Schismic-counterpyth equivalence continuum

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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

Optimal GPV sequence5, 31c, 36c, 41, 46, 87, 220, 307

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

Optimal GPV sequence41, 58, 99, 239, 338, 915b, 1253bc

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)

Optimal GPV sequence41, 111, 152

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

Optimal GPV sequence41, 123, 164, 205, 369, 574, 779, 2132bc

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)

Optimal GPV sequence41, 135c, 176, 217, 475, 1167, 1642, 2117b

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

Optimal GPV sequence41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 3592bc, 5523bbc

Badness: 1.528465