Schismic-Mercator equivalence continuum

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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 these 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 Tricot (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…

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 sequence53, 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 sequence53, 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 sequence53, 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 sequence53, 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 sequence53, 4190, 4243, 4296

Badness: 0.173433