Catalog of rank-4 temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (3/)
Switch to Sintel's badness, WE & CWE tunings (4/)
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.576{{c}}, ~5/4 = 386.314{{c}}, ~7/4 = 968.126{{c}}
* [[WE]]: ~2 = 1199.3774{{c}}, ~3/2 = 703.4693{{c}}, ~5/4 = 387.5573{{c}}, ~7/4 = 969.3690{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.743{{c}}, ~5/4 = 387.245{{c}}, ~7/4 = 969.048{{c}}
: [[error map]]: {{val| -0.623 +0.892 -0.002 -0.702 +1.061 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7426{{c}}, ~5/4 = 387.2445{{c}}, ~7/4 = 969.0476{{c}}
: error map: {{val| 0.000 +1.788 +0.931 +0.222 +2.759 }}


{{Optimal ET sequence|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}
{{Optimal ET sequence|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}


[[Badness]] (Smith): 0.0658 × 10<sup>-6</sup>
[[Badness]] (Sintel): 1.20


=== Tridecimal pentacircle a.k.a. gentle ===
=== Tridecimal pentacircle (gentle) ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


Line 547: Line 549:
{{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 -4 -7 | 0 0 1 0 0 0 | 0 0 0 1 1 1 }}
{{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 -4 -7 | 0 0 1 0 0 0 | 0 0 0 1 1 1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 703.786{{c}}, ~5/4 = 386.314{{c}}, ~7/4 = 967.665{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3706{{c}}, ~3/2 = 703.4872{{c}}, ~5/4 = 387.5708{{c}}, ~7/4 = 969.3987{{c}}
: [[error map]]: {{val| -0.629 +0.903 -0.002 -0.686 +0.985 +0.055 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.8328{{c}}, ~5/4 = 387.2881{{c}}, ~7/4 = 969.1612{{c}}
: error map: {{val| 0.000 +1.878 +0.974 +0.335 +2.512 +1.804 }}


{{Optimal ET sequence|legend=1| 12f, 17c, 22, 29, 34d, 41, 46, 58, 80, 87, 121, 167, 179ef, 208, 266ef, 433bceef, 641bbceeeff, 699bbcdeeeff }}
{{Optimal ET sequence|legend=1| 12f, 17c, 22, 29, 34d, 41, 46, 58, 80, 87, 121, 167, 179ef, 208, 266ef, 433bceef, 641bbceeeff, 699bbcdeeeff }}


[[Badness]] (Smith): 3.375 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.885


== Topsy (847/845, 1001/1000) ==
== Topsy (847/845, 1001/1000) ==
Line 558: Line 564:
[[Comma list]]: 847/845, 1001/1000
[[Comma list]]: 847/845, 1001/1000


{{Mapping|legend=1| 1 0 0 2 0 1 | 0 1 0 0 0 0 | 0 0 1 1 1 1 | 0 0 0 4 -3 1 }}
{{Mapping|legend=1| 1 0 0 -2 3 2 | 0 1 0 0 0 0 | 0 0 1 1 1 1 | 0 0 0 4 -3 -1 }}
: Mapping generators: ~2, ~3, ~5, ~13/10
: Mapping generators: ~2, ~3, ~5, ~20/13
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0343{{c}}, ~3/2 = 701.9205{{c}}, ~5/4 = 386.6098{{c}}, ~20/13 = 745.4658{{c}}
: [[error map]]: {{val| +0.034 -0.000 +0.365 -0.353 -0.934 +0.753 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9261{{c}}, ~5/4 = 386.6458{{c}}, ~20/13 = 745.4431{{c}}
: error map: {{val| 0.000 -0.029 +0.332 -0.408 -1.001 +0.675 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.955{{c}}, ~5/4 = 386.699{{c}}, ~13/10 = 454.558{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.955{{c}}, ~5/4 = 386.699{{c}}, ~13/10 = 454.558{{c}}


{{Optimal ET sequence|legend=1| 16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e }}
{{Optimal ET sequence|legend=1| 16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e }}
[[Badness]] (Sintel): 1.04


== Lehmerismic (3025/3024) ==
== Lehmerismic (3025/3024) ==
Line 574: Line 588:


