Pergen names: Difference between revisions
Wikispaces>TallKite **Imported revision 624089979 - 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>2017-12-20 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-20 02:29:18 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>624090545</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[toc | <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;white-space: pre-wrap ! important" class="old-revision-html">= = | ||
[[toc]] | |||
=__**Definition**__= | =__**Definition**__= | ||
A **pergen** (pronounced "peer-gen") is a way of identifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. | A **pergen** (pronounced "peer-gen") is a way of identifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. | ||
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For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multi-gen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), the same as Blackwood. | For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multi-gen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), the same as Blackwood. | ||
To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix, it's the reduced mapping. Next make a **square | To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix, it's the reduced mapping. Next make a **square mapping** by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5,y3) example, to ensure that the diagonal has no zeros. Lower primes > 3 may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents. | ||
For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multi-gen, P is the period P8/m and G is the generator M/n | For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multi-gen, P is the period P8/m and G is the generator M/n | ||
2/1 = P8 = xP, thus P = P8/x | 2/1 = P8 = xP, thus P = P8/x | ||
3/1 = P12 = yP + zG, thus G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz | 3/1 = P12 = yP + zG, thus G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz | ||
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G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz | G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz | ||
<span style="display: block; text-align: center;">**<span style="font-size: 110%;">The rank-2 pergen from the [(x, 0) | <span style="display: block; text-align: center;">**<span style="font-size: 110%;">The rank-2 pergen from the [(x, 0) (y, z)] square mapping is (P8/x, (nz-y,x)/xz), with -x <= n <= x</span>** | ||
</span> | </span> | ||
For example, 250/243... | For example, 250/243 has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 <= n <= 1. No value of n reduces the fraction, so the best generator is the one with the least cents. The pergen is (P8, P4/3). | ||
Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7|x31.com]] gives us this matrix: | Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7|x31.com]] gives us this matrix: | ||
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//[Change R to P and G]// | //[Change R to P and G]// | ||
To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, simply find a ratio | To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, simply find a ratio for P or G that contains only one higher prime, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work for P, too many and too few primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*G - P4 = (10/9)^3 / (4/3) = 250/243. | ||
Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multi-gen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4). | Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multi-gen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4). | ||
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For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2*P8 - 3*P5)/(3*2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6*G - m3. The comma splits both the octave and the fifth. | For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2*P8 - 3*P5)/(3*2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6*G - m3. The comma splits both the octave and the fifth. | ||
This | This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2*P8 - 4*P4)/(2*4) = (2*M2)/8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. | ||
A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48 | A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively. | ||
Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an **alternate** generator. A generator or period plus or minus any number of enharmonics makes an **equivalent** generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. | Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an **alternate** generator. A generator or period plus or minus any number of enharmonics makes an **equivalent** generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. | ||
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[//Question: what if there are highs and lows?//]</pre></div> | [//Question: what if there are highs and lows?//]</pre></div> | ||
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<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 names</title></head><body><!-- ws:start:WikiTextTocRule: | <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 names</title></head><body><!-- ws:start:WikiTextHeadingRule:21:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:21 --> </h1> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:43:&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:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | |||
<!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div> | |||
<!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 1em;"><a href="#Applications">Applications</a></div> | |||
<!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div> | |||
<!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multi-gens">Extremely large multi-gens</a></div> | |||
<!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div> | |||
<!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div> | |||
<!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><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:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div> | |||
<!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate keyspans">Alternate keyspans</a></div> | |||
<!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --></div> | |||
<!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextHeadingRule:23:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:23 --><u><strong>Definition</strong></u></h1> | |||
