Pergen names: Difference between revisions
Wikispaces>TallKite **Imported revision 624093217 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624132641 - 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 19:25:03 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>624132641</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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=__Applications__= | =__Applications__= | ||
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/ | 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), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), 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. Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit. Meantone (mean = average, tone = major 2nd) implies y3/2. Semisixth (aka sensei) implies y6/2. Sometimes the multi-gen isn't even a multi-gen: y3/2 isn't actually a generator, although y6/2 is. | ||
Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit. Meantone (mean = average, tone = major 2nd) implies y3/2. Semisixth (aka sensei) implies y6/2. | |||
Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a JI-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly. | Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a JI-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly. | ||
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==Alternate enharmonics== | ==Alternate enharmonics== | ||
For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 < |x| <= m/2 and 0 < y <= n/2 | For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 < |x| <= m/2 and 0 < y <= n/2 | ||
x and y are the counts of the two enharmonics. Negative values are allowed, and are preferred if smaller. | x and y are the counts of the two enharmonics. Negative values are allowed, and are preferred if smaller. | ||
For false doubles using single-pair notation, E = E'. | For false doubles using single-pair notation, E = E'. | ||
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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. | 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. | ||
== == | |||
==Combining pergens== | |||
250/243 makes third-fourth, and 49/48 makes half-fourth, and tempering out both commas makes sixth-fourth. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). General rules for combining pergens: | |||
(P8/m, P5) + (P8/m', P5) = (P8/m", P5), where m" = LCM (m,m') | |||
(P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n') | |||
(P8/m, P5) + (P8, M/n) = (P8/m, M/n) | |||
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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 names</title></head><body><!-- ws:start:WikiTextHeadingRule:21:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:21 --> </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 names</title></head><body><!-- ws:start:WikiTextHeadingRule:21:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:21 --> </h1> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:47:&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:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 1em;"><a href="#Applications">Applications</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multi-gens">Extremely large multi-gens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><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:55 --><!-- ws:start:WikiTextTocRule:56: --><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:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate keyspans and stepspans">Alternate keyspans and stepspans</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><div style="margin-left: 2em;"><a href="#toc11"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><div style="margin-left: 2em;"><a href="#Further Discussion-Combining pergens">Combining pergens</a></div> | ||
<!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --></div> | |||
<!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextHeadingRule:23:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:23 --><u><strong>Definition</strong></u></h1> | |||
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<!-- ws:start:WikiTextHeadingRule:27:&lt;h1&gt; --><h1 id="toc3"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:27 --><u>Applications</u></h1> | <!-- 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/ | 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), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), 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. Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit. Meantone (mean = average, tone = major 2nd) implies y3/2. Semisixth (aka sensei) implies y6/2. Sometimes the multi-gen isn't even a multi-gen: y3/2 isn't actually a generator, although y6/2 is.<br /> | ||
Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit. Meantone (mean = average, tone = major 2nd) implies y3/2. Semisixth (aka sensei) implies y6/2. | |||
<br /> | <br /> | ||
Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a JI-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.<br /> | Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a JI-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:39:&lt;h2&gt; --><h2 id="toc9"><a name="Further Discussion-Alternate enharmonics"></a><!-- ws:end:WikiTextHeadingRule:39 -->Alternate enharmonics</h2> | <!-- 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 (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 &lt; |x| &lt;= m/2 and 0 &lt; y &lt;= n/2 <br /> | For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 &lt; |x| &lt;= m/2 and 0 &lt; y &lt;= n/2<br /> | ||
x and y are the counts of the two enharmonics. Negative values are allowed, and are preferred if smaller.<br /> | x and y are the counts of the two enharmonics. Negative values are allowed, and are preferred if smaller.<br /> | ||
For false doubles using single-pair notation, E = E'.<br /> | For false doubles using single-pair notation, E = E'.<br /> | ||
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<br /> | <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 /> | 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 /> | ||
<br /> | <!-- ws:start:WikiTextHeadingRule:43:&lt;h2&gt; --><h2 id="toc11"><!-- ws:end:WikiTextHeadingRule:43 --> </h2> | ||
<!-- ws:start:WikiTextHeadingRule:45:&lt;h2&gt; --><h2 id="toc12"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:45 -->Combining pergens</h2> | |||
<br /> | |||
250/243 makes third-fourth, and 49/48 makes half-fourth, and tempering out both commas makes sixth-fourth. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). General rules for combining pergens:<br /> | |||
(P8/m, P5) + (P8/m', P5) = (P8/m&quot;, P5), where m&quot; = LCM (m,m')<br /> | |||
(P8, M/n) + (P8, M/n') = (P8, M/n&quot;), where n&quot; = LCM (n,n')<br /> | |||
(P8/m, P5) + (P8, M/n) = (P8/m, M/n)<br /> | |||
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