[[Optimal tuning]]:  
[[Optimal tuning]]:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.998{{c}}, ~5/4 = 386.252{{c}}, ~55/36 = 733.436{{c}}
* [[WE]]: ~2 = 1200.0197{{c}}, ~3/2 = 701.9725{{c}}, ~5/4 = 386.2210{{c}}, ~55/36 = 733.4363{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.978{{c}}, ~5/4 = 386.232{{c}}, ~55/36 = 733.430{{c}}
: [[error map]]: {{val| +0.020 +0.037 -0.053 +0.039 -0.118 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9782{{c}}, ~5/4 = 386.2319{{c}}, ~55/36 = 733.4299{{c}}
: error map: {{val| 0.000 +0.023 -0.082 +0.012 -0.164 }}


{{Optimal ET sequence|legend=1| 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
{{Optimal ET sequence|legend=1| 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
 
[[Badness]] (Sintel): 0.0802


=== Commas 2080/2079, 3025/3024 ===
=== Commas 2080/2079, 3025/3024 ===
Line 587: Line 605:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.076{{c}}, ~5/4 = 386.208{{c}}, ~55/36 = 733.438{{c}}
* [[WE]]: ~2 = 1199.9717{{c}}, ~3/2 = 702.0958{{c}}, ~5/4 = 386.2618{{c}}, ~55/36 = 733.4380{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.099{{c}}, ~5/4 = 386.248{{c}}, ~55/36 = 733.449{{c}}
: [[error map]]: {{val| -0.028 +0.113 -0.108 +0.118 -0.007 -0.134 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0990{{c}}, ~5/4 = 386.2477{{c}}, ~55/36 = 733.4490{{c}}
: error map: {{val| 0.000 +0.144 -0.066 +0.171 +0.081 -0.082 }}


{{Optimal ET sequence|legend=1| 15, 26, 41, 46, 72, 87, 111, 183, 224, 270, 311, 494 }}
{{Optimal ET sequence|legend=1| 15, 24d, 26, 31f, 39df, 41, 46, 65d, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}
 
[[Badness]] (Sintel): 0.272


== Trimitone (8019/8000) ==
== Trimitone (8019/8000) ==
Line 601: Line 623:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.5449{{c}}, ~5/4 = 386.7538{{c}}, ~7/4 = 968.8259{{c}}
* [[WE]]: ~2 = 1200.1317{{c}}, ~3/2 = 701.4920{{c}}, ~5/4 = 386.4054{{c}}, ~7/4 = 968.5622{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4729{{c}}, ~5/4 = 386.5374{{c}}, ~7/4 = 968.6210{{c}}
: [[error map]]: {{val| +0.132 -0.331 +0.355 -0.000 -0.263 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4730{{c}}, ~5/4 = 386.5375{{c}}, ~7/4 = 968.6211{{c}}
: error map: {{val| 0.000 -0.482 +0.224 -0.205 -0.543 }}


{{Optimal ET sequence|legend=1| 12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be }}
{{Optimal ET sequence|legend=1| 12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be }}


[[Badness]] (Smith): 0.0820 × 10<sup>-6</sup>
[[Badness]] (Sintel): 1.50


=== Commas 729/728, 1001/1000 ===
=== Commas 729/728, 1001/1000 ===
Line 617: Line 641:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.5537{{c}}, ~5/4 = 386.7680{{c}}, ~7/4 = 968.8144{{c}}
* [[WE]]: ~2 = 1200.1318{{c}}, ~3/2 = 701.4926{{c}}, ~5/4 = 386.4063{{c}}, ~7/4 = 968.5613{{c}}
: [[error map]]: {{val| +0.132 -0.331 +0.356 -0.001 -0.264 -0.002 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4770{{c}}, ~5/4 = 386.5437{{c}}, ~7/4 = 968.6150{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4770{{c}}, ~5/4 = 386.5437{{c}}, ~7/4 = 968.6150{{c}}
: error map: {{val| 0.000 -0.478 +0.230 -0.211 -0.549 -0.280 }}


{{Optimal ET sequence|legend=1| 53, 58, 72, 111, 130, 183, 190, 243e, 248, 301, 373, 804, 1177be }}
{{Optimal ET sequence|legend=1| 53, 58, 72, 111, 130, 183, 190, 243e, 248, 301, 373, 804, 1177be }}


