Kite's thoughts on pergens: Difference between revisions

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For example, small triple amber tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the enharmonic is v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2, but if 11/8 is notated as a vA4, the enharmonic is ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2.

This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.

==Alternate keyspans and stepspans==  

A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49, or rryy&amp;gT) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.

||||||~ Tetratonic MOS scales ||~ secondary examples ||

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.

==Combining pergens==  

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For example, small triple amber tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the enharmonic is v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2, but if 11/8 is notated as a vA4, the enharmonic is ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2.&lt;br /&gt;

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This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.&lt;br /&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:76:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Further Discussion-Alternate keyspans and stepspans"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:76 --&gt;Alternate keyspans and stepspans&lt;/h2&gt;

A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49, or rryy&amp;amp;gT) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.&lt;br /&gt;

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Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.&lt;br /&gt;

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Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).&lt;br /&gt;

However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. Further study is needed.&lt;br /&gt;

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Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.&lt;br /&gt;

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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.&lt;br /&gt;

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Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;

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Screenshot of the first 38 pergens:&lt;br /&gt;

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Given:&lt;br /&gt;