Garischismic family: Difference between revisions

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m Units & misc. cleanup
Switch to Sintel's badness, WE & CWE tunings
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2224{{c}}, ~5/4 = 386.3137{{c}}
* [[WE]]: ~2 = 1199.9155{{c}}, ~3/2 = 702.1584{{c}}, ~5/4 = 386.4827{{c}}
: [[error map]]: {{val| -0.085 +0.119 -0.000 +0.027 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2124{{c}}, ~5/4 = 386.4496{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2124{{c}}, ~5/4 = 386.4496{{c}}
: error map: {{val| 0.000 +0.257 +0.136 +0.201 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2224{{c}}, ~5/4 = 386.3137{{c}} -->


{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd }}
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd }}


[[Badness]] (Smith): 1.31 × 10<sup>-3</sup>
[[Badness]] (Sintel): 5.79


== Cassaschismic ==
== Cassaschismic ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2280{{c}}, ~5/4 = 386.3137{{c}}
* [[WE]]: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}}, ~5/4 = 386.3874{{c}}
: [[error map]]: {{val| -0.037 +0.216 -0.000 -0.139 -0.173 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}, ~5/4 = 386.3819{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}, ~5/4 = 386.3819{{c}}
: error map: {{val| 0.000 +0.274 +0.068 -0.032 -0.051 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2280{{c}}, ~5/4 = 386.3137{{c}} -->


{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }}
{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }}


[[Badness]] (Smith): 1.41 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Line 43: Line 49:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2289{{c}}, ~5/4 = 386.2869{{c}}
* WE: ~2 = 1199.9785{{c}}, ~3/2 = 702.2180{{c}}, ~5/4 = 386.2991{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2303{{c}}, ~5/4 = 386.3031{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2303{{c}}, ~5/4 = 386.3031{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2289{{c}}, ~5/4 = 386.2869{{c}} -->


{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}
{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}


Badness (Smith): 0.872 × 10<sup>-3</sup>
Badness (Sintel): 0.815


=== 2.3.5.7.11.13.19 subgroup ===
=== 2.3.5.7.11.13.19 subgroup ===
Line 57: Line 64:
Subgroup-val mapping: {{mapping| 1 0 0 25 -33 -13 -6 | 0 1 0 -14 23 12 5 | 0 0 1 0 0 -1 1 }}
Subgroup-val mapping: {{mapping| 1 0 0 25 -33 -13 -6 | 0 1 0 -14 23 12 5 | 0 0 1 0 0 -1 1 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.2293{{c}}, ~5/4 = 386.3021{{c}}
Optimal tunings:
* WE: ~2 = 1199.9817{{c}}, ~3/2 = 702.2203{{c}}, ~5/4 = 386.3225{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2307{{c}}, ~5/4 = 386.3245{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2293{{c}}, ~5/4 = 386.3021{{c}} -->


{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851 }}
{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851 }}


Badness (Smith): 0.496 × 10<sup>-3</sup>
Badness (Sintel): 0.486


== Androschismic ==
== Androschismic ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2308{{c}}, ~5/4 = 386.3806{{c}}
* [[WE]]: ~2 = 1199.9118{{c}}, ~3/2 = 702.1606{{c}}, ~5/4 = 386.5301{{c}}
: [[error map]]: {{val| -0.088 +0.117 +0.040 -0.045 +0.044 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2178{{c}}, ~5/4 = 386.5048{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2178{{c}}, ~5/4 = 386.5048{{c}}
: error map: {{val| 0.000 +0.263 +0.191 +0.125 +0.266 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2308{{c}}, ~5/4 = 386.3806{{c}} -->


{{Optimal ET sequence|legend=1| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }}
{{Optimal ET sequence|legend=1| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }}


[[Badness]] (Smith): 1.64 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.97


=== 13-limit ===
=== 13-limit ===
Line 86: Line 99:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2305{{c}}, ~5/4 = 386.3748{{c}}
* WE: ~2 = 1199.9121{{c}}, ~3/2 = 702.1603{{c}}, ~5/4 = 386.5212{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2174{{c}}, ~5/4 = 386.4968{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2174{{c}}, ~5/4 = 386.4968{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2305{{c}}, ~5/4 = 386.3748{{c}} -->


{{Optimal ET sequence|legend=0| 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff }}
{{Optimal ET sequence|legend=0| 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff }}


Badness (Smith): 1.01 × 10<sup>-3</sup>
Badness (Sintel): 0.942


[[Category:Temperament families]]
[[Category:Temperament families]]

Revision as of 11:56, 6 February 2026

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

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