Schismic: Difference between revisions
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Added more information on the 19-limit extension, Nestoria. |
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'''Schismic''' or '''Schismatic'''<sup>[a]</sup>, [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. | '''Schismic''' or '''Schismatic'''<sup>[a]</sup>, [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 | 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, tempering out 513/512, which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma. Tempering out both 513/512 and 1216/1215 results in an extension called ''Nestoria''. | ||
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]]. | 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]]. | ||
Revision as of 00:13, 29 March 2026
| Schismic |
361/360, 513/512 (2.3.5.19)
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, tempering out 513/512, which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma. Tempering out both 513/512 and 1216/1215 results in an extension called Nestoria.
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 (preferring flatter schismic tunings) 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 |
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 |
Footnotes
[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).
| 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 |
