Kite's thoughts on pergens: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-09 23:11:01 UTC</tt>.<br>

: The original revision id was <tt>626231179</tt>.<br>

The interval P8/2 has a "ratio" of the square root of 2 = 2&lt;span style="vertical-align: super;"&gt;1/2&lt;/span&gt;, and its monzo could be written (1/2, 0). Likewise, 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.  

Let x = m, y = -am/b, and z = n/b. Because we started with a valid pergen, the square mapping must be an integer matrix, and x, y and z are all integers. 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) &gt;= |b|.  

Let r be a positive integer such that GCD (m,n) = r·|b|. Let m = prb and n = qrb, where p and q are coprime integers, 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 r parts.

Assume r &gt; 1 and (a,b)/n splits P8 into b parts

Does (a,b)/n split P8 into m periods?

(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5

M/b = (n/b)·M/n = (n/b)·G

(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)

Let c and d be the bezout pair of a+b and b, with c·(a+b) + d·b = 1

c·(a+b)·P8 = c·b·((n/b)·G - P5)

(1 - d·b)·P8 = c·b·((n/b)·G - P5)

P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5)

P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)

Therefore P8 is split into m periods

Therefore if m = |b|, the pergen is explicitly false

&lt;!-- ws:start:WikiTextUrlRule:5526: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:5526 --&gt;&lt;br /&gt;

&lt;!-- ws:start:WikiTextUrlRule:5527: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:5527 --&gt;&lt;br /&gt;

The interval P8/2 has a &amp;quot;ratio&amp;quot; of the square root of 2 = 2&lt;span style="vertical-align: super;"&gt;1/2&lt;/span&gt;, and its monzo could be written (1/2, 0). Likewise, 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;

Let x = m, y = -am/b, and z = n/b. Because we started with a valid pergen, the square mapping must be an integer matrix, and x, y and z are all integers. 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) &amp;gt;= |b|. &lt;br /&gt;

Assume r &amp;gt; 1 and (a,b)/n splits P8 into b parts&lt;br /&gt;

(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5&lt;br /&gt;

Because n is a multiple of b, n/b is an integer&lt;br /&gt;

M/b = (n/b)·M/n = (n/b)·G&lt;br /&gt;

(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)&lt;br /&gt;

Let c and d be the bezout pair of a+b and b, with c·(a+b) + d·b = 1&lt;br /&gt;

c·(a+b)·P8 = c·b·((n/b)·G - P5)&lt;br /&gt;

(1 - d·b)·P8 = c·b·((n/b)·G - P5)&lt;br /&gt;

P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5)&lt;br /&gt;

P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)&lt;br /&gt;

Therefore P8 is split into m periods&lt;br /&gt;

Therefore if m = |b|, the pergen is explicitly false&lt;br /&gt;