Schismic

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Schismic
Subgroups 2.3.5, 2.3.5.19
Reduced mapping ⟨1; 1 -8 -3]
ET join 12 & 29
Generators (CWE) ~3/2 = 701.7¢
MOS scales 5L 2s, 5L 7s, 12L 5s
Ploidacot monocot
Comma basis 32805/32768 (2.3.5);
361/360, 513/512 (2.3.5.19)
Minimax error 5-odd-limit: 0.217¢
Target scale size 5-odd-limit: 12 notes

Schismic or Schismatic[a], [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the schisma, which is about 2 cents; this means that Schismic can be tuned to perfect Pythagorean tuning (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.

The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third (and consequently 19/15 with the diatonic major third), tempering out 513/512, which is sometimes called Boethius' comma.

The primary MOS to use for Schismic is Schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting porcupine.

48edo and 58edo are the first two edos to observe the schisma - that is, to not support Schismic - while tuning the fifth within the aforementioned Schismic tuning range.

Extensions

Prime 7

The primary 7-limit extension of Schismic is Garibaldi (also 12 & 29), which equates 64/63 with the syntonic comma. This is not considered canonical due to a significant loss in accuracy (tuned best with a fifth slightly sharp of just) - that is, Garibaldi is not a microtemperament - but it is still more accurate than Meantone as well as distinguishing 5-limit, 7-limit, and Pythagorean intervals in any given interval category. Garibaldi is an intuitive way of organizing just intonation as it tempers together the defining ~20-30c commas of the 7-limit.

Primes 11 and 13

A reasonable extension to the 11-limit assuming Garibaldi is Cassandra (41 & 53), which sets 33/32 to twice the Garibaldi comma; alternatively there is Andromeda, which instead sets 33/32 to the difference between that and the chroma and is best tuned sharp of 41edo. In either case, 11/9 is set to the opposite neutral third to 16/13 to extend to the 13-limit.

Prime 17

Schismic generally does not have 17; two options are setting 17/16 equal to 16/15 and 15/14 and tempering together 17/16 and 18/17 (which results in a weak extension including 17/12 as the semioctave).

Intervals

These interpretations assume Cassandra (in Pythagorean tuning).

Up from the unison Down from the octave
# Cents JI # Cents JI
0 0.00 1/1 0 1,200.00 2/1
1 701.96 3/2 1 498.04 4/3
2 203.91 9/8 2 996.09 16/9
3 905.87 32/19, 27/16 3 294.13 19/16
4 407.82 19/15 4 792.18 19/12
5 1,109.78 19/10 5 90.22 19/18, 20/19, 21/20
6 611.73 10/7 6 588.27 7/5
7 113.69 16/15, 15/14 7 1,086.31 15/8
8 815.64 8/5 8 384.36 5/4
9 317.60 5/3 9 882.40 6/5
10 1,019.55 9/5 10 180.45 10/9
11 521.51 27/20 11 678.49 40/27
12 23.46 50/49 12 1,176.54 49/25
13 725.42 32/21 13 474.58 21/16
14 227.37 8/7 14 972.63 7/4
15 929.33 12/7 15 270.67 7/6
16 431.28 9/7 16 768.72 14/9
17 1,133.24 28/27 17 66.76 27/14
18 635.19 13/9 18 564.81 18/13
19 137.15 13/12 19 1,062.85 24/13
20 839.10 13/8 20 360.90 16/13
21 341.06 11/9 21 858.94 18/11
22 1,043.01 11/6 22 156.99 12/11
23 544.97 11/8 23 655.03 16/11
24 46.92 33/32 24 1,153.08 64/33
25 748.88 54/35 25 451.12 35/27
26 250.83 81/70 26 949.17 140/81
27 952.79 26/15 27 247.21 15/13
28 454.74 13/10 28 745.26 20/13
29 1,156.70 39/20 29 43.30 40/39

[a] Despite ending in -isma, schisma is a 3rd-declension comma name, therefore its temperaments are Schismatic and Schismic (which, as it is a 2.3.5 comma, refer to the same temperament).

Patent vals

The following patent vals up to 272edo support Schismic.

