Schismic–Mercator equivalence continuum

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
		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	Submajor	(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
		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
53 & 3684	11/6 = 1.83	11/5 = 2.2
53 & 4190	13/7 = 1.857142	13/6 = 2.16
Countritonic	9/2 = 4.5	9/7 = 1.285714
Quartonic	11/2 = 5.5	11/9 = 1.2
Ditonic	13/2 = 6.5	13/11 = 1.18

Mercator

See also: Mercator's comma and Mercator family

Comma list: [-84 53⟩

Mapping: [⟨53 84 123], ⟨0 0 1]]

Wedgie: ⟨⟨ 0 53 84 ]]

Optimal tuning (CTE): ~531441/524288 = 1\53, ~5/4 = 386.264

Optimal ET sequence: 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650

Badness: 0.2843

Counterschismic

See also: Counterschisma

Counterschismic is much like schismic, but the harmonic 5 is located at +45 fifths instead of schismic's -8. They unite in 53edo, of course.

Subgroup: 2.3.5

Comma list: [-69 45 -1⟩

Mapping: [⟨1 2 21], ⟨0 -1 -45]]

Wedgie: ⟨⟨ 1 45 69 ]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9175

Optimal ET sequence: 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056

Badness: 0.09123

Countritonic

See also: High badness temperaments § Countritonic

For extensions, see Ragismic microtemperaments #Countritonic and Hemifamity temperaments #Countriton.

Subgroup: 2.3.5

Comma list: [33 -34 9⟩

Mapping: [⟨1 6 19], ⟨0 -9 -34]]

Optimal tuning (CTE): ~2 = 1\1, ~14348907/10240000 = 588.636

Optimal ET sequence: 53, 263, 316, 369, 422, 475, 528, 2587b, 3115b, 3643b

Badness: 0.256

53 & 3684

Subgroup: 2.3.5

Comma list: [-339 230 -11⟩

Mapping: [⟨1 2 11], ⟨0 -11 -230]]

Wedgie: ⟨⟨ 11 230 339 ]]

Optimal tuning (CTE): ~2 = 1\1, ~10737418240/10460353203 = 45.2769

Optimal ET sequence: 53, 3684, 11105

Badness: 0.276036

53 & 4190

Subgroup: 2.3.5

Comma list: [393 -267 13⟩

Mapping: [⟨1 6 93], ⟨0 -13 -267]]

Wedgie: ⟨⟨ 13 267 393 ]]

Optimal tuning (CTE): ~2 = 1\1, ~[-60 41 -2⟩ = 407.5419

Optimal ET sequence: 53, 4190, 4243, 4296

Badness: 0.173433