[[Badness]] (Smith): 3.33 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.875


== Kalismic (9801/9800) ==
== Kalismic (9801/9800) ==
Line 633: Line 659:


[[Optimal tuning]]:  
[[Optimal tuning]]:  
* [[CTE]]: ~99/70 = 600.000{{c}}, ~3/2 = 701.942{{c}}, ~5/4 = 386.327{{c}}, ~7/4 = 968.846{{c}}
* [[WE]]: ~99/70 = 600.0018{{c}}, ~3/2 = 701.9397{{c}}, ~5/4 = 386.3193{{c}}, ~7/4 = 968.8374{{c}}
* [[CWE]]: ~99/70 = 600.000{{c}}, ~3/2 = 701.940{{c}}, ~5/4 = 386.323{{c}}, ~7/4 = 968.840{{c}}
: [[error map]]: {{val| +0.004 -0.012 +0.013 +0.019 -0.028 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9399{{c}}, ~5/4 = 386.3226{{c}}, ~7/4 = 968.8399{{c}}
: error map: {{val| 0.000 -0.015 +0.009 +0.014 -0.035 }}
 
{{Optimal ET sequence|legend=1| 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e }}


{{Optimal ET sequence|legend=1| 8d, 10, 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e }}
[[Badness]] (Sintel): 0.0415


=== Commas 1716/1715, 2080/2079 ===
=== Commas 1716/1715, 2080/2079 ===
Line 648: Line 678:
: 3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299
: 3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299


[[Optimal tuning]] ([[CTE]]): ~99/70 = 600.000{{c}}, ~3/2 = 702.024{{c}}, ~5/4 = 386.294{{c}}, ~7/4 = 969.114{{c}}
[[Optimal tuning]]:
* [[WE]]: ~99/70 = 599.9732{{c}}, ~3/2 = 702.0465{{c}}, ~5/4 = 386.4214{{c}}, ~7/4 = 969.1757{{c}}
: [[error map]]: {{val| -0.054 +0.038 +0.001 +0.243 -0.001 -0.210 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0554{{c}}, ~5/4 = 386.3788{{c}}, ~7/4 = 969.1756{{c}}
: error map: {{val| 0.000 +0.100 +0.065 +0.350 +0.126 -0.121 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~99/70 = {{monzo| 1/2 0 0 0 0 0 }}, ~3 = {{monzo| 7/10 4/5 0 -2/5 0 1/5 }}, ~5 = {{monzo| 7/10 -1/5 1 -2/5 0 1/5 }}, ~7 = {{monzo| 7/5 -2/5 0 1/5 0 2/5 }}
* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~99/70 = {{monzo| 1/2 0 0 0 0 0 }}, ~3 = {{monzo| 7/10 4/5 0 -2/5 0 1/5 }}, ~5 = {{monzo| 7/10 -1/5 1 -2/5 0 1/5 }}, ~7 = {{monzo| 7/5 -2/5 0 1/5 0 2/5 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-intervals) basis]]: 2, 6/5, 16/13, 9/7
: [[eigenmonzo basis|eigenmonzo (unchanged-intervals) basis]]: 2.5/3.9/7.13
 
{{Optimal ET sequence|legend=1| 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d }}


{{Optimal ET sequence|legend=1| 10e, 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d }}
[[Badness]] (Sintel): 0.256


== Unisquary (12005/11979) ==
== Unisquary (12005/11979) ==

Revision as of 13:25, 6 March 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.

A rank-4 temperament has a period and three additional independent generators, for a total of four dimensions. Typical examples include 7-limit JI, full 11-limit temperament with a one-dimensional comma basis, and full 13-limit temperament with a two-dimensional comma basis.

