User:Overthink/Ploidacot/Triploid Tricot
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| Pergen | [P8/3, P4/3] |
| Numeral form | 3-ploid 3-cot |
| Pure generator size | 166.01 ¢ |
| Pure period size | 400 ¢ |
| Forms | 6, 9, 15, 21 |
| Characteristic multival entry | 9 |
Triploid tricot is a temperament archetype where the generator is a supermajor second, three of which stack to a 3/2 perfect fifth, and the period is a third of a 2/1 octave, or 400 ¢. Alternatively, the generator can be a submajor second, three of which stack to a 4/3 perfect fourth. Triploid tricot temperaments usually generate the 3L 3s, 6L 3s, and 6L 9s mos scales.
Intervals and notation
There is no universally agreed apon notation for triploid tricot. The table below uses the following notation:
| Amount | Sharp | Flat |
|---|---|---|
| Chromatic semitone/3 (≈ 38 ¢) | Lift (/) | Drop (\) |
| Diatonic semitone/3 (≈ 30 ¢) | Up (^) | Down (v) |
| Pythagorean comma/3 (≈ 8 ¢) | Plus (+) | Minus (-) |
| # | Ploid 0 | Ploid 1 | Ploid 2 | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Cents | Notation | Name | Cents | Notation | Name | Cents | Notation | Name | |
| −6 | 203.91 | D | Major second | 603.91 | F#- | — | 1003.91 | Bb+ | — |
| −5 | 369.93 | \E | — | 769.93 | vAb+ | — | 1169.93 | Cv | — |
| −4 | 535.94 | /F | — | 935.94 | ^A | — | 135.94 | /Db+ | — |
| −3 | 701.96 | G | Perfect fifth | 1101.96 | B- | — | 301.96 | Eb+ | — |
| −2 | 867.97 | \A | — | 67.97 | vDb+ | — | 467.97 | vF | — |
| −1 | 1033.99 | /Bb | — | 233.99 | D^ | — | 633.99 | ^F#- | — |
| 0 | 0.00 | C | Unison | 400.00 | vE | — | 800.00 | ^Ab | — |
| 1 | 166.01 | \D | — | 566.01 | vGb+ | — | 966.01 | vBb | — |
| 2 | 332.03 | /Eb | — | 732.03 | vF# | — | 1132.03 | ^B- | — |
| 3 | 498.04 | F | Perfect fourth | 898.04 | A- | — | 98.04 | Db+ | — |
| 4 | 664.06 | \G | — | 1064.06 | \B- | — | 264.06 | vEb | — |
| 5 | 830.07 | /Ab | — | 30.07 | ^C | — | 430.07 | ^E- | — |
| 6 | 996.09 | Bb | Minor seventh | 196.09 | D- | — | 596.09 | Gb+ | — |
Temperament interpretations
A third of 3/2 is close to 8/7, and a third of 4/3 is close to 11/10. Making both of these equivalences in the 2.3.7.11/5 subgroup, thus tempering out 1029/1024 and 4000/3993, leads to the subgroup temperament trisect. Prime 13 can then be added by setting 13/9 to be a third of 3/1, tempering out 2197/2187.
Extending this temperament to the full 13-limit by setting the third-octave to 5/4 leads to the trisected temperament, though much accuracy is lost compared to the subgroup temperament.