Ploidacot/Omega-hexacot
Omega-hexacot is a temperament archetype where the generator is a small semitone of about 82–84 ¢, six of which make a perfect fourth of 4/3, and the period is a 2/1 octave. Omega-hexacot temperaments also include all alpha-dicot and omega-tricot intervals. Omega-hexacot temperaments typically generate the 14L 1s and 14L 15s MOS scales.
Intervals and notation
Due to dividing the fourth into so many steps, standard notation becomes almost useless for omega-hexacot. Regardless, notation has been provided for where omega-hexacot intervals align with standard monocot intervals (which use chain-of-fifths notation).
| # | Cents | Notation | Name |
|---|---|---|---|
| −24 | 407.820 | E | major third |
| −23 | 490.828 | ||
| −22 | 573.835 | ||
| −21 | 656.843 | ||
| −20 | 739.850 | ||
| −19 | 822.858 | ||
| −18 | 905.865 | A | major sixth |
| −17 | 988.873 | ||
| −16 | 1071.880 | ||
| −15 | 1154.888 | ||
| −14 | 37.895 | ||
| −13 | 120.903 | ||
| −12 | 203.910 | D | major second |
| −11 | 286.918 | ||
| −10 | 369.925 | ||
| −9 | 452.933 | ||
| −8 | 535.940 | ||
| −7 | 618.948 | ||
| −6 | 701.955 | G | perfect fifth |
| −5 | 784.963 | ||
| −4 | 867.970 | ||
| −3 | 950.978 | ||
| −2 | 1033.985 | ||
| −1 | 1116.993 | ||
| 0 | 0.000 | C | perfect unison |
| 1 | 83.007 | ||
| 2 | 166.015 | ||
| 3 | 249.022 | ||
| 4 | 332.030 | ||
| 5 | 415.037 | ||
| 6 | 498.045 | F | perfect fourth |
| 7 | 581.052 | ||
| 8 | 664.060 | ||
| 9 | 747.067 | ||
| 10 | 830.075 | ||
| 11 | 913.082 | ||
| 12 | 996.090 | Bb | minor seventh |
| 13 | 1079.097 | ||
| 14 | 1162.105 | ||
| 15 | 45.112 | ||
| 16 | 128.120 | ||
| 17 | 211.127 | ||
| 18 | 294.135 | Eb | minor third |
| 19 | 377.142 | ||
| 20 | 460.150 | ||
| 21 | 543.157 | ||
| 22 | 626.165 | ||
| 23 | 709.172 | ||
| 24 | 792.180 | Ab | minor sixth |
Temperament interpretations
An obvious interpretation for omega-hexacot is historical, 2.3.7/5.11/5.13/5 retraction of history, where the generator is 21/20~22/21, two of which make 11/10, three make 15/13, five make 14/11, six make 4/3, seven make 7/5, nine make 20/13, eleven make 22/13, sixteen make 14/13 above an octave. There are some extensions for full 13-limit: sextilifourths (29 & 130), marvolo (29 & 43), and nautilus (15 & 29).
Sextilifourths
In sextilifourths, the generator is 21/20~22/21, two of which make 11/10, three make 15/13, five make 14/11, six make 4/3, seven make 7/5, nine make 20/13, eleven make 22/13, sixteen make 14/13 above an octave, 48 make tenth harmonic, 50 make eleventh harmonic, and 55 make fourteenth harmonic.
Marvolo
In marvolo, the generator is 21/20~22/21, two of which make 11/10, three make 15/13, five make 14/11, six make 4/3, seven make 7/5, nine make 20/13, ten make 13/8, eleven make 22/13, and sixteen make 13/12~14/13.
Nautilus
In nautilus, the generator is 21/20~22/21, two of which make 10/9~11/10, three make 7/6~8/7, five make 14/11, six make 4/3, seven make 7/5, eight make 16/11, and ten make 8/5.