EDO Generator tuning Extension info
12 700.0000 Garibaldi
101 700.9901
89 701.1236
77 701.2987
219 701.3699
142 701.4085
207 701.4493
272 701.4706
65 701.5385
248 701.6129
183 701.6393
118 701.6949
171 701.7544
224 701.7857 (no Schismic 19)
53 701.8868 Garibaldi
253 701.9763 (no Schismic 19)
200 702.0000 (no Schismic 19)
147 702.0408 Garibaldi
94 702.1277 Garibaldi
135 702.2222 Garibaldi
41 702.4390 Garibaldi
29 703.4483 Garibaldi
17 705.8824


The full list of patent vals supporting schismic is below:

EDO Generator tuning Extension info
12 700.0000 Garibaldi
101 700.9901
89 701.1236
77 701.2987
219 701.3699
142 701.4085
207 701.4493
272 701.4706
65 701.5385
508 701.5748 (no Schismic 19)
443 701.5801 (no Schismic 19)
378 701.5873 (no Schismic 19)
313 701.5974 (no Schismic 19)
561 701.6043 (no Schismic 19)
248 701.6129
679 701.6200 (no Schismic 19)
431 701.6241 (no Schismic 19)
614 701.6287 (no Schismic 19)
797 701.6311 (no Schismic 19)
183 701.6393
1033 701.6457 (no Schismic 19)
850 701.6471 (no Schismic 19)
667 701.6492 (no Schismic 19)
1151 701.6507 (no Schismic 19)
484 701.6529 (no Schismic 19)
1269 701.6548 (no Schismic 19)
785 701.6561 (no Schismic 19)
1086 701.6575 (no Schismic 19)
1387 701.6583 (no Schismic 19)
301 701.6611 (no Schismic 19)
1322 701.6641 (no Schismic 19)
1021 701.6650 (no Schismic 19)
720 701.6667 (no Schismic 19)
1139 701.6681 (no Schismic 19)
1558 701.6688 (no Schismic 19)
419 701.6706 (no Schismic 19)
1794 701.6722 (no Schismic 19)
1375 701.6727 (no Schismic 19)
956 701.6736 (no Schismic 19)
1493 701.6745 (no Schismic 19)
2030 701.6749 (no Schismic 19)
537 701.6760 (no Schismic 19)
1729 701.6773 (no Schismic 19)
1192 701.6779 (no Schismic 19)
1847 701.6784 (no Schismic 19)
655 701.6794 (no Schismic 19)
2083 701.6803 (no Schismic 19)
1428 701.6807 (no Schismic 19)
773 701.6818 (no Schismic 19)
1664 701.6827 (no Schismic 19)
891 701.6835 (no Schismic 19)
1900 701.6842 (no Schismic 19)
1009 701.6848 (no Schismic 19)
2136 701.6854 (no Schismic 19)
1127 701.6859 (no Schismic 19)
1245 701.6867 (no Schismic 19)
1363 701.6875 (no Schismic 19)
1481 701.6880 (no Schismic 19)
1599 701.6886 (no Schismic 19)
1717 701.6890 (no Schismic 19)
1835 701.6894 (no Schismic 19)
1953 701.6897 (no Schismic 19)
2071 701.6900 (no Schismic 19)
2189 701.6903 (no Schismic 19)
118 701.6949
2295 701.6993 (no Schismic 19)
2177 701.6996 (no Schismic 19)
2059 701.6999 (no Schismic 19)
1941 701.7002 (no Schismic 19)
1823 701.7005 (no Schismic 19)
1705 701.7009 (no Schismic 19)
1587 701.7013 (no Schismic 19)
1469 701.7018 (no Schismic 19)
1351 701.7024 (no Schismic 19)
1233 701.7032 (no Schismic 19)
2348 701.7036 (no Schismic 19)
1115 701.7040 (no Schismic 19)
2112 701.7045 (no Schismic 19)
997 701.7051 (no Schismic 19)
1876 701.7058 (no Schismic 19)
879 701.7065 (no Schismic 19)
1640 701.7073 (no Schismic 19)
2401 701.7076 (no Schismic 19)
761 701.7083 (no Schismic 19)
2165 701.7090 (no Schismic 19)
1404 701.7094 (no Schismic 19)
2047 701.7098 (no Schismic 19)
643 701.7107 (no Schismic 19)
2454 701.7115 (no Schismic 19)
1811 701.7118 (no Schismic 19)
1168 701.7123 (no Schismic 19)
1693 701.7129 (no Schismic 19)
2218 701.7133 (no Schismic 19)
525 701.7143 (no Schismic 19)
1982 701.7154 (no Schismic 19)
1457 701.7159 (no Schismic 19)
2389 701.7162 (no Schismic 19)
932 701.7167 (no Schismic 19)
2271 701.7173 (no Schismic 19)
1339 701.7177 (no Schismic 19)
1746 701.7182 (no Schismic 19)
2153 701.7185 (no Schismic 19)
407 701.7199 (no Schismic 19)
2324 701.7212 (no Schismic 19)
1917 701.7214 (no Schismic 19)
1510 701.7219 (no Schismic 19)
1103 701.7226 (no Schismic 19)
1799 701.7232 (no Schismic 19)
2495 701.7234 (no Schismic 19)
696 701.7241 (no Schismic 19)
2377 701.7249 (no Schismic 19)
1681 701.7252 (no Schismic 19)
985 701.7259 (no Schismic 19)
2259 701.7264 (no Schismic 19)
1274 701.7268 (no Schismic 19)
1563 701.7274 (no Schismic 19)
1852 701.7279 (no Schismic 19)
2141 701.7282 (no Schismic 19)
2430 701.7284 (no Schismic 19)
289 701.7301 (no Schismic 19)
2483 701.7318 (no Schismic 19)
2194 701.7320 (no Schismic 19)
1905 701.7323 (no Schismic 19)
1616 701.7327 (no Schismic 19)
1327 701.7332 (no Schismic 19)
2365 701.7336 (no Schismic 19)
1038 701.7341 (no Schismic 19)
1787 701.7348 (no Schismic 19)
2536 701.7350 (no Schismic 19)
749 701.7356 (no Schismic 19)
2707 701.7362 (no Schismic 19)
1958 701.7365 (no Schismic 19)
1209 701.7370 (no Schismic 19)
1669 701.7376 (no Schismic 19)
2129 701.7379 (no Schismic 19)
2589 701.7381 (no Schismic 19)
460 701.7391 (no Schismic 19)
2471 701.7402 (no Schismic 19)
2011 701.7404 (no Schismic 19)
1551 701.7408 (no Schismic 19)
1091 701.7415 (no Schismic 19)
1722 701.7422 (no Schismic 19)
2353 701.7425 (no Schismic 19)
631 701.7433 (no Schismic 19)
2064 701.7442 (no Schismic 19)
1433 701.7446 (no Schismic 19)
802 701.7456 (no Schismic 19)
1775 701.7465 (no Schismic 19)
973 701.7472 (no Schismic 19)
1144 701.7483 (no Schismic 19)
1315 701.7490 (no Schismic 19)
1486 701.7497 (no Schismic 19)
1657 701.7502 (no Schismic 19)
1828 701.7505 (no Schismic 19)
171 701.7544
1421 701.7593 (no Schismic 19)
1250 701.7600 (no Schismic 19)
1079 701.7609 (no Schismic 19)
908 701.7621 (no Schismic 19)
737 701.7639 (no Schismic 19)
1303 701.7652 (no Schismic 19)
566 701.7668 (no Schismic 19)
961 701.7690 (no Schismic 19)
395 701.7722 (no Schismic 19)
1014 701.7751 (no Schismic 19)
619 701.7771 (no Schismic 19)
843 701.7794 (no Schismic 19)
1067 701.7807 (no Schismic 19)
224 701.7857 (no Schismic 19)
725 701.7931 (no Schismic 19)
501 701.7964 (no Schismic 19)
778 701.7995 (no Schismic 19)
277 701.8051 (no Schismic 19)
607 701.8122 (no Schismic 19)
330 701.8182 (no Schismic 19)
383 701.8277 (no Schismic 19)
436 701.8349 (no Schismic 19)
489 701.8405 (no Schismic 19)
542 701.8450 (no Schismic 19)
53 701.8868 Garibaldi
253 701.9763 (no Schismic 19)
200 702.0000 (no Schismic 19)
147 702.0408 Garibaldi
94 702.1277 Garibaldi
135 702.2222 Garibaldi
41 702.4390 Garibaldi
29 703.4483 Garibaldi
17 705.8824