<br /> | <br /> | ||
<br /> | |||
A <strong>pergen</strong> (pronounced &quot;peer-gen&quot;) is a way of identifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.<br /> | A <strong>pergen</strong> (pronounced &quot;peer-gen&quot;) is a way of identifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.<br /> | ||
<br /> | <br /> | ||
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The preferred pergen for untempered just intonation is the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the monzo must be 1 or -1. These commas are called <strong>notational commas</strong>. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is widespread agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th.<br /> | The preferred pergen for untempered just intonation is the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the monzo must be 1 or -1. These commas are called <strong>notational commas</strong>. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is widespread agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:25:&lt;h1&gt; --><h1 id="toc2"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:25 --><u>Derivation</u></h1> | ||
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For any comma containing primes 2 and 3, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents. The comma will split the octave into m parts, and if n &gt; m, it will split some other 3-limit interval into n parts. Therefore a comma with only two primes, one of which is 2, always splits the octave (unless the other prime's exponent is ±1, e.g. 32/31). And a comma with only one higher prime will always split something, unless that prime's exponent is ±1.<br /> | For any comma containing primes 2 and 3, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents. The comma will split the octave into m parts, and if n &gt; m, it will split some other 3-limit interval into n parts. Therefore a comma with only two primes, one of which is 2, always splits the octave (unless the other prime's exponent is ±1, e.g. 32/31). And a comma with only one higher prime will always split something, unless that prime's exponent is ±1.<br /> | ||
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For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multi-gen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), the same as Blackwood.<br /> | For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multi-gen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), the same as Blackwood.<br /> | ||
<br /> | <br /> | ||
To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder <a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank">x31eq.com/temper/uv.html</a> that will find such a matrix, it's the reduced mapping. Next make a <strong>square | To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder <a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank">x31eq.com/temper/uv.html</a> that will find such a matrix, it's the reduced mapping. Next make a <strong>square mapping</strong> by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5,y3) example, to ensure that the diagonal has no zeros. Lower primes &gt; 3 may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.<br /> | ||
<br /> | <br /> | ||
For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multi-gen, P is the period P8/m and G is the generator M/n<br /> | For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multi-gen, P is the period P8/m and G is the generator M/n<br /> | ||
2/1 = P8 = xP, thus P = P8/x <br /> | 2/1 = P8 = xP, thus P = P8/x<br /> | ||
3/1 = P12 = yP + zG, thus G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz<br /> | 3/1 = P12 = yP + zG, thus G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz<br /> | ||
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G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz<br /> | G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz<br /> | ||
<br /> | <br /> | ||
<span style="display: block; text-align: center;"><strong><span style="font-size: 110%;">The rank-2 pergen from the [(x, 0) | <span style="display: block; text-align: center;"><strong><span style="font-size: 110%;">The rank-2 pergen from the [(x, 0) (y, z)] square mapping is (P8/x, (nz-y,x)/xz), with -x &lt;= n &lt;= x</span></strong><br /> | ||
</span><br /> | </span><br /> | ||
For example, 250/243...<br /> | For example, 250/243 has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &lt;= n &lt;= 1. No value of n reduces the fraction, so the best generator is the one with the least cents. The pergen is (P8, P4/3).<br /> | ||
<br /> | <br /> | ||
Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7" rel="nofollow">x31.com</a> gives us this matrix:<br /> | Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7" rel="nofollow">x31.com</a> gives us this matrix:<br /> | ||
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Again, period = P8 and gen1 = P5/2. Gen2 = (-3, -1, 2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multi-gen, the multi-gen2. A double half-fifth is a fifth = (-1, 1, 0), and this gives us (-4, 0, 2)/2 = 7/4. The fraction disappears, the multi-gen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-fifths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-fifth with red. Assuming 7's notational comma is 64/63, the pergen is (P8, P5/2, ^1), half-fifth with ups. This is far better than (P8, P5/2, gg7/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.<br /> | Again, period = P8 and gen1 = P5/2. Gen2 = (-3, -1, 2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multi-gen, the multi-gen2. A double half-fifth is a fifth = (-1, 1, 0), and this gives us (-4, 0, 2)/2 = 7/4. The fraction disappears, the multi-gen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-fifths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-fifth with red. Assuming 7's notational comma is 64/63, the pergen is (P8, P5/2, ^1), half-fifth with ups. This is far better than (P8, P5/2, gg7/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:27:&lt;h1&gt; --><h1 id="toc3"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:27 --><u>Applications</u></h1> | ||
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One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), triforce is (P8/3, P4/2), tetracot is (P8, P5/4), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split.<br /> | One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), triforce is (P8/3, P4/2), tetracot is (P8, P5/4), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:29:&lt;h1&gt; --><h1 id="toc4"><a name="Further Discussion"></a><!-- ws:end:WikiTextHeadingRule:29 --><u>Further Discussion</u></h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:31:&lt;h2&gt; --><h2 id="toc5"><a name="Further Discussion-Extremely large multi-gens"></a><!-- ws:end:WikiTextHeadingRule:31 -->Extremely large multi-gens</h2> | ||
