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⟩). The head of this family is garischismic, which is generated by a perfect fifth and an independent generator for 5/4. Two 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–Cbb).

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.

Garischismic

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

Optimal tunings:

• 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

Cassaschismic

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]]

Optimal tunings:

• 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]]

Optimal tunings:

• 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