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Kite's thoughts on pergens: Difference between revisions

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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. 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. Thus __true doubles require commas of at least 7-limit__, whereas false doubles require only 5-limit. To summarize:
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. 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. Thus __true doubles require commas of at least 7-limit__, whereas false doubles require only 5-limit. To summarize:
* **A double-split pergen is __explicitly false__ if and only if m = |b|.**
* **A double-split pergen is __explicitly false__ if and only if m = |b|.**
* **A double-split pergen is a __true double__ if and only if GCD (m, n) &gt; |b|.**
* **A double-split pergen is a __true double__ if and only if neither it nor its unreduced form is explicitly false.**
* **A double-split pergen is a __true double__ if and only if neither it nor its unreduced form is explicitly false.**
* **A double-split pergen is a __true double__ if GCD (m, n) &gt; |b|, and a false double if GCD (m, n) = |b|.**


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.
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.
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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. 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. Thus &lt;u&gt;true doubles require commas of at least 7-limit&lt;/u&gt;, whereas false doubles require only 5-limit. To summarize:&lt;br /&gt;
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. 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. Thus &lt;u&gt;true doubles require commas of at least 7-limit&lt;/u&gt;, whereas false doubles require only 5-limit. To summarize:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;&lt;strong&gt;A double-split pergen is &lt;u&gt;explicitly false&lt;/u&gt; if and only if m = |b|.&lt;/strong&gt;&lt;/li&gt;&lt;li&gt;&lt;strong&gt;A double-split pergen is a &lt;u&gt;true double&lt;/u&gt; if and only if GCD (m, n) &amp;gt; |b|.&lt;/strong&gt;&lt;/li&gt;&lt;li&gt;&lt;strong&gt;A double-split pergen is a &lt;u&gt;true double&lt;/u&gt; if and only if neither it nor its unreduced form is explicitly false.&lt;/strong&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;&lt;strong&gt;A double-split pergen is &lt;u&gt;explicitly false&lt;/u&gt; if and only if m = |b|.&lt;/strong&gt;&lt;/li&gt;&lt;li&gt;&lt;strong&gt;A double-split pergen is a &lt;u&gt;true double&lt;/u&gt; if and only if neither it nor its unreduced form is explicitly false.&lt;/strong&gt;&lt;/li&gt;&lt;li&gt;&lt;strong&gt;A double-split pergen is a &lt;u&gt;true double&lt;/u&gt; if GCD (m, n) &amp;gt; |b|, and a false double if GCD (m, n) = |b|.&lt;/strong&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
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.&lt;br /&gt;
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.&lt;br /&gt;
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