Ennealimmal–enneadecal equivalence continuum
The ennealimmal-enneadecal equivalence continuum is a continuum of 5-limit temperaments which equate a number of ennealimmas ([1 -27 18⟩) with enneadeca comma ([-14 -19 19⟩). This continuum is theoretically interesting in that these are all 5-limit microtemperaments.
All temperaments in the continuum satisfy (2･3-27･518)n ~ (2-14･3-19･519). Varying n results in different temperaments listed in the table below. It converges to ennealimmal as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 171edo (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of n is approximately 3.2669545024..., and temperaments having n near this value tend to be the most accurate ones.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -4 | 171 & 2429 | [-10 -127 91⟩ | |
| -3 | 171 & 1817 | [-11 -100 73⟩ | |
| -2 | Semidimi | [-12 -73 55⟩ | |
| -1 | Supermajor | [-13 -46 37⟩ | |
| 0 | Enneadecal | 19073486328125/19042491875328 | [-14 -19 19⟩ |
| 1 | Schismic | 32805/32768 | [-15 8 1⟩ |
| 2 | Minortone | 50031545098999707/50000000000000000 | [-16 35 -17⟩ |
| 3 | Senior | [-17 62 -35⟩ | |
| 4 | 171 & 1783 | [18 -89 53⟩ | |
| 5 | 171 & 2395 | [19 -116 71⟩ | |
| … | … | … | … |
| ∞ | Ennealimmal | 7629394531250/7625597484987 | [1 -27 18⟩ |
Examples of temperaments with fractional values of n:
- 171 & 4125 (n = 10/3 = 3.3)