp;quot;u, y + k&amp;quot;v, k&amp;quot;w) &amp;gt;= r for some k&lt;br /&gt;
-Assume GCD (x,y) = 1&lt;br /&gt;
 &lt;br /&gt;
 To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts&lt;br /&gt;