Garischismic family
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The garischismic family of rank-3 temperaments tempers out the garischisma (ratio: 33554432/33480783, monzo: [25 -14 0 -1⟩).
Garischismic
The head of this family is garischismic, which is generated by a perfect fifth and an independent generator for 5/4. Two Pythagorean apotomes i.e. 14 fifths octave-reduced make a septimal major second (8/7). Equivalently stated, the harmonic seventh (7/4) is found at the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step which stands in for both the Pythagorean comma and the septimal comma.
Garischismic can be easily notated with chain-of-fifths notation with two additional set of accidentals, one for the generic comma step, and the other for the generic aberschisma step which stands in for the schisma and the aberschisma.
Subgroup: 2.3.5.7
Comma list: 33554432/33480783
Mapping: [⟨1 0 0 25], ⟨0 1 0 -14], ⟨0 0 1 0]]
- mapping generators: ~2, ~3, ~5
- WE: ~2 = 1199.9155 ¢, ~3/2 = 702.1584 ¢, ~5/4 = 386.4827 ¢
- error map: ⟨-0.085 +0.119 -0.000 +0.027]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2124 ¢, ~5/4 = 386.4496 ¢
- error map: ⟨0.000 +0.257 +0.136 +0.201]
Optimal ET sequence: 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd
Badness (Sintel): 5.79
Overview to extensions
The best extension to the 11-limit identifies the 11/8 at +23 fifths. This is also the mapping used in cassandra, so we call it cassaschismic. An alternative, supported by andromeda, is androschismic.
Cassaschismic
Cassaschismic is naturally a no-17 19-limit temperament, where the undevicesimal schisma of 513/512 is also added to the generic aberschisma step.
Subgroup: 2.3.5.7.11
Comma list: 19712/19683, 41503/41472
Mapping: [⟨1 0 0 25 -33], ⟨0 1 0 -14 23], ⟨0 0 1 0 0]]
- WE: ~2 = 1199.9631 ¢, ~3/2 = 702.2077 ¢, ~5/4 = 386.3874 ¢
- error map: ⟨-0.037 +0.216 -0.000 -0.139 -0.173]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2290 ¢, ~5/4 = 386.3819 ¢
- error map: ⟨0.000 +0.274 +0.068 -0.032 -0.051]
Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851, 1121
Badness (Sintel): 1.69
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 4096/4095, 19712/19683
Mapping: [⟨1 0 0 25 -33 -13], ⟨0 1 0 -14 23 12], ⟨0 0 1 0 0 -1]]
Optimal tunings:
- WE: ~2 = 1199.9785 ¢, ~3/2 = 702.2180 ¢, ~5/4 = 386.2991 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2303 ¢, ~5/4 = 386.3031 ¢
Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851, 2283b
Badness (Sintel): 0.815
2.3.5.7.11.13.19 subgroup
Subgroup: 2.3.5.7.11.13.19
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079
Subgroup-val mapping: [⟨1 0 0 25 -33 -13 -6], ⟨0 1 0 -14 23 12 5], ⟨0 0 1 0 0 -1 1]]
Optimal tunings:
- WE: ~2 = 1199.9817 ¢, ~3/2 = 702.2203 ¢, ~5/4 = 386.3225 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2307 ¢, ~5/4 = 386.3245 ¢
Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851
Badness (Sintel): 0.486
Androschismic
Subgroup: 2.3.5.7.11
Comma list: 151263/151250, 200704/200475
Mapping: [⟨1 0 0 25 62], ⟨0 1 0 -14 -34], ⟨0 0 1 0 -2]]
- WE: ~2 = 1199.9118 ¢, ~3/2 = 702.1606 ¢, ~5/4 = 386.5301 ¢
- error map: ⟨-0.088 +0.117 +0.040 -0.045 +0.044]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2178 ¢, ~5/4 = 386.5048 ¢
- error map: ⟨0.000 +0.263 +0.191 +0.125 +0.266]
Optimal ET sequence: 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee
Badness (Sintel): 1.97
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 43904/43875, 154880/154791
Mapping: [⟨1 0 0 25 62 82], ⟨0 1 0 -14 -34 -43], ⟨0 0 1 0 -2 -3]]
Optimal tunings:
- WE: ~2 = 1199.9121 ¢, ~3/2 = 702.1603 ¢, ~5/4 = 386.5212 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2174 ¢, ~5/4 = 386.4968 ¢
Optimal ET sequence: 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff
Badness (Sintel): 0.942