Cake (45/44)

Subgroup: 2.3.5.7.11

Comma list: 45/44

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

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1202.4372 ¢, ~3/2 = 693.2306 ¢, ~5/4 = 374.6677 ¢, ~7/4 = 963.8387 ¢
error map: ⟨+2.437 -6.287 -6.772 -0.113 +14.685]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.1966 ¢, ~5/4 = 376.0821 ¢, ~7/4 = 965.0600 ¢
error map: ⟨0.000 -8.758 -10.232 -3.766 +11.157]

Optimal ET sequence: 7d, 9, 10, 12, 19, 26, 45

Badness (Sintel): 0.400

Mothwellsmic (99/98)

Subgroup: 2.3.5.7.11

Comma list: 99/98

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

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1200.3175 ¢, ~3/2 = 700.0109 ¢, ~5/4 = 385.6677 ¢, ~7/4 = 973.2571 ¢
error map: ⟨+0.318 -1.627 -0.011 +5.066 -3.873]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.0599 ¢, ~5/4 = 385.9307 ¢, ~7/4 = 973.4880 ¢
error map: ⟨0.000 -1.895 -0.383 +4.662 -4.462]

Optimal ET sequence: 5, 8d, 9, 12, 17c, 19e, 22, 31, 53, 84e, 96, 127

Badness (Sintel): 0.545

Ptolemismic (100/99)

Subgroup: 2.3.5.7.11

Comma list: 100/99

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

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1199.2633 ¢, ~3/2 = 704.5205 ¢, ~5/4 = 383.8317 ¢, ~7/4 = 970.2843 ¢
error map: ⟨-0.737 +1.829 -3.955 -0.015 +4.358]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.4751 ¢, ~5/4 = 383.0756 ¢, ~7/4 = 969.8904 ¢
error map: ⟨0.000 +2.520 -3.238 +1.065 +5.883]

Optimal ET sequence: 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e *

* optimal patent val: 104

Badness (Sintel): 0.411

Biyatismic (121/120)

Subgroup: 2.3.5.7.11

Comma list: 121/120

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

Mapping generators: ~2, ~3, ~11/10, ~7

Optimal tunings:

  • WE: ~2 = 1200.6628 ¢, ~3/2 = 701.8452 ¢, ~11/10 = 157.8337 ¢, ~7/4 = 967.4929 ¢
error map: ⟨+0.663 +0.553 +1.190 -0.007 -5.318]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9848 ¢, ~11/10 = 157.6099 ¢, ~7/4 = 967.8187 ¢
error map: ⟨0.000 +0.030 +0.451 -1.007 -6.943]

Optimal ET sequence: 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee

Badness (Sintel): 0.629

Valinorsmic (176/175)

Subgroup: 2.3.5.7.11

Comma list: 176/175

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

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1199.3113 ¢, ~3/2 = 702.6414 ¢, ~5/4 = 389.5404 ¢, ~7/4 = 971.5534 ¢
error map: ⟨-0.689 -0.022 +1.849 +1.350 -2.061]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5869 ¢, ~5/4 = 389.3935 ¢, ~7/4 = 971.5196 ¢
error map: ⟨0.000 +0.632 +3.080 +2.694 -1.011]

Optimal ET sequence: 22, 27e, 31, 46, 53, 58, 80, 111, 237cd, 268cd

Badness (Sintel): 0.339

Tridecimal valinorsmic

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350

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

Optimal tunings:

  • WE: ~2 = 1199.3397 ¢, ~3/2 = 702.5360 ¢, ~5/4 = 389.5451 ¢, ~7/4 = 971.5415 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6943 ¢, ~5/4 = 389.3823 ¢, ~7/4 = 971.5305 ¢

Optimal ET sequence: 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 268cd

Badness (Sintel): 0.603

Rastmic (243/242)

Rastmic halves 3/2 into two 11/9's, 32/27 into two 12/11's, and 25/24 into two 45/44~55/54's.

Subgroup: 2.3.5.7.11

Comma list: 243/242

Mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]

Mapping generators: ~2, ~11/9, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1200.0635 ¢, ~11/9 = 350.5439 ¢, ~5/4 = 386.1857 ¢, ~7/4 = 968.6977 ¢
error map: ⟨+0.064 -0.804 -0.001 -0.001 +1.529]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.5486 ¢, ~5/4 = 386.2387 ¢, ~7/4 = 968.7352 ¢
error map: ⟨0.000 -0.858 -0.075 -0.091 +1.425]

Optimal ET sequence: 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202

Badness (Sintel): 0.928

Frostmic (245/242)