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So far, the largest multi-gen has been a 12th. As the multi-gen fractions get larger, the multi-gen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave and a multi-gen split into n parts, valid multi-gens include any voicing of the fifth that is less than n/2 octaves. For (P8, multi-gen/6), the multi-gen can be P4, P5, P11, P12, WWP4 or WWP5.<br /> | So far, the largest multi-gen has been a 12th. As the multi-gen fractions get larger, the multi-gen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave and a multi-gen split into n parts, valid multi-gens include any voicing of the fifth that is less than n/2 octaves. For (P8, multi-gen/6), the multi-gen can be P4, P5, P11, P12, WWP4 or WWP5.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:33:&lt;h2&gt; --><h2 id="toc6"><a name="Further Discussion-Singles and doubles"></a><!-- ws:end:WikiTextHeadingRule:33 -->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 pergens 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.<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 pergens 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.<br /> | ||
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A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.<br /> | A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:35:&lt;h2&gt; --><h2 id="toc7"><a name="Further Discussion-Finding an example temperament"></a><!-- ws:end:WikiTextHeadingRule:35 -->Finding an example temperament</h2> | ||
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<em>[Change R to P and G]</em><br /> | <em>[Change R to P and G]</em><br /> | ||
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To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, simply find a ratio | To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, simply find a ratio for P or G that contains only one higher prime, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work for P, too many and too few primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*G - P4 = (10/9)^3 / (4/3) = 250/243.<br /> | ||
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Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multi-gen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is <strong>explicitly false</strong>. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).<br /> | Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multi-gen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is <strong>explicitly false</strong>. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).<br /> | ||
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For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2*P8 - 3*P5)/(3*2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This <u>is</u> explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6*G - m3. The comma splits both the octave and the fifth.<br /> | For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2*P8 - 3*P5)/(3*2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This <u>is</u> explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6*G - m3. The comma splits both the octave and the fifth.<br /> | ||
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This | This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2*P8 - 4*P4)/(2*4) = (2*M2)/8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48.<br /> | ||
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A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48 | A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.<br /> | ||
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Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an <strong>alternate</strong> generator. A generator or period plus or minus any number of enharmonics makes an <strong>equivalent</strong> generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently.<br /> | Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an <strong>alternate</strong> generator. A generator or period plus or minus any number of enharmonics makes an <strong>equivalent</strong> generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently.<br /> | ||
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There are also equivalent enharmonics, see below.<br /> | There are also equivalent enharmonics, see below.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:37:&lt;h2&gt; --><h2 id="toc8"><a name="Further Discussion-Finding a notation for a pergen"></a><!-- ws:end:WikiTextHeadingRule:37 -->Finding a notation for a pergen</h2> | ||
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There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some basic concepts:<br /> | There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some basic concepts:<br /> | ||
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A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is a 3rd or a 4th, as with (P8/2, P4/3).<br /> | A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is a 3rd or a 4th, as with (P8/2, P4/3).<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:39:&lt;h2&gt; --><h2 id="toc9"><a name="Further Discussion-Alternate enharmonics"></a><!-- ws:end:WikiTextHeadingRule:39 -->Alternate enharmonics</h2> | ||
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For single-comma pergens, the enharmonic should equal the comma's mapping. For example, (P8, P5/2) might arise from 243/242, which splits the 5th into two 11/9 halves.<br /> | For single-comma pergens, the enharmonic should equal the comma's mapping. For example, (P8, P5/2) might arise from 243/242, which splits the 5th into two 11/9 halves.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:41:&lt;h2&gt; --><h2 id="toc10"><a name="Further Discussion-Alternate keyspans"></a><!-- ws:end:WikiTextHeadingRule:41 -->Alternate keyspans</h2> | ||
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One might wonder, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to 7-edo on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo.<br /> | One might wonder, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to 7-edo on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo.<br /> | ||