Kite's thoughts on pergens: Difference between revisions
added an addenda for late 2024 |
→Addenda (late 2024): added a section on chord names |
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== Addenda (late 2024) == | == Addenda (late 2024) == | ||
=== Chord names === | |||
When naming chords, it's very convenient to have the freedom to rename an aug 4th as a dim 5th, or a minor 10th as an aug ninth. Thus for some pergens, an extra pair of accidentals is used. Some examples: | |||
* [[Chords of meantone]] (P8, P5) (^1 = -d2 = pythagorean comma) | |||
* [[Chords of hemififths]] (P8, P5/2) (/1 = vm2 = ~81/80 = ~64/63) | |||
* [[Chords of porcupine]] (P8, P4/2) | |||
* [[Chords of magic]] (P8, P12/5) (/1 = ^^d2) | |||
=== Frequency of imperfect pergens === | === Frequency of imperfect pergens === | ||
Imperfect pergens occur when there are multiple genchains (i.e. the octave is split), and the fifth is on a different genchain than the tonic, and also on a different perchain. How often do they occur? In order to answer that, we need to survey all pergens in order. But the question of how to do that | Imperfect pergens occur when there are multiple genchains (i.e. the octave is split), and the fifth is on a different genchain than the tonic, and also on a different perchain. How often do they occur? In order to answer that, we need to survey all pergens in order. But the question of how to do that depends on how they are sorted. The pergenLister app sorts them by the size of the larger denominator. Using this order, pergenLister finds about 4% of all pergens are imperfect. But they can also be sorted by their canonical mappings [(a b) (0 c)]. If sorted by a (octave fraction), and then by |c| (perfect multigen's fraction), more complex pergens appear sooner, and the percentage rises to about 25%. | ||
This table lists all pergens with an unsplit octave up to the fifth-splits. In each column, the pergens are sorted by the size of the generator. The generator is listed followed by a, b and c from its mapping. All pergens with an unsplit octave are perfect. | This table lists all pergens with an unsplit octave up to the fifth-splits. In each column, the pergens are sorted by the size of the generator. The generator is listed followed by a, b and c from its mapping. All pergens with an unsplit octave are perfect. | ||
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| | | | ||
|P11/3 (2 6 -3) | |P11/3 (2 6 -3) | ||
|P5/4 (2 2 4) | |P5/4 (2 2 4) | ||
|P11/5 (2 6 -5) | |P11/5 (2 6 -5) | ||
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| | | | ||
|P4/2 (4 8 -2) | |P4/2 (4 8 -2) | ||
|P5/3 (4 4 3) | |P5/3 (4 4 3) | ||
|'''m6/16 (4 7 -4)''' | |'''m6/16 (4 7 -4)''' | ||
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|'''M10/16 (4 5 4)''' | |'''M10/16 (4 5 4)''' | ||
|P11/5 (4 12 -5) | |P11/5 (4 12 -5) | ||
|P11/6 (4 12 -6) | |P11/6 (4 12 -6) | ||
|P11/7 (4 12 -7) | |P11/7 (4 12 -7) | ||
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|'''m6/24 (4 7 -6)''' | |'''m6/24 (4 7 -6)''' | ||
|P12/7 (4 0 7) | |P12/7 (4 0 7) | ||
| | |P4/8 (4 8 -8) | ||
|- | |- | ||
| | | | ||
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|'''M10/24 (4 5 6)''' | |'''M10/24 (4 5 6)''' | ||
|ccP4/7 (4 16 -7) | |ccP4/7 (4 16 -7) | ||
| | |P5/8 (4 4 8) | ||
|- | |- | ||
| | | | ||
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|'''ccm6/24 (4 9 -6)''' | |'''ccm6/24 (4 9 -6)''' | ||
|ccP5/7 (4 -4 7) | |ccP5/7 (4 -4 7) | ||
|''' | |'''ccm6/32 (4 9 -8)''' | ||
|- | |- | ||
| | | | ||
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| | | | ||
|c<sup>3</sup>P4/7 (4 20 -7) | |c<sup>3</sup>P4/7 (4 20 -7) | ||
| | |'''cm7/16 (4 10 -8)''' | ||
|- | |- | ||
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| | | | ||
| | |'''c<sup>3</sup>M3/32 (4 3 8)''' | ||
|} | |} | ||
Percentage of imperfect pergens in each category: | Percentage of imperfect pergens in each category: | ||