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Septischismic family

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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 septischismic family of rank-3 temperaments tempers out the septischisma (monzo: [25 -14 0 -1⟩, ratio: 33 554 432 / 33 480 783).

Septischismic

Main article: Septischismic

The head of this family is septischismic (sometimes 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.

Septischismic can be easily notated with chain-of-fifths notation with two additional sets 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

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

Overview to extensions

The best extension to the 11-limit identifies the 11/8 at +23 fifths. This is also the mapping used in the related lower accuracy temperament cassandra, so we also call it cassaschismic. An alternative, supported by andromeda, is androschismic.

Undecimal septischismic (cassaschismic)

Main article: Septischismic

Undecimal septischismic, a.k.a. cassaschismic, maps prime 11 to +23 perfect fifths, so it is an expansion of the 2.3.7.11-subgroup version of gary. It 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]]

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 ¢
error map: ⟨-0.022 +0.241 -0.058 -0.114 -0.089 -0.146]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2303 ¢, ~5/4 = 386.3031 ¢
error map: ⟨0.000 +0.275 -0.011 -0.050 -0.021 -0.067]

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 ¢
error map: ⟨-0.018 +0.247 -0.028 -0.111 -0.069 -0.152 -0.107]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2307 ¢, ~5/4 = 386.3245 ¢
error map: ⟨0.000 +0.276 +0.011 -0.056 -0.011 -0.084 -0.035]

Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851

Badness (Sintel): 0.486

Septiplanar

Septischismic has two reasonable extensions to add 29 and 31 into the system, both of which coalesce in the structurally very similar temperament newt. Septiplanar finds them in the chain of fifths, without need of third generators, thus they in the same plane to the rank-2 chain of fifths, mapped respectively to a major seventh – three commas (3sM7) and a perfect octave minus two commas (hP8). Because this extension is reached only through the septischismic chain of fifths, it is also available in gary.

Subgroup: 2.3.5.7.11.13.19.29

Comma list: 784/783, 1216/1215, 1540/1539, 1729/1728, 2080/2079

Subgroup-val mapping: [⟨1 0 0 25 -33 -13 -6 23], ⟨0 1 0 -14 23 12 5 -31], ⟨0 0 1 0 0 -1 1 0]]

Optimal tunings:

  • WE: ~2 = 1199.948 ¢, ~3/2 = 702.213 ¢, ~5/4 = 386.379 ¢
error map: ⟨-0.052 +0.206 -0.039 -0.389 +0.113 -0.190 -0.119 +0.603]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.244 ¢, ~5/4 = 386.388 ¢
error map: ⟨0.000 +0.289 +0.075 -0.242 +0.293 +0.012 +0.095 +0.860]

Optimal ET sequence: TBA

Badness (Sintel): 0.758

2.3.5.7.11.13.19.29.31 subgroup

Subgroup: 2.3.5.7.11.13.19.29.31

Comma list: 784/783, 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2233/2232

Subgroup-val mapping: [⟨1 0 0 25 -33 -13 -6 23 19], ⟨0 1 0 -14 23 12 5 -31 -24], ⟨0 0 1 0 0 -1 1 0 0]]

Optimal tunings:

  • WE: ~2 = 1199.930 ¢, ~3/2 = 702.208 ¢, ~5/4 = 386.406 ¢
error map: ⟨-0.070 +0.183 -0.048 -0.513 +0.173 -0.225 -0.136 +0.353 +0.633]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.250 ¢, ~5/4 = 386.420 ¢
error map: ⟨0.000 +0.295 +0.106 -0.319 +0.421 +0.047 +0.154 +0.688 +0.976]

Optimal ET sequence: TBA

Badness (Sintel): 0.793

Septifantasious

Septifantasious tempers out 1625/1624 – the fantasia, and thus finds 29/16 as 3 aberschismas above 9/5 – that is to say, superminor seventh + 2 aberschismas (^^Sm7), and 31/16 one apotome higher as a hypermajor seventh + 2 aberschismas (^^HM7).

Subgroup: 2.3.5.7.11.13.19.29

Comma list: 1216/1215, 1540/1539, 1625/1624, 1729/1728, 2080/2079

Subgroup-val mapping: [⟨1 0 0 25 -33 -13 -6 -41], ⟨0 1 0 -14 23 12 5 26], ⟨0 0 1 0 0 -1 1 2]]

Optimal tunings:

  • WE: ~2 = 1199.983 ¢, ~3/2 = 702.213 ¢, ~5/4 = 386.172 ¢
error map: ⟨-0.017 +0.242 -0.175 +0.005 -0.246 -0.091 -0.291 +0.495]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.233 ¢, ~5/4 = 386.174 ¢
error map: ⟨0.000 +0.268 -0.140 +0.055 -0.193 -0.028 -0.225 +0.564]

Optimal ET sequence: TBA

Badness (Sintel): 0.716

2.3.5.7.11.13.19.29.31 subgroup

Subgroup: 2.3.5.7.11.13.19.29.31

Comma list: 1216/1215, 1540/1539, 1625/1624, 1729/1728, 2080/2079, 2233/2232

Subgroup-val mapping: [⟨1 0 0 25 -33 -13 -6 -41 -52], ⟨0 1 0 -14 23 12 5 26 33], ⟨0 0 1 0 0 -1 1 2 2]]

Optimal tunings:

  • WE: ~2 = 1199.985 ¢, ~3/2 = 702.210 ¢, ~5/4 = 386.120 ¢
error map: ⟨-0.015 +0.241 -0.223 +0.066 -0.331 -0.079 -0.356 +0.296 +0.370]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.219 ¢, ~5/4 = 386.122 ¢
error map: ⟨0.000 +0.264 -0.192 +0.110 -0.284 -0.023 -0.297 +0.358 +0.431]

Optimal ET sequence: TBA

Badness (Sintel): 0.796

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, 10648/10647, 43904/43875

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

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 2080/2079, 3136/3135, 10648/10647

Mapping: [⟨1 0 0 25 62 82 -6], ⟨0 1 0 -14 -34 -43 5], ⟨0 0 1 0 -2 -3 1]]

Optimal tunings:

  • WE: ~2 = 1199.9282 ¢, ~3/2 = 702.1718 ¢, ~5/4 = 386.5228 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2181 ¢, ~5/4 = 386.5012 ¢

Optimal ET sequence: 12f, 29, 41, …, 229, 241, 270, 552, 581, 851, 1133, 1403, 1984, 3117bcef, 3387bcef

Badness (Sintel): 0.580

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