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The '''sensamagic family''' of [[rank-3 temperament]]s tempers out [[245/243]]. For a list of rank-2 temperaments, see [[Sensamagic clan]].  
{{Technical data page}}
The '''sensamagic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the sensamagic comma, [[245/243]].  
 
For a list of rank-2 temperaments, see [[Sensamagic clan]].  


== Sensamagic ==
== Sensamagic ==
{{Main| Sensamagic }}
{{Main| Sensamagic }}
Sensamagic is generated by a perfect fifth and a wide supermajor third of ~[[9/7]], two of which make ~[[5/3]]. Among the good edo tunings are [[87edo]] and [[128edo]], as well as the [[optimal patent val]] [[283edo]].
Another notable tuning is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 703.7424{{c}}, 440.9020{{c}} with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from [[CWE]] is practically unnoticeable.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 8: Line 15:
[[Comma list]]: 245/243
[[Comma list]]: 245/243


[[Mapping]]: [{{val| 1 0 0 0 }}, {{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]
{{Mapping|legend=1| 1 0 0 0 | 0 1 1 2 | 0 0 2 -1 }}
 
: mapping generators: ~2, ~3, ~9/7
Mapping generators: ~2, ~3, ~9/7


[[Mapping to lattice]]: [{{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]
[[Mapping to lattice]]: [{{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]
Line 18: Line 24:
: Angle (3/2, 9/7) = 86.5288°
: Angle (3/2, 9/7) = 86.5288°


[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.7424, ~9/7 = 440.9020
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9983{{c}}, ~3/2 = 703.7414{{c}}, ~9/7 = 440.9014{{c}}
: [[error map]]: {{val| -0.002 +1.785 -0.771 -2.248 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7411{{c}}, ~9/7 = 440.9017{{c}}
: error map: {{val| 0.000 +1.786 -0.769 -2.245 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 0 1/5 2/5 }}, {{monzo| 0 0 1 0 }}, {{monzo| 0 0 0 1 }}]
: {{monzo list| 1 0 0 0 | 0 0 1/5 2/5 | 0 0 1 0 | 0 0 0 1 }}
: [[Eigenmonzo]]s (unchanged intervals): 2, 8/7, 5/4
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
* [[9-odd-limit]]
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 0 5/3 2/3 -2/3 }}, {{monzo| 0 5/3 -1/3 1/3 }}]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 5/3 2/3 -2/3 | 0 5/3 -1/3 1/3 }}
: [[Eigenmonzo]]s (unchanged intervals): 2, 4/3, 7/5  
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5  


{{Val list|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }}
{{Optimal ET sequence|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }}


[[Badness]]: 0.129 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.570


[[Projection pair]]: 5 243/49 to 2.3.7
[[Projection pair]]: 5 243/49 to 2.3.7


{{databox|[[Minkowski blocks]]|
{{Databox|[[Minkowski block]]s|
{2, 3, 7} subgroup
2.3.7 subgroup
* 12: 729/686, 64/63
* 12: 729/686, 64/63
* 17: 64/63, 19683/19208
* 17: 64/63, 19683/19208
Line 44: Line 54:


=== Overview to extensions ===
=== Overview to extensions ===
Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.  
The second comma in the comma list defines which [[11-limit]] family member we are looking at. Undecimal sensamagic adds [[385/384]], sensawer adds [[441/440]], octarod adds [[100/99]], shrusus adds [[176/175]]. These temperaments use the same generators as sensamagic. Bisector adds [[121/120]] with a half-octave period.
 
Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]) and [[sensigh]] (→ [[Sengic family #Sensigh|Sengic family]]). The rest are considered below.  


== Undecimal sensamagic ==
== Undecimal sensamagic ==
{{Main| Sensamagic }}
{{Main| Sensamagic }}
Undecimal sensamagic tempers out not only [[385/384]], but [[896/891]], making itself a [[strong extension]] of [[parapyth]].


