Ploidacot/Omega-pentacot
Omega-pentacot is a temperament archetype where the generator is a semitone, five of which stack to form a perfect fourth of 4/3, and the period is a 2/1 octave. Omega-pentacot temperaments usually generate the 1L 11s and 12L 1s MOS structures. Regular temperaments of omega-pentacot are cluster temperaments with 12 clusters of notes in an octave.
Intervals and notation
While there is no agreed-upon notation system for omega-pentacot, the following is based on interpreting the generator as a semitone (1/5 of a fourth), allowing for an ^ or v to stand for 1/5 of an inversed diminished second (the difference between diatonic semitone and chromatic semitone, equivalent to the Pythagorean comma), so vvvC# and ^^Db are enharmonic.
| # | Cents | Notation | Name |
|---|---|---|---|
| −30 | 611.730 | F# | augmented fourth |
| −29 | 711.339 | ^^G | |
| −28 | 810.948 | vG# | |
| −27 | 910.557 | ^A | |
| −26 | 1010.166 | vvA# | |
| −25 | 1109.775 | B | major seventh |
| −24 | 9.384 | ^^C | |
| −23 | 108.993 | vC# | |
| −22 | 208.602 | ^D | |
| −21 | 308.211 | vvD# | |
| −20 | 407.820 | E | major third |
| −19 | 507.429 | ^^F | |
| −18 | 607.038 | vF# | |
| −17 | 706.647 | ^G | |
| −16 | 806.256 | vvG# | |
| −15 | 905.865 | A | major sixth |
| −14 | 1005.474 | ^^Bb | |
| −13 | 1105.083 | vB | |
| −12 | 4.692 | ^C | |
| −11 | 104.301 | vvC# | |
| −10 | 203.910 | D | major second |
| −9 | 303.519 | ^^Eb | |
| −8 | 403.128 | vE | |
| −7 | 502.737 | ^F | |
| −6 | 602.346 | vvF# | |
| −5 | 701.955 | G | perfect fifth |
| −4 | 801.564 | ^^Ab | |
| −3 | 901.173 | vA | |
| −2 | 1000.782 | ^Bb | |
| −1 | 1100.391 | vvB | |
| 0 | 0.000 | C | perfect unison |
| 1 | 99.609 | ^^Db | |
| 2 | 199.218 | vD | |
| 3 | 298.827 | ^Eb | |
| 4 | 398.436 | vvE | |
| 5 | 498.045 | F | perfect fourth |
| 6 | 597.654 | ^^Gb | |
| 7 | 697.263 | vG | |
| 8 | 796.872 | ^Ab | |
| 9 | 896.481 | vvA | |
| 10 | 996.090 | Bb | minor seventh |
| 11 | 1095.699 | ^^Cb | |
| 12 | 1195.308 | vC | |
| 13 | 94.917 | ^Db | |
| 14 | 194.526 | vvD | |
| 15 | 294.135 | Eb | minor third |
| 16 | 393.744 | ^^Fb | |
| 17 | 493.353 | vF | |
| 18 | 592.962 | ^Gb | |
| 19 | 692.571 | vvG | |
| 20 | 792.180 | Ab | minor sixth |
| 21 | 891.789 | ^^Bbb | |
| 22 | 991.398 | vBb | |
| 23 | 1091.007 | ^Cb | |
| 24 | 1190.616 | vvC | |
| 25 | 90.225 | Db | minor second |
| 26 | 189.834 | ^^Ebb | |
| 27 | 289.443 | vEb | |
| 28 | 389.052 | ^Fb | |
| 29 | 488.661 | vvF | |
| 30 | 588.270 | Gb | diminished fifth |
A notable feature of omega-pentacot is the small comma, encountered after 12 steps, which represents one-fifth of a Pythagorean comma (or its equivalence, inversed diminished second). This makes omega-pentacot scales cluster around 12edo.
Temperament interpretations
Quinticular
Omega-pentacot temperaments are generally interpretated as quinticular temperaments; the generator is 18/17, five of them gives 4/3, so the quinticular comma (1419857/1417176) is tempered out.
Quintilischis
In quintilischis, the generator is 18/17, three of which make 19/16, five make 4/3, and 40 make 10th harmonic in the 2.3.5.17.19 subgroup, so 4624/4617, 6144/6137, and 6885/6859 are tempered out. This temperament is a weak extension of schismic, splitting the fourth in five. In the 2.3.5.7.17.19 subgroup, tempering out 400/399 (equating 20/19 and 21/20) leads to quintilipyth (12 & 253), and tempering out 476/475 (equating 19/17 with 28/25) leads to quintaschis (12 & 289).
Quindromeda
In quindromeda, the generator is 18/17, three of which make 19/16, five make 4/3, and 28 make 5th harmonic in the 2.3.5.17.19 subgroup, so 1216/1215, 1445/1444, and 6144/6137 are tempered out. Equating 225/224 with 256/255 leads to quintakwai (12 & 193), which tempers out 400/399 (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads to quintagar (12 & 217), which tempers out 476/475 (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.
Quintaleap
In quintaleap, the generator is 18/17, three of which make 19/16, five make 4/3, and 16 make 5/2 in the 2.3.5.17.19 subgroup, so 256/255, 361/360, and 4624/4617 are tempered out. In the 2.3.5.7.17.19 subgroup, tempering out 400/399 leads to quintupole (12 & 121), and tempering out 476/475 leads to quinticosiennic (12 & 145).
Passion
In passion, the generator is 16/15, four of which make 5/4, and five make 4/3. It is best tuned with a slightly flat generator of about 98.7 ¢, and follows that both 3 and 5 should be tuned sharp. The canonical mapping of 7 places 7/4 at 10 generators, and follows that the generator should be tuned flatter (about 98.1 ¢).
Ripple
In ripple, the generator is 27/25, five of which make 4/3, and eight make 8/5. It is best tuned with a sharp generator of about 101–102 ¢, giving the 11L 1s MOS structure (rather than 1L 11s), and follows that 3 should be tuned flat.