Schismic–countercommatic equivalence continuum

The schismic–countercommatic equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with the Pythagorean countercomma ([65 -41⟩). This continuum is theoretically interesting in that these are all 5-limit microtemperaments supported by 41edo.

All temperaments in the continuum satisfy (32805/32768)ⁿ ~ [65 -41⟩. Varying n results in different temperaments listed in the table below. It converges to schismic as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 41edo due to it being the unique equal temperament that tempers out both commas and thus tempers out all combinations of them. The just value of n is approximately 10.1575233481…, and temperaments having n near this value tend to be the most accurate ones.

The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, and has many advantages as a target. In each case, n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of harmonic 3 in the generator chain. For example:

Cotoneum (n = 1) is generated by a fifth;
Newt (n = 2) splits its fifth in two;
Etc.

For a similar but perhaps more intuitive and practical concept, see Schismic–commatic equivalence continuum.

Temperaments of integer n
n	Temperament	Comma
n	Temperament	Ratio	Monzo
−7	Merman	1121008359375/1099511627776	[-40 15 7⟩
−6	Ampersand	34171875/33554432	[-25 7 6⟩
−5	Magic	3125/3072	[-10 -1 5⟩
−4	Tetracot	20000/19683	[5 -9 4⟩
−3	Rodan	131072000/129140163	[20 -17 3⟩
−2	Hemififths	858993459200/847288609443	[35 -25 2⟩
−1	Kwai	(32 digits)	[50 -33 1⟩
0	Countercomp	(40 digits)	[65 -41⟩
1	Cotoneum	(50 digits)	[80 -49 -1⟩
2	Newt	(58 digits)	[95 -57 -2⟩
3	41 & 282	(68 digits)	[110 -65 -3⟩
4	41 & 335	(76 digits)	[125 -73 -4⟩
5	41 & 388	(86 digits)	[140 -81 -5⟩
6	41 & 441	(94 digits)	[155 -89 -6⟩
7	41 & 453	(104 digits)	[170 -97 -7⟩
8	41 & 506	(112 digits)	[185 -105 -8⟩
9	41 & 559	(122 digits)	[200 -113 -9⟩
10	41 & 571	(130 digits)	[215 -121 -10⟩
11	41 & 624	(140 digits)	[-230 129 11⟩
12	41 & 677	(148 digits)	[-245 137 12⟩
13	41 & 730	(158 digits)	[-260 145 13⟩
…	…	…	…
∞	Schismic	32805/32768	[-15 8 1⟩