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 53: Line 67:
[[Comma list]]: 245/243, 385/384
[[Comma list]]: 245/243, 385/384


[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 1 2 -2 }}, {{val| 0 0 2 -1 -1 }}]
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 2 -2 | 0 0 2 -1 -1 }}


[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.8004, ~9/7 = 440.9178
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9667{{c}}, ~3/2 = 703.7809{{c}}, ~9/7 = 440.9056{{c}}
: [[error map]]: {{val| -0.033 +1.793 -0.755 -2.236 +0.048 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7948{{c}}, ~9/7 = 440.9180{{c}}
: error map: {{val| 0.000 +1.840 -0.683 -2.154 +0.175 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 21/13 6/13 -1/13 1/13 -3/13 }}, {{monzo| 35/13 10/13 7/13 -7/13 -5/13 }}, {{monzo| 35/13 10/13 -6/13 6/13 -5/13 }}, {{monzo| 42/13 -14/13 -2/13 2/13 7/13 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 21/13 6/13 -1/13 1/13 -3/13 }}, {{monzo| 35/13 10/13 7/13 -7/13 -5/13 }}, {{monzo| 35/13 10/13 -6/13 6/13 -5/13 }}, {{monzo| 42/13 -14/13 -2/13 2/13 7/13 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 11/9, 7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9


{{Val list|legend=1| 17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd }}
{{Optimal ET sequence|legend=1| 17, 19, 22, 41, 68, 87, 196, 283 }}


[[Badness]]: 0.722 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.868


[[Projection pair]]s: 5 243/49 11 896/81 to 2.3.7
[[Projection pair]]s: 5 243/49 11 896/81 to 2.3.7
Line 74: Line 91:
Comma list: 245/243, 352/351, 364/363
Comma list: 245/243, 352/351, 364/363


Mapping: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 1 2 -2 -5 }}, {{val| 0 0 2 -1 -1 -1 }}]
Mapping: {{mapping| 1 0 0 0 7 12 | 0 1 1 2 -2 -5 | 0 0 2 -1 -1 -1 }}


Optimal GPV sequence: {{Val list| 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd }}
Optimal tunings:  
* WE: ~2 = 1199.9905{{c}}, ~3/2 = 703.7325{{c}}, ~9/7 = 440.9149{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.7381{{c}}, ~9/7 = 440.9184{{c}}


Badness: 1.20 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 17, 19f, 22, 41, 46, 63, 87, 237, 283 }}
 
Badness (Sintel): 1.12


== Sensawer ==
== Sensawer ==
Line 85: Line 106:
[[Comma list]]: 245/243, 441/440
[[Comma list]]: 245/243, 441/440


[[Mapping]]: [{{val| 1 0 0 0 -3 }}, {{val| 0 1 1 2 5 }}, {{val| 0 0 2 -1 -4 }}]
{{Mapping|legend=1| 1 0 0 0 -3 | 0 1 1 2 5 | 0 0 2 -1 -4 }}


[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.1900, ~9/7 = 441.1359
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1654{{c}}, ~3/2 = 703.2870{{c}}, ~9/7 = 441.1967{{c}}
: [[error map]]: {{val| -0.033 +1.793 -0.755 -2.236 +0.048 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2917{{c}}, ~9/7 = 441.1849{{c}}
: error map: {{val| 0.000 +1.840 -0.683 -2.1554 +0.175 }}


{{Val list|legend=1| 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd }}
{{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 60e, 87 }}


[[Badness]]: 0.796 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.957


=== 13-limit ===
=== 13-limit ===
Line 98: Line 123:
Comma list: 196/195, 245/243, 352/351
Comma list: 196/195, 245/243, 352/351


Mapping: [{{val| 1 0 0 0 -3 2 }}, {{val| 0 1 1 2 5 2 }}, {{val| 0 0 2 -1 -4 -4 }}]
Mapping: {{mapping| 1 0 0 0 -3 2 | 0 1 1 2 5 2 | 0 0 2 -1 -4 -4 }}
 
Optimal tunings:
* WE: ~2 = 1199.9800{{c}}, ~3/2 = 703.4468{{c}}, ~9/7 = 441.3705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.4494{{c}}, ~9/7 = 441.3758{{c}}


Optimal GPV sequence: {{Val list| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }}
{{Optimal ET sequence|legend=0| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }}


Badness: 0.928 × 10<sup>-3</sup>
Badness (Sintel): 0.868


== Octarod ==
== Octarod ==
Octarod tempers out [[100/99]] and the interval class of [[11/1|11]] is found as a stack of four ~9/7's. The name ''octarod'' was the former name of the sensamagic comma before being reused for this 11-limit extension, and comes from [[octacot]] and [[rodan]]; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 100/99, 245/243
[[Comma list]]: 100/99, 245/243


[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 1 2 0 }}, {{val| 0 0 2 -1 4 }}]
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 2 0 | 0 0 2 -1 4 }}


[[Optimal tuning]] ([[POTE]]): ~3/2 = 705.0464, ~9/7 = 439.5050
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2854{{c}}, ~3/2 = 704.6266{{c}}, ~9/7 = 439.2433{{c}}
: [[error map]]: {{val| -0.715 +1.957 -3.915 -0.245 +4.226 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5246{{c}}, ~9/7 = 439.2798{{c}}
: error map: {{val| 0.000 +2.570 -3.230 +0.944 +5.801 }}