Subgroup: 2.3.5.7.11

Comma list: 245/242

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

Mapping generators: ~2, ~3, ~11/7, ~7

Optimal tunings:

  • WE: ~2 = 1200.2442 ¢, ~3/2 = 701.7038 ¢, ~11/7 = 792.3594 ¢, ~7/4 = 964.4060 ¢
error map: ⟨+0.244 -0.007 -1.351 -3.931 +5.936]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7533 ¢, ~11/7 = 792.3351 ¢, ~7/4 = 964.5274 ¢
error map: ⟨0.000 -0.202 -1.644 -4.298 +5.545]

Optimal ET sequence: 9, 12, 15, 23de, 24, 26, 27e, 38d, 41, 91, 106d

Badness (Sintel): 1.64

Akua (352/351, 847/845)

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845

Mapping: [⟨1 0 0 10 0 5], ⟨0 1 0 -6 0 -3], ⟨0 0 1 1 0 0], ⟨0 0 0 0 1 1]]

Mapping generators: ~2, ~3, ~5, ~11

Optimal tunings:

  • WE: ~2 = 1199.6918 ¢, ~3/2 = 702.7270 ¢, ~5/4 = 386.9729 ¢, ~11/8 = 551.3122 ¢
error map: ⟨-0.308 +0.464 +0.043 -0.064 -0.930 +1.062]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8960 ¢, ~5/4 = 386.7835 ¢, ~11/8 = 551.0679 ¢
error map: ⟨0.000 +0.941 +0.470 +0.581 -0.025 +1.852]

Optimal ET sequence: 12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f

Badness (Sintel): 0.669

Keenanismic (385/384)

Main article: Keenanismic family

Subgroup: 2.3.5.7.11

Comma list: 385/384

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

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1200.4104 ¢, ~3/2 = 701.6916 ¢, ~5/4 = 385.1761 ¢, ~7/4 = 967.5421 ¢
error map: ⟨+0.410 +0.147 -0.317 -0.463 -0.703]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6933 ¢, ~5/4 = 385.1491 ¢, ~7/4 = 967.3004 ¢
error map: ⟨0.000 -0.262 -1.165 -1.525 -2.074]

Minimax tuning:

  • 11-odd-limit: ~2 = [1 0 0 0 0⟩, ~3 = [0 1 0 0 0⟩, ~5 = [7/3 1/3 2/3 -1/3 -1/3⟩, ~7 = [7/3 1/3 -1/3 2/3 -1/3⟩
Unchanged-interval (eigenmonzo) basis: 2.3.7/5.11/5

Optimal ET sequence: 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de

Badness (Sintel): 0.277

Martwin

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384

Mapping: [⟨1 0 0 0 7 2], ⟨0 1 0 0 1 4], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 -1 0]]

Lattice basis:

4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247

Optimal tunings:

  • WE: ~2 = 1200.3494 ¢, ~3/2 = 702.2131 ¢, ~5/4 = 384.8666 ¢, ~7/4 = 967.8487 ¢
error map: ⟨+0.349 +0.608 -0.748 -0.278 -0.422 -0.710]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1659 ¢, ~5/4 = 384.8721 ¢, ~7/4 = 967.6109 ¢
error map: ⟨0.000 +0.211 -1.442 -1.215 -1.635 -1.608]

Minimax tuning:

  • 13- and 15-odd-limit: ~2 = ⟨1 0 0 0 0 0], ~3 = ⟨0 1 0 0 0 0], ~5 = ⟨2/3 4/3 1/3 0 0 -1/3], ~7 = ⟨19/6 -1/6 -1/6 1/2 -1/2 1/6]
unchanged-interval (eigenmonzo) basis: 2.3.11/7.13/5

Optimal ET sequence: 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff

Badness (Sintel): 0.579

Ancient

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624

Mapping: [⟨1 0 0 0 7 -4], ⟨0 1 0 0 1 -1], ⟨0 0 1 0 -1 4], ⟨0 0 0 1 -1 0]]

Optimal tunings:

  • WE: ~2 = 1200.4130 ¢, ~3/2 = 701.6863 ¢, ~5/4 = 385.2319 ¢, ~7/4 = 967.5188 ¢
error map: ⟨+0.413 +0.144 -0.256 -0.481 -0.730 -0.047]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6838 ¢, ~5/4 = 385.2497 ¢, ~7/4 = 967.2556 ¢
error map: ⟨0.000 -0.271 -1.064 -1.570 -2.139 -1.213]

Optimal ET sequence: 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def

Badness (Sintel): 0.675

Commas 351/350, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384

Mapping: [⟨1 0 0 0 7 1], ⟨0 1 0 0 1 -3], ⟨0 0 1 0 -1 2], ⟨0 0 0 1 -1 1]]

Optimal tunings:

  • WE: ~2 = 1200.6793 ¢, ~3/2 = 700.7455 ¢, ~5/4 = 385.5609 ¢, ~7/4 = 967.5816 ¢
error map: ⟨+0.679 -0.530 +0.606 +0.114 -0.998 -1.343]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.5141 ¢, ~5/4 = 385.6072 ¢, ~7/4 = 967.1521 ¢
error map: ⟨0.000 -1.441 -0.706 -1.674 -3.563 -3.703]

Optimal ET sequence: 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff

Badness (Sintel): 0.781

Zaxa

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384

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

Optimal tunings:

  • WE: ~2 = 1200.1117 ¢, ~3/2 = 702.6588 ¢, ~5/4 = 385.6821 ¢, ~7/4 = 968.0061 ¢
error map: ⟨+0.112 +0.815 -0.408 -0.596 -1.901 +1.137]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.6029 ¢, ~5/4 = 385.6441 ¢, ~7/4 = 967.9030 ¢
error map: ⟨0.000 +0.648 -0.670 -0.923 -2.262 +0.720]

Optimal ET sequence: 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e

Badness (Sintel): 0.880

Commas 364/363, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 385/384

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 1 3], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 -1 -3]]

Optimal tunings:

  • WE: ~2 = 1200.4593 ¢, ~3/2 = 701.9714 ¢, ~5/4 = 384.8903 ¢, ~7/4 = 966.4090 ¢
error map: ⟨+0.459 +0.476 -0.505 -1.498 +1.191 -1.325]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.9809 ¢, ~5/4 = 384.8523 ¢, ~7/4 = 966.1074 ¢
error map: ⟨0.000 +0.026 -1.461 -2.719 -0.297 -2.612]

Optimal ET sequence: 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159

Badness (Sintel): 0.871

Commas 385/384, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 847/845

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

Mapping generators: ~2, ~3, ~5, ~13/11

Optimal tunings:

  • WE: ~2 = 1200.3530 ¢, ~3/2 = 701.7283 ¢, ~5/4 = 385.4633 ¢, ~13/11 = 291.0352 ¢
error map: ⟨+0.353 +0.126 -0.144 -0.586 -1.175 +0.651]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7205 ¢, ~5/4 = 385.3656 ¢, ~13/11 = 290.9769 ¢
error map: ⟨0.000 -0.235 -0.948 -1.507 -2.282 -0.515]

Optimal ET sequence: 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee

Badness (Sintel): 1.09

Werckismic (441/440)

Subgroup: 2.3.5.7.11

Comma list: 441/440

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

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1200.1734 ¢, ~3/2 = 701.4907 ¢, ~5/4 = 386.2784 ¢, ~7/4 = 967.5667 ¢
error map: ⟨+0.173 -0.291 +0.311 -0.912 +0.692]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4963 ¢, ~5/4 = 386.4317 ¢, ~7/4 = 967.5883 ¢
error map: ⟨0.000 -0.459 +0.118 -1.238 +0.420]

Minimax tuning:

  • 11-odd-limit: ~2 = [1 0 0 0 0⟩, ~3 = [1/2 2/3 1/6 -1/3 1/6⟩, ~5 = [0 0 1 0 0⟩, ~7 = [1 -2/3 1/3 1/3 1/3⟩
unchanged-interval (eigenmonzo) basis: 2.5.9/7.11

Optimal ET sequence: 10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d

Badness (Sintel): 0.189

Commas 364/363, 441/440

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440

Mapping: [⟨1 0 0 0 -3 -8], ⟨0 1 0 0 2 5], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 2 3]]

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

Lattice basis:

3/2 length = 1.2263, 14/11 length = 1.4629, 21/16 length = 1.4657

Optimal tunings:

  • WE: ~2 = 1200.0145 ¢, ~3/2 = 701.8745 ¢, ~5/4 = 386.4853 ¢, ~7/4 = 967.8244 ¢
error map: ⟨+0.014 -0.066 +0.201 -0.972 +1.609 -0.667]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8716 ¢, ~5/4 = 386.4975 ¢, ~7/4 = 967.8241 ¢
error map: ⟨0.000 -0.083 +0.184 -1.002 +1.576 -0.692]

Minimax tuning:

  • 13-odd-limit: ~2 = [1 0 0 0 0 0⟩, ~3 = [5/3 0 1/3 -1/3 -1/3 1/3⟩, ~5 = [1/6 0 5/6 2/3 -5/6 1/3⟩, ~7 = [0 0 0 1 0 0⟩
eigenmonzo (unchanged-interval) basis: 2.7.11/5.13
  • 15-odd-limit: ~2 = [1 0 0 0 0 0⟩, ~3 = [5/4 1/4 1/4 -1/4 -1/4 1/4⟩, ~5 = [5/4 -3/4 5/4 -1/4 -1/4 1/4⟩, ~7 = [17/8 -11/8 5/8 -1/8 3/8 1/8⟩
eigenmonzo (unchanged-interval) basis: 2.5/3.11/9.13/7

Optimal ET sequence: 12f, 14cf, 15, 17c, 26, 29, 31f, 41, 46, 58, 72, 87, 130, 217, 289

Badness (Sintel): 0.452

Commas 351/350, 441/440

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440

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

Optimal tunings:

  • WE: ~2 = 1200.2006 ¢, ~3/2 = 701.0354 ¢, ~5/4 = 386.8579 ¢, ~7/4 = 968.2226 ¢
error map: ⟨+0.201 -0.719 +0.945 -0.202 +0.541 -0.893]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.0371 ¢, ~5/4 = 387.0417 ¢, ~7/4 = 968.2546 ¢
error map: ⟨0.000 -0.918 +0.728 -0.571 +0.224 -1.301]

Optimal ET sequence: 12f, 14cf, 19e, 26, 27e, 31, 45ef, 46, 58, 72, 103, 130, 233, 279, 409, 512bf, 642bf

Badness (Sintel): 0.524

Commas 196/195, 352/351

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351

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

Optimal tunings:

  • WE: ~2 = 1199.6830 ¢, ~3/2 = 702.6689 ¢, ~5/4 = 387.7765 ¢, ~7/4 = 967.0260 ¢
error map: ⟨-0.317 +0.397 +0.829 -2.434 -0.022 +2.128]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.7536 ¢, ~5/4 = 387.5834 ¢, ~7/4 = 966.9373 ¢
error map: ⟨0.000 +0.799 +1.270 -1.889 +0.480 +3.010]

Optimal ET sequence: 10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232

Badness (Sintel): 0.548

Tannic

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1287/1280

Mapping: [⟨1 0 0 0 -3 11], ⟨0 1 0 0 2 -4], ⟨0 0 1 0 -1 2], ⟨0 0 0 1 2 -2]]

Optimal tunings:

  • WE: ~2 = 1200.7398 ¢, ~3/2 = 700.6733 ¢, ~5/4 = 385.4125 ¢, ~7/4 = 966.9648 ¢
error map: ⟨+0.740 -0.542 +0.578 -0.381 -0.714 -1.147]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.3274 ¢, ~5/4 = 385.8974 ¢, ~7/4 = 966.8107 ¢
error map: ⟨0.000 -1.628 -0.416 -2.015 -2.939 -3.664]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 700.555 ¢, ~5/4 = 387.816 ¢, ~7/4 = 967.984 ¢

Optimal ET sequence: 17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff

Badness (Sintel): 1.23

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 441/440, 561/560

Mapping: [⟨1 0 0 0 -3 11 7], ⟨0 1 0 0 2 -4 -3], ⟨0 0 1 0 -1 2 2], ⟨0 0 0 1 2 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1200.7320 ¢, ~3/2 = 700.7374 ¢, ~5/4 = 385.2303 ¢, ~7/4 = 966.8369 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.3909 ¢, ~5/4 = 385.7215 ¢, ~7/4 = 966.6922 ¢

