User:Royalmilktea/Keemic equivalence continuum

The keemic equivalence continuum is a continuum of temperaments which equate a number of ptolemismas (100/99) with the keema (875/864).

All temperaments in the continuum satisfy (100/99)ⁿ ~ 875/864. Varying n results in different temperaments listed in the table below. It converges to the 11-limit temperament of ptolemismic as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 11-limit temperaments supported by undecimal keemic due to it being the unique temperament that tempers out both commas and thus tempers out all combinations of them. The just value of n is approximately 1.2588, and temperaments near this tend to be the most accurate ones.

Temperament families in the continuum

n	Temperament	Comma
		Ratio	Monzo
-1	Nokotanmic(?)	21875/21384	[-3 -5 5 1 -1⟩
0	Keemic extension	875/864	[-5 -3 3 1 0⟩
1	Keenanismic	385/384	[-7 -1 1 1 1⟩
2	Absinthmic(?)	2560/2541	[9 -1 1 -1 -2⟩
...	...	...	...
∞	Ptolemismic	100/99	[2 -2 2 0 -1⟩

Temperaments with fractional n

n	Temperament	Comma
5/4 = 1.25	270 & 836 & 2190 & 311	[-30 -2 2 4 5⟩
9/7 = 1.285714	494 & 472 & 41 & 981	[-53 -3 3 7 9⟩
4/3 = 1.3	224 & 41 & 239 & 22	[-23 -1 1 3 4⟩
3/2 = 1.5	Orgonismic	[16 0 0 -2 -3⟩