{{Val list|legend=1| 8d, 14c, 19, 22, 27e, 41, 104, 131e }}
{{Optimal ET sequence|legend=1| 14c, 19, 22, 27e, 41, 90e, 131e}}*


[[Badness]]: 0.581 × 10<sup>-3</sup>
<nowiki/>*[[Optimal patent val]]: [[104edo|104]]
 
[[Badness]] (Sintel): 0.698


Scales: [[octarod1]], [[octarod2]], [[octarod3]], [[octarod4]], [[octarod5]]
Scales: [[octarod1]], [[octarod2]], [[octarod3]], [[octarod4]], [[octarod5]]
Line 124: Line 161:
[[Comma list]]: 176/175, 245/243
[[Comma list]]: 176/175, 245/243


[[Mapping]]: [{{val| 1 0 0 0 -4 }}, {{val| 0 1 1 2 4 }}, {{val| 0 0 2 -1 3 }}]
{{Mapping|legend=1| 1 0 0 0 -4 | 0 1 1 2 4 | 0 0 2 -1 3 }}


[[Optimal tuning]] ([[POTE]]): ~3/2 = 706.3702, ~9/7 = 442.1147
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.9114{{c}}, ~3/2 = 705.7294{{c}}, ~9/7 = 441.7137{{c}}
: [[error map]]: {{val| -1.089 +2.686 +1.754 -1.258 -3.259 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.8402{{c}}, ~9/7 = 442.1064{{c}}
: error map: {{val| 0.000 +3.885 +3.739 +0.748 -1.638 }}


{{Val list|legend=1| 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc }}
{{Optimal ET sequence|legend=1| 19e, 22, 27e, 46, 68, 95, 141bc, 163bc }}


[[Badness]]: 0.877 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.05


=== Shrusic ===
=== Shrusic ===
Line 137: Line 178:
Comma list: 91/90, 176/175, 245/243
Comma list: 91/90, 176/175, 245/243


Mapping: [{{val| 1 0 0 0 -4 1 }}, {{val| 0 1 1 2 4 1 }}, {{val| 0 0 2 -1 3 3 }}]
Mapping: {{mapping| 1 0 0 0 -4 1 | 0 1 1 2 4 1 | 0 0 2 -1 3 3 }}
 
Optimal tunings:
* WE: ~2 = 1199.7256{{c}}, ~3/2 = 704.9071{{c}}, ~9/7 = 443.1303{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9572{{c}}, ~9/7 = 443.2018{{c}}


Optimal GPV sequence: {{Val list| 22, 46, 211bcf, 233bcf, 257bcf, 279bcf }}
{{Optimal ET sequence|legend=0| 19e, 22, 27e, 46 }}


Badness: 1.125 × 10<sup>-3</sup>
Badness (Sintel): 1.05


== Bisector ==
== Bisector ==
Line 148: Line 193:
[[Comma list]]: 121/120, 245/243
[[Comma list]]: 121/120, 245/243


[[Mapping]]: [{{val| 2 0 0 0 3 }}, {{val| 0 1 1 2 1 }}, {{val| 0 0 2 -1 1 }}]
{{Mapping|legend=1| 2 0 0 0 3 | 0 1 1 2 1 | 0 0 2 -1 1 }}
 
: mapping generators: ~77/54, ~3, ~9/7
Mapping generators: ~77/54, ~3, ~9/7
 
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.0884, ~9/7 = 441.1060
 
{{Val list|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}
 
[[Badness]]: 1.089 × 10<sup>-3</sup>
 
== Sensigh ==
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 169/168
 
Mapping: [{{val| 1 6 8 11 0 10 }}, {{val| 0 -7 -9 -13 0 -10 }}, {{val| 0 0 0 0 1 0 }}]
 
Mapping generators: ~2, ~9/7, ~11
 
Optimal GPV sequence: {{Val list| 19, 27, 46, 111df, 157df }}
 
Badness: 0.939 × 10<sup>-3</sup>
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Comma list: 91/90, 126/125, 154/153, 169/168
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 600.3096{{c}}, ~3/2 = 703.4512{{c}}, ~9/7 = 441.3336{{c}}
: [[error map]]: {{val| +0.619 +2.115 +0.424 -2.019 -4.985 }}
* [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 703.5671{{c}}, ~9/7 = 441.2436{{c}}
: error map: {{val| 0.000 +1.612 -0.259 -2.935 -6.507 }}


