Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626121387 - Original comment: ** |
Wikispaces>TallKite **Imported revision 626231179 - 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>2018-02- | : 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> | : The original revision id was <tt>626231179</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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==Singles and doubles== | ==Singles and doubles== | ||
If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a **single-split** pergen. If it has two fractions, it's a **double-split** pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called **single-pair** notation because it adds only a single pair of accidentals to conventional notation. **Double-pair** notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs. | If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a **single-split** pergen. If it has two fractions, it's a **double-split** pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called **single-pair** notation because it adds only a single pair of accidentals to conventional notation. **Double-pair** notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs. | ||
In this article, for double-pair notation, the period uses ups and downs, and the generator uses highs and lows. But the choice of which pair of accidentals is used for what is arbitrary, and ups/downs could be exchanged with highs/lows. | In this article, for double-pair notation, the period uses ups and downs, and the generator uses highs and lows. But the choice of which pair of accidentals is used for what is arbitrary, and ups/downs could be exchanged with highs/lows. | ||
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||= 15-edo ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = m2, E' = v<span style="vertical-align: super;">3</span>A1 = v<span style="vertical-align: super;">3</span>M2 || | ||= 15-edo ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = m2, E' = v<span style="vertical-align: super;">3</span>A1 = v<span style="vertical-align: super;">3</span>M2 || | ||
||= 24-edo ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = d2, E' = vvA1 = vvm2 || | ||= 24-edo ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = d2, E' = vvA1 = vvm2 || | ||
||= pythagorean ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= ---- || | ||= pythagorean ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= | ||
||= meantone ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= ---- || | ---- || | ||
||= meantone ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= | |||
---- || | |||
||= semaphore ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = vvm2 || | ||= semaphore ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = vvm2 || | ||
||= decimal ||= rank-2 ||= double-pair ||= rank-4 ||= 2 ||= E = vvd2, E' = \\m2 || | ||= decimal ||= rank-2 ||= double-pair ||= rank-4 ||= 2 ||= E = vvd2, E' = \\m2 || | ||
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||= marvel ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- || | ||= marvel ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- || | ||
||= breedsmic ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 || | ||= breedsmic ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 || | ||
||= 7-limit JI ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= ---- || | ||= 7-limit JI ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= | ||
---- || | |||
A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. | A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. | ||
All the previous rank-3 examples had the 2nd generator be a mapping comma, represented by an up. However, this isn't always possible. | All the previous rank-3 examples had the 2nd generator be a mapping comma, represented by an up. However, this isn't always possible. | ||
||~ 7-limit temperament ||~ comma ||~ pergen ||~ spoken pergen ||~ period ||~ gen1 ||~ gen2 ||~ enharmonic || | ||~ 7-limit temperament ||~ comma ||~ pergen ||~ spoken pergen ||~ period ||~ gen1 ||~ gen2 ||~ enharmonic || | ||
||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= unsplit with ups ||= P8 ||= P5 ||= ^1 = 81/80 ||= --- || | ||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= unsplit with ups ||= P8 ||= P5 ||= ^1 = 81/80 ||= --- || | ||
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==Various proofs (unfinished)== | ==Various proofs (unfinished)== | ||
The interval P8/2 has a "ratio" of the square root of 2 = 2<span style="vertical-align: super;">1/2</span>, 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) >= |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 > 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 | |||
Because n is a multiple of b, n/b is an integer | |||
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 | |||
Since the pergen is a double-split, m > 1, therefore |b| > 1, therefore c ≠ 0 | |||
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 | |||
Given: | Given: | ||
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<!-- ws:start:WikiTextHeadingRule:71:&lt;h2&gt; --><h2 id="toc8"><a name="Further Discussion-Singles and doubles"></a><!-- ws:end:WikiTextHeadingRule:71 -->Singles and doubles</h2> | <!-- ws:start:WikiTextHeadingRule:71:&lt;h2&gt; --><h2 id="toc8"><a name="Further Discussion-Singles and doubles"></a><!-- ws:end:WikiTextHeadingRule:71 -->Singles and doubles</h2> | ||
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If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a <strong>single-split</strong> pergen. If it has two fractions, it's a <strong>double-split</strong> pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called <strong>single-pair</strong> notation because it adds only a single pair of accidentals to conventional notation. <strong>Double-pair</strong> notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs. <br /> | If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a <strong>single-split</strong> pergen. If it has two fractions, it's a <strong>double-split</strong> pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called <strong>single-pair</strong> notation because it adds only a single pair of accidentals to conventional notation. <strong>Double-pair</strong> notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.<br /> | ||
<br /> | <br /> | ||
In this article, for double-pair notation, the period uses ups and downs, and the generator uses highs and lows. But the choice of which pair of accidentals is used for what is arbitrary, and ups/downs could be exchanged with highs/lows.<br /> | In this article, for double-pair notation, the period uses ups and downs, and the generator uses highs and lows. But the choice of which pair of accidentals is used for what is arbitrary, and ups/downs could be exchanged with highs/lows.<br /> | ||
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A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &quot;superfalse&quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more.<br /> | A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &quot;superfalse&quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more.<br /> | ||
<br /> | <br /> | ||
All the previous rank-3 examples had the 2nd generator be a mapping comma, represented by an up. However, this isn't always possible. <br /> | All the previous rank-3 examples had the 2nd generator be a mapping comma, represented by an up. However, this isn't always possible.<br /> | ||
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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:5526: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:5526 --><br /> | ||
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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. Written in Jesusonic, runs inside Reaper.<br /> | 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.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:5527:http://www.tallkite.com/misc_files/alt-pergenLister.zip --><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><!-- ws:end:WikiTextUrlRule:5527 --><br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:97:&lt;h2&gt; --><h2 id="toc21"><a name="Further Discussion-Various proofs (unfinished)"></a><!-- ws:end:WikiTextHeadingRule:97 -->Various proofs (unfinished)</h2> | <!-- ws:start:WikiTextHeadingRule:97:&lt;h2&gt; --><h2 id="toc21"><a name="Further Discussion-Various proofs (unfinished)"></a><!-- ws:end:WikiTextHeadingRule:97 -->Various proofs (unfinished)</h2> | ||
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The interval P8/2 has a &quot;ratio&quot; of the square root of 2 = 2<span style="vertical-align: super;">1/2</span>, 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 <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 /> | |||
<br /> | |||
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|. <br /> | |||
<br /> | |||
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.<br /> | |||
<br /> | |||
Assume r &gt; 1 and (a,b)/n splits P8 into b parts<br /> | |||
<br /> | |||
Does (a,b)/n split P8 into m periods?<br /> | |||
(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5<br /> | |||
Because n is a multiple of b, n/b is an integer<br /> | |||
M/b = (n/b)·M/n = (n/b)·G<br /> | |||
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)<br /> | |||
Let c and d be the bezout pair of a+b and b, with c·(a+b) + d·b = 1<br /> | |||
Since the pergen is a double-split, m &gt; 1, therefore |b| &gt; 1, therefore c ≠ 0<br /> | |||
c·(a+b)·P8 = c·b·((n/b)·G - P5)<br /> | |||
(1 - d·b)·P8 = c·b·((n/b)·G - P5)<br /> | |||
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5)<br /> | |||
P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)<br /> | |||
Therefore P8 is split into m periods<br /> | |||
Therefore if m = |b|, the pergen is explicitly false<br /> | |||
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Given:<br /> | Given:<br /> | ||
A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x &gt; 0, z ≠ 0, and |i| &lt;= x<br /> | A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x &gt; 0, z ≠ 0, and |i| &lt;= x<br /> | ||