Schismic–Mercator equivalence continuum

The syntonic-chromatic equivalence continuum is a continuum of temperaments which equate a number of schismas (32805/32768) with Mercator's comma ([-84 53⟩).

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⟩
…	…	…	…
∞	Schismic	32805/32768	[-15 8 1⟩

Counterschismic

Comma: [-69 45 -1⟩

Map: [<1 2 21|, <0 -1 -45|]

Wedgie: <<1 45 69||

POTE generator: ~3/2 = 701.9175

EDOs: 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056

Badness: 0.09123

3684 & 11105

Comma: [-339 230 -11⟩

Map: ⟨1 2 11], ⟨0 -11 -230]

Wedgie: ⟨⟨11 230 339]]

POTE generator: 45.2769

EDOs: Template:Val list