Schismic–countercommatic equivalence continuum

The schismic-counterpyth equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with counterpyth comma ([65 -41⟩). This continuum is theoretically interesting in that these are all 5-limit microtemperaments.

All temperaments in the continuum satisfy (32805/32768)ⁿ ~ [65 -41⟩. 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 41edo (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 10.1575233481..., 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 Schismic-Pythagorean equivalence continuum.

Temperaments in the continuum

n	Temperament	Comma
		Ratio	Monzo
-7	Merman	1121008359375 / 1099511627776	[-40 15 7⟩
-6	Ampersand	34171875 / 33554432	[-25 7 6⟩
-5	Magic	3125 / 3072	[-10 -1 5⟩
-4	Tetracot	20000 / 19683	[5 -9 4⟩
-3	Rodan	131072000 / 129140163	[20 -17 3⟩
-2	Hemififths	858993459200 / 847288609443	[35 -25 2⟩
-1	Kwai		[50 -33 1⟩
0	Counterpyth		[65 -41⟩
1	Cotoneum		[80 -49 -1⟩
2	Newt		[95 -57 -2⟩
3	41&282		[110 -65 -3⟩
4	41&335		[125 -73 -4⟩
5	41&388		[140 -81 -5⟩
6	41&441		[155 -89 -6⟩
7	41&453		[170 -97 -7⟩
8	41&506		[185 -105 -8⟩
9	41&559		[200 -113 -9⟩
10	41&571		[215 -121 -10⟩
11	41&624		[-230 129 11⟩
12	41&677		[-245 137 12⟩
13	41&730		[-260 145 13⟩
…	…	…	…
∞	Schismic	32805/32768	[-15 8 1⟩

Examples of temperaments with fractional values of n:

• Septimin (n = -11/2 = -5.5)
• Shibboleth (n = -9/2 = -4.5)
• Pluto (n = -7/2 = -3.5)
• 3737 & 5585 (n = 31/3 = 10.3)
• 1277 & 2513 (n = 21/2 = 10.5)

Rodan (5-limit)

See also: Gamelismic clan § Rodan

Comma: 131072000/129140163

Mapping: [⟨1 1 -1], ⟨0 3 17]]

POTE generator: ~729/640 = 234.528

Badness: 0.168264

Hemififths (5-limit)

See also: Breedsmic temperaments § Hemififths

Comma: 858993459200/847288609443

Mapping: [⟨1 1 -5], ⟨0 2 25]]

POTE generator: ~655360/531441 = 351.476

Badness: 0.372848

Kwai (5-limit)

See also: Mirkwai clan § Kwai

Comma: [50 -33 1⟩ = 5629499534213120/5559060566555523

Mapping: [⟨1 2 16], ⟨0 -1 -33]]

POTE generator: ~4/3 = 497.370 (or ~3/2 = 702.630)

Badness: 0.636715

Counterpyth

See also: Counterpyth family and 41-comma

Comma list: [65 -41⟩

Mapping: [⟨41 65 0], ⟨0 0 1]]

POTE generator: ~5/4 = 386.668

Badness: 0.934310

Cotoneum (5-limit)

See also: Hemimage temperaments § Cotoneum

Comma: [80 -49 -1⟩

Mapping: [⟨1 2 -18], ⟨0 -1 49]]

POTE generator: ~4/3 = 497.685 (or ~3/2 = 702.315)

Badness: 1.240078

Newt (5-limit)

See also: Breedsmic temperaments § Newt

Comma: [95 -57 -2⟩

Mapping: [⟨1 1 19], ⟨0 2 -57]]

POTE generator: ~[47 -28 -1⟩ = 351.114

Badness: 1.528465