Schismic–countercommatic equivalence continuum

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The schismic–countercommatic equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with the Pythagorean countercomma ([65 -41). This continuum is theoretically interesting in that these are all 5-limit microtemperaments supported by 41edo.

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 out both commas and thus tempers out 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.

The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, and has many advantages as a target. In each case, n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of harmonic 3 in the generator chain. For example:

  • Cotoneum (n = 1) is generated by a fifth;
  • Newt (n = 2) splits its fifth in two;
  • Etc.

For a similar but perhaps more intuitive and practical concept, see Schismic–Pythagorean equivalence continuum.

Temperaments of integer n
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 (32 digits) [50 -33 1
0 Countercomp (40 digits) [65 -41
1 Cotoneum (50 digits) [80 -49 -1
2 Newt (58 digits) [95 -57 -2
3 41 & 282 (68 digits) [110 -65 -3
4 41 & 335 (76 digits) [125 -73 -4
5 41 & 388 (86 digits) [140 -81 -5
6 41 & 441 (94 digits) [155 -89 -6
7 41 & 453 (104 digits) [170 -97 -7
8 41 & 506 (112 digits) [185 -105 -8
9 41 & 559 (122 digits) [200 -113 -9
10 41 & 571 (130 digits) [215 -121 -10
11 41 & 624 (140 digits) [-230 129 11
12 41 & 677 (148 digits) [-245 137 12
13 41 & 730 (158 digits) [-260 145 13
Schismic 32805/32768 [-15 8 1

Examples of temperaments with fractional values of n:

  • Septimin (n = −11/2)
  • Shibboleth (n = −9/2)
  • Pluto (n = −7/2)
  • 3737 & 5585 (n = 31/3 = 10.3)
  • 1277 & 2513 (n = 21/2)

Rodan (5-limit)

Subgroup: 2.3.5

Comma list: 131072000/129140163

Mapping[1 1 -1], 0 3 17]]

Optimal tuning (POTE): ~729/640 = 234.528

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

Badness: 0.168264

Hemififths (5-limit)

Subgroup: 2.3.5

Comma list: 858993459200/847288609443

Mapping[1 1 -5], 0 2 25]]

Optimal tuning (POTE): ~655360/531441 = 351.476

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

Badness: 0.372848

Kwai (5-limit)

Subgroup: 2.3.5

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

Mapping[1 0 -50], 0 1 33]]

Optimal tuning (POTE): ~3/2 = 702.630

Optimal ET sequence41, 111, 152

Badness: 0.636715

Countercomp

See also: Countercomp family and 41-comma

Subgroup: 2.3.5

Comma list: [65 -41

Mapping[41 65 0], 0 0 1]]

Optimal tuning (POTE): ~5/4 = 386.668

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

Badness: 0.934310

Cotoneum (5-limit)

Subgroup: 2.3.5

Comma list: [80 -49 -1

Mapping[1 0 80], 0 1 -49]]

Optimal tuning (POTE): ~3/2 = 702.315

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

Badness: 1.240078

Newt (5-limit)

Subgroup: 2.3.5

Comma list: [95 -57 -2

Mapping[1 1 19], 0 2 -57]]

Optimal tuning (POTE): ~[47 -28 -1 = 351.114

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

Badness: 1.528465