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.

Temperaments in the continuum

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 & 3193 (n = −5/2 = −2.5)
• 171 & 2140 (n = −3/2 = −1.5)
• 171 & 1087 (n = −1/2 = −0.5)
• Pnict (n = 1/3 = 0.3)
• Gammic (n = 1/2 = 0.5)
• Mutt (n = 2/3 = 0.6)
• Septichrome (n = 4/3 = 1.3)
• Geb (n = 3/2 = 1.5)
• 171 & 1901 (n = 5/2 = 2.5)
• 171 & 4125 (n = 10/3 = 3.3)
• 171 & 3125 (n = 7/2 = 3.5)