Mapping: [{{val| 1 6 8 11 0 10 0 }}, {{val| 0 -7 -9 -13 0 -10 1 }}, {{val| 0 0 0 0 1 0 1 }}]
{{Optimal ET sequence|legend=1| 8d, 14c, 22, 38d, 46, 60e, 68, 106de, 128e, 174e }}


Optimal GPV sequence: {{Val list| 19, 27, 46 }}
[[Badness]] (Sintel): 1.31


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Sensamagic family| ]] <!-- main article -->
[[Category:Sensamagic family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 10:14, 11 April 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 sensamagic family of rank-3 temperaments tempers out the sensamagic comma, 245/243.

For a list of rank-2 temperaments, see Sensamagic clan.

Sensamagic

Main article: Sensamagic

Sensamagic is generated by a perfect fifth and a wide supermajor third of ~9/7, two of which make ~5/3. Among the good edo tunings are 87edo and 128edo, as well as the optimal patent val 283edo.

Another notable tuning is given by TE, CTE and POTE, all coinciding at 703.7424 ¢, 440.9020 ¢ with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from CWE is practically unnoticeable.

Subgroup: 2.3.5.7

Comma list: 245/243

Mapping: [⟨1 0 0 0], ⟨0 1 1 2], ⟨0 0 2 -1]]

mapping generators: ~2, ~3, ~9/7

Mapping to lattice: [⟨0 1 1 2], ⟨0 0 2 -1]]

Lattice basis:

3/2 length = 0.9644, 9/7 length = 1.0807
Angle (3/2, 9/7) = 86.5288°

Optimal tunings:

  • WE: ~2 = 1199.9983 ¢, ~3/2 = 703.7414 ¢, ~9/7 = 440.9014 ¢
error map: ⟨-0.002 +1.785 -0.771 -2.248]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7411 ¢, ~9/7 = 440.9017 ¢
error map: ⟨0.000 +1.786 -0.769 -2.245]

Minimax tuning:

[[1 0 0 0⟩, [0 0 1/5 2/5⟩, [0 0 1 0⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.5.7
[[1 0 0 0⟩, [0 1 0 0⟩, [0 5/3 2/3 -2/3⟩, [0 5/3 -1/3 1/3⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7/5

Optimal ET sequence: 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283

Badness (Sintel): 0.570

Projection pair: 5 243/49 to 2.3.7

Minkowski blocks

2.3.7 subgroup

  • 12: 729/686, 64/63
  • 17: 64/63, 19683/19208
  • 19: 49/48, 177147/175616
  • 22: 64/63, 537824/531441
  • 24: 64/63, 15059072/14348907

Overview to extensions

The second comma in the comma list defines which 11-limit family member we are looking at. Undecimal sensamagic adds 385/384, sensawer adds 441/440, octarod adds 100/99, shrusus adds 176/175. These temperaments use the same generators as sensamagic. Bisector adds 121/120 with a half-octave period.

Temperaments discussed elsewhere include supernatural (→ Keemic family) and sensigh (→ Sengic family). The rest are considered below.

Undecimal sensamagic

Main article: Sensamagic

Undecimal sensamagic tempers out not only 385/384, but 896/891, making itself a strong extension of parapyth.

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384

Mapping: [⟨1 0 0 0 7], ⟨0 1 1 2 -2], ⟨0 0 2 -1 -1]]

Optimal tunings:

  • WE: ~2 = 1199.9667 ¢, ~3/2 = 703.7809 ¢, ~9/7 = 440.9056 ¢
error map: ⟨-0.033 +1.793 -0.755 -2.236 +0.048]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7948 ¢, ~9/7 = 440.9180 ¢
error map: ⟨0.000 +1.840 -0.683 -2.154 +0.175]

Minimax tuning:

[[1 0 0 0 0⟩, [21/13 6/13 -1/13 1/13 -3/13⟩, [35/13 10/13 7/13 -7/13 -5/13⟩, [35/13 10/13 -6/13 6/13 -5/13⟩, [42/13 -14/13 -2/13 2/13 7/13⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11/9

Optimal ET sequence: 17, 19, 22, 41, 68, 87, 196, 283

Badness (Sintel): 0.868

Projection pairs: 5 243/49 11 896/81 to 2.3.7

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 245/243, 352/351, 364/363

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 1 2 -2 -5], ⟨0 0 2 -1 -1 -1]]

Optimal tunings:

  • WE: ~2 = 1199.9905 ¢, ~3/2 = 703.7325 ¢, ~9/7 = 440.9149 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7381 ¢, ~9/7 = 440.9184 ¢

Optimal ET sequence: 17, 19f, 22, 41, 46, 63, 87, 237, 283

Badness (Sintel): 1.12

Sensawer

Subgroup: 2.3.5.7.11

Comma list: 245/243, 441/440

Mapping: [⟨1 0 0 0 -3], ⟨0 1 1 2 5], ⟨0 0 2 -1 -4]]

Optimal tunings:

  • WE: ~2 = 1200.1654 ¢, ~3/2 = 703.2870 ¢, ~9/7 = 441.1967 ¢
error map: ⟨-0.033 +1.793 -0.755 -2.236 +0.048]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.2917 ¢, ~9/7 = 441.1849 ¢
error map: ⟨0.000 +1.840 -0.683 -2.1554 +0.175]

Optimal ET sequence: 14c, 19e, 27e, 41, 60e, 87

Badness (Sintel): 0.957

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 352/351

Mapping: [⟨1 0 0 0 -3 2], ⟨0 1 1 2 5 2], ⟨0 0 2 -1 -4 -4]]

Optimal tunings:

  • WE: ~2 = 1199.9800 ¢, ~3/2 = 703.4468 ¢, ~9/7 = 441.3705 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.4494 ¢, ~9/7 = 441.3758 ¢

Optimal ET sequence: 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd

Badness (Sintel): 0.868

Octarod

Octarod tempers out 100/99 and the interval class of 11 is found as a stack of four ~9/7's. The name octarod was the former name of the sensamagic comma before being reused for this 11-limit extension, and comes from octacot and rodan; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243

Mapping: [⟨1 0 0 0 2], ⟨0 1 1 2 0], ⟨0 0 2 -1 4]]

Optimal tunings:

  • WE: ~2 = 1199.2854 ¢, ~3/2 = 704.6266 ¢, ~9/7 = 439.2433 ¢
error map: ⟨-0.715 +1.957 -3.915 -0.245 +4.226]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5246 ¢, ~9/7 = 439.2798 ¢
error map: ⟨0.000 +2.570 -3.230 +0.944 +5.801]

Optimal ET sequence: 14c, 19, 22, 27e, 41, 90e, 131e*

*Optimal patent val: 104

Badness (Sintel): 0.698

Scales: octarod1, octarod2, octarod3, octarod4, octarod5

Shrusus

Subgroup: 2.3.5.7.11

Comma list: 176/175, 245/243

Mapping: [⟨1 0 0 0 -4], ⟨0 1 1 2 4], ⟨0 0 2 -1 3]]

Optimal tunings:

  • WE: ~2 = 1198.9114 ¢, ~3/2 = 705.7294 ¢, ~9/7 = 441.7137 ¢
error map: ⟨-1.089 +2.686 +1.754 -1.258 -3.259]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.8402 ¢, ~9/7 = 442.1064 ¢
error map: ⟨0.000 +3.885 +3.739 +0.748 -1.638]

Optimal ET sequence: 19e, 22, 27e, 46, 68, 95, 141bc, 163bc

Badness (Sintel): 1.05

Shrusic

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 176/175, 245/243

Mapping: [⟨1 0 0 0 -4 1], ⟨0 1 1 2 4 1], ⟨0 0 2 -1 3 3]]

Optimal tunings:

  • WE: ~2 = 1199.7256 ¢, ~3/2 = 704.9071 ¢, ~9/7 = 443.1303 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.9572 ¢, ~9/7 = 443.2018 ¢

Optimal ET sequence: 19e, 22, 27e, 46

Badness (Sintel): 1.05

Bisector

Subgroup: 2.3.5.7.11

Comma list: 121/120, 245/243

Mapping: [⟨2 0 0 0 3], ⟨0 1 1 2 1], ⟨0 0 2 -1 1]]

mapping generators: ~77/54, ~3, ~9/7

Optimal tunings:

  • WE: ~2 = 600.3096 ¢, ~3/2 = 703.4512 ¢, ~9/7 = 441.3336 ¢
error map: ⟨+0.619 +2.115 +0.424 -2.019 -4.985]
  • CWE: ~2 = 600.0000 ¢, ~3/2 = 703.5671 ¢, ~9/7 = 441.2436 ¢
error map: ⟨0.000 +1.612 -0.259 -2.935 -6.507]

Optimal ET sequence: 8d, 14c, 22, 38d, 46, 60e, 68, 106de, 128e, 174e

Badness (Sintel): 1.31

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