View • Talk • EditRegular temperaments
Rank-2
Acot Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave)
Monocot Meantone • Schismic • Leapday • Archy
Complexity 2 Diaschismic (diploid monocot) • Pajara (diploid monocot) • Injera (diploid monocot) • Rastmatic (dicot) • Mohajira (dicot) • Intertridecimal (dicot) • Interseptimal (alpha-dicot)
Complexity 3 Augmented (triploid) • Misty (triploid) • Slendric (tricot) • Porcupine (omega-tricot)
Complexity 4 Diminished (tetraploid) • Tetracot (tetracot) • Buzzard (alpha-tetracot) • Squares (beta-tetracot) • Negri (omega-tetracot)
Complexity 5-6 Magic (alpha-pentacot) • Amity (gamma-pentacot) • Kleismic (alpha-hexacot) • Miracle (hexacot)
Higher complexity Orwell (alpha-heptacot) • Sensi (beta-heptacot) • Octacot (octacot) • Wurschmidt (beta-octacot) • Valentine (enneacot) • Ammonite (epsilon-enneacot) • Myna (beta-decacot) • Ennealimmal (enneaploid dicot)
Straddle-3 A-Team (alter-tricot) • Machine (alter-monocot)
No-3 Trismegistus (alpha-triseph) • Orgone (trimech) • Didacus (diseph)
No-octaves Sensamagic (monogem)
Exotemperament Dicot • Mavila • Father
Higher-rank
Rank-3 Hemifamity • Marvel • Parapyth
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