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
Due to dividing the fifth into so many steps, standard notation becomes almost useless for omega-pentacot. Regardless, notation has been provided for where monocot intervals appear in this system.
| # | Cents | Notation | Name |
|---|---|---|---|
| −16 | 806.256 | ||
| −15 | 905.865 | A | major sixth |
| −14 | 1005.474 | ||
| −13 | 1105.083 | ||
| −12 | 4.692 | ||
| −11 | 104.301 | ||
| −10 | 203.910 | D | major second |
| −9 | 303.519 | ||
| −8 | 403.128 | ||
| −7 | 502.737 | ||
| −6 | 602.346 | ||
| −5 | 701.955 | G | perfect fifth |
| −4 | 801.564 | ||
| −3 | 901.173 | ||
| −2 | 1000.782 | ||
| −1 | 1100.391 | ||
| 0 | 0.000 | C | perfect unison |
| 1 | 99.609 | ||
| 2 | 199.218 | ||
| 3 | 298.827 | ||
| 4 | 398.436 | ||
| 5 | 498.045 | F | perfect fourth |
| 6 | 597.654 | ||
| 7 | 697.263 | ||
| 8 | 796.872 | ||
| 9 | 896.481 | ||
| 10 | 996.090 | Bb | minor seventh |
| 11 | 1095.699 | ||
| 12 | 1195.308 | ||
| 13 | 94.917 | ||
| 14 | 194.526 | ||
| 15 | 294.135 | Eb | minor third |
| 16 | 393.744 |
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.
Quindromeda
In quindromeda, the generator is 18/17, three generators 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. This temperament is supported by 12, 169, 181, 193, 205, 217, 229, and 241 edos.
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 generators 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. This temperament is supported by 12, 109, 121, 133, 145, and 157 edos.
In the 2.3.5.7.17.19 subgroup, tempering out 400/399 (equating 20/19 and 21/20) leads to quintupole (12 & 121), and tempering out 476/475 (equating 19/17 with 28/25) leads to quinticosiennic (12 & 145).
Passion
In passion, the generator is 16/15, four generators 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 ¢).