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 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]]

Badness: 0.2843

Counterschismic

See also: Counterschisma

Comma list: [-69 45 -1⟩

POTE generator: ~3/2 = 701.9175

Mapping: [⟨1 2 21], ⟨0 -1 -45]]

Wedgie: ⟨⟨1 45 69]]

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]]

Badness: 0.276036

53 & 4190

Comma list: [393 -267 13⟩

Wedgie: ⟨⟨13 267 393]]