Schismic–Mercator equivalence continuum

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The schismic–Mercator equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with Mercator's comma ([-84 53⟩). This continuum is theoretically interesting in that the temperaments associated with its various commas are all 5-limit microtemperaments.

All temperaments in the continuum satisfy (32805/32768)ⁿ ~ [-84 53⟩. 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 53edo (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 1.8503390493…, and temperaments having n near this value tend to be the most accurate ones.

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

Temperaments with integer n
n	Temperament	Comma
n	Temperament	Ratio	Monzo
0	Mercator	(52 digits)	[-84 53⟩
1	Counterschismic	(44 digits)	[-69 45 -1⟩
2	Monzismic	(36 digits)	[54 -37 2⟩
3	Alphatricot	(28 digits)	[39 -29 3⟩
4	Vulture	(22 digits)	[24 -21 4⟩
5	Amity	1600000/1594323	[9 -13 5⟩
6	Kleismic	15625/15552	[-6 -5 6⟩
7	Orson	2109375/2097152	[-21 3 7⟩
8	Demibuzzard	(22 digits)	[-36 11 8⟩
9	Untriton	(32 digits)	[-51 19 9⟩
…	…	…	…
∞	Schismic	32805/32768	[-15 8 1⟩

We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the counterschismic–Mercator equivalence continuum, which is essentially the same thing. The just value of m is 2.17600… While the counterschisma is of comparable size as the schisma, it is way more complex, so this continuum does not contain as many useful temperaments at integer points.

Temperaments with integer m
m	Temperament	Comma
m	Temperament	Ratio	Monzo
0	Mercator	(52 digits)	[-84 53⟩
1	Schismic	32805/32768	[-15 8 1⟩
2	Monzismic	(36 digits)	[54 -37 2⟩
…	…	…	…
∞	Counterschismic	(44 digits)	[-69 45 -1⟩

Temperaments with fractional n and m
Temperament	n	m	Comma
53 & 3684	11/6 = 1.83	11/5 = 2.2	[-339 230 -11⟩
53 & 4296	13/7 = 1.857142	13/6 = 2.16	[393 -267 13⟩
Countritonic	9/2 = 4.5	9/7 = 1.285714	[33 -34 9⟩
Quartonic	11/2 = 5.5	11/9 = 1.2	[3 -18 11⟩
Maja	17/3 = 5.6	17/14 = 1.2142857	[-3 -23 17⟩
Ditonic	13/2 = 6.5	13/11 = 1.18	[-27 -2 13⟩

Counterschismic

See also: Counterschisma

Counterschismic is generated by a perfect fifth, like schismic, but the 5th harmonic is located at +45 fifths instead of schismic's -8. They unite at 53edo, of course. 730edo may be recommended as a tuning.

Subgroup: 2.3.5

Comma list: [-69 45 -1⟩

Mapping: [⟨1 0 -69], ⟨0 1 45]]

mapping generators: ~2, ~3

Optimal tunings: