Pergen names: Difference between revisions
Wikispaces>TallKite **Imported revision 621983257 - Original comment: ** |
Wikispaces>TallKite **Imported revision 621983405 - 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>2017-11-19 20: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-19 20:39:53 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>621983405</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> | ||
<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">= | <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|flat]] | ||
=__**Definition**__= | |||
== == | == == | ||
A **pergen** set (pronounced "peer-gen") is a way of identifying a rank-2 or rank-3 regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. The pergen set is chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. | A **pergen** set (pronounced "peer-gen") is a way of identifying a rank-2 or rank-3 regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. The pergen set is 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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A rank-4 temperament has a pergen set of four intervals. A rank-1 temperament could have a pergen set of one, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so. | A rank-4 temperament has a pergen set of four intervals. A rank-1 temperament could have a pergen set of one, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so. | ||
=__Derivation__= | |||
In a multi-comma temperament, it's possible that one comma will contain only the 1st and 2nd primes. The 2nd prime is directly related to the 1st prime. If this happens, the multi-gen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth. | In a multi-comma temperament, it's possible that one comma will contain only the 1st and 2nd primes. The 2nd prime is directly related to the 1st prime. If this happens, the multi-gen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth. | ||
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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. 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 set is {P8, P5/2, r1} = half-fifth with red. This is far better than {P8, P5/2, gg7/4}. The pergen set sometimes uses a larger prime in place of a smaller one, in order to avoid splitting gen2, __IF__ both primes are > 3. | 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. 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 set is {P8, P5/2, r1} = half-fifth with red. This is far better than {P8, P5/2, gg7/4}. The pergen set sometimes uses a larger prime in place of a smaller one, in order to avoid splitting gen2, __IF__ both primes are > 3. | ||
= | =__Applications__= | ||
Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. | Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. | ||
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(to be continued)</pre></div> | (to be continued)</pre></div> | ||
<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:0:&lt; | <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:8:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Derivation">Derivation</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Applications">Applications</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | ||
<!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 --><u><strong>Definition</strong></u></h1> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><!-- ws:end:WikiTextHeadingRule:2 --> </h2> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><!-- ws:end:WikiTextHeadingRule:2 --> </h2> | ||
A <strong>pergen</strong> set (pronounced &quot;peer-gen&quot;) is a way of identifying a rank-2 or rank-3 regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. The pergen set is 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> set (pronounced &quot;peer-gen&quot;) is a way of identifying a rank-2 or rank-3 regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. The pergen set is chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.<br /> | ||
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A rank-4 temperament has a pergen set of four intervals. A rank-1 temperament could have a pergen set of one, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.<br /> | A rank-4 temperament has a pergen set of four intervals. A rank-1 temperament could have a pergen set of one, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt; | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:4 --><u>Derivation</u></h1> | ||
<br /> | <br /> | ||
In a multi-comma temperament, it's possible that one comma will contain only the 1st and 2nd primes. The 2nd prime is directly related to the 1st prime. If this happens, the multi-gen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.<br /> | In a multi-comma temperament, it's possible that one comma will contain only the 1st and 2nd primes. The 2nd prime is directly related to the 1st prime. If this happens, the multi-gen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.<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. 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 set is {P8, P5/2, r1} = half-fifth with red. This is far better than {P8, P5/2, gg7/4}. The pergen set sometimes uses a larger prime in place of a smaller one, in order to avoid splitting gen2, <u>IF</u> both primes are &gt; 3.<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. 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 set is {P8, P5/2, r1} = half-fifth with red. This is far better than {P8, P5/2, gg7/4}. The pergen set sometimes uses a larger prime in place of a smaller one, in order to avoid splitting gen2, <u>IF</u> both primes are &gt; 3.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt; | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:6 --><u>Applications</u></h1> | ||
<br /> | <br /> | ||
Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically.<br /> | Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically.<br /> | ||
<br /> | <br /> | ||
(to be continued)</body></html></pre></div> | (to be continued)</body></html></pre></div> | ||