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 in the continuum
n Temperament Comma
Ratio Monzo
0 Mercator [-84 53
1 Counterschismic [-69 45 -1
2 Monzismic [54 -37 2
3 Tricot [39 -29 3
4 Vulture [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 [-36 11 8
9 Untriton [-51 19 9
Schismic 32805/32768 [-15 8 1

Examples of temperaments with fractional values of n:

  • 53 & 3684 (n = 11/6 = 1.83)
  • 53 & 4190 (n = 13/7 = 1.857142)

Mercator

See also: Mercator's comma and Mercator family

Comma list: [-84 53

POTE generator: ~5/4 = 386.264

Mapping: [53 84 123], 0 0 1]]

Wedgie: ⟨⟨0 53 84]]

Vals53, 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.

Comma list: [-69 45 -1

POTE generator: ~3/2 = 701.9175

Mapping: [1 2 21], 0 -1 -45]]

Wedgie⟨⟨1 45 69]]

Vals53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056

Badness: 0.09123

53 & 3684

Comma list: [-339 230 -11

POTE generator: ~10737418240/10460353203 = 45.2769

Mapping: [1 2 11], 0 -11 -230]]

Wedgie: ⟨⟨11 230 339]]

Vals53, 3684, 11105

Badness: 0.276036

53 & 4190

Comma list: [393 -267 13

POTE generator: ~[-60 41 -2 = 407.5419

Mapping: [1 6 93], 0 -13 -267]]

Wedgie: ⟨⟨13 267 393]]

Vals53, 4190, 4243, 4296

Badness: 0.173433