Optimal ET sequence: 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef

Badness (Sintel): 0.701

Commas 441/440, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 847/845

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

Mapping generators: ~2, ~3, ~5, ~13/11

Optimal tunings:

  • WE: ~2 = 1200.1778 ¢, ~3/2 = 701.4785 ¢, ~5/4 = 385.4755 ¢, ~13/11 = 290.3844 ¢
error map: ⟨+0.178 -0.299 +0.517 -1.226 -0.170 +1.005]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4843 ¢, ~5/4 = 386.6332 ¢, ~13/11 = 290.3162 ¢
error map: ⟨0.000 -0.471 +0.319 -1.560 -0.451 +0.655]

Optimal ET sequence: 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d

Badness (Sintel): 1.11

Swetismic (540/539)

Subgroup: 2.3.5.7.11

Comma list: 540/539

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

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1199.9149 ¢, ~3/2 = 701.7196 ¢, ~5/4 = 386.2545 ¢, ~7/4 = 969.6659 ¢
error map: ⟨-0.085 -0.321 -0.229 +0.670 +0.508]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.6950 ¢, ~5/4 = 386.1795 ¢, ~7/4 = 969.6366 ¢
error map: ⟨0.000 -0.260 -0.134 +0.811 +0.673]

Optimal ET sequence: 8d, 9, 10, 12e, 14c, 17c, 19, 22, 27e, 31, 41, 53, 58, 72, 130, 152, 224, 354, 506e, 578, 730de, 761d, 985d, 1115de, 1267dde

Badness (Sintel): 0.191

Commas 540/539, 729/728

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728

Mapping: [⟨1 0 0 0 2 -3], ⟨0 1 0 0 3 6], ⟨0 0 1 0 1 0], ⟨0 0 0 1 -2 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1199.9170 ¢, ~3/2 = 701.7128 ¢, ~5/4 = 386.2521 ¢, ~7/4 = 969.6598 ¢
error map: ⟨-0.083 -0.325 -0.228 +0.668 +0.504 +0.006]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7231 ¢, ~5/4 = 386.1819 ¢, ~7/4 = 969.6608 ¢
error map: ⟨0.000 -0.232 -0.132 +0.835 +0.712 +0.150]

Optimal ET sequence: 12e, 14cf, 17c, 19, 22f, 31f, 39df, 41, 53, 58, 72, 111, 130, 183, 224, 354, 578, 985d, 1267ddef

Badness (Sintel): 0.454

Commas 540/539, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 847/845

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

Mapping generators: ~2, ~3, ~5, ~13/11

Optimal tunings:

  • WE: ~2 = 1199.8339 ¢, ~3/2 = 701.4961 ¢, ~5/4 = 386.5624 ¢, ~13/11 = 291.6840 ¢
error map: ⟨-0.166 -0.625 -0.084 +0.772 -0.627 +1.848]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4439 ¢, ~5/4 = 386.4201 ¢, ~13/11 = 291.7287 ¢
error map: ⟨0.000 -0.511 +0.106 +1.052 -0.321 +2.198]

Optimal ET sequence: 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f

Badness (Sintel): 1.04

Commas 540/539, 625/624

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624

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

Optimal tunings:

  • WE: ~2 = 1200.0035 ¢, ~3/2 = 701.6973 ¢, ~5/4 = 385.6264 ¢, ~7/4 = 969.4616 ¢
error map: ⟨+0.004 -0.254 -0.680 +0.643 +0.488 +0.291]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6983 ¢, ~5/4 = 385.6278 ¢, ~7/4 = 969.4622 ¢
error map: ⟨0.000 -0.257 -0.686 +0.636 +0.480 +0.285]

Optimal ET sequence: 19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224

Badness (Sintel): 0.943

Commas 540/539, 676/675

Subgroup: 2.3.5.7.11

Comma list: 540/539, 676/675

Mapping: [⟨1 0 0 0 2 -1], ⟨0 2 0 0 6 3], ⟨0 0 1 0 1 1], ⟨0 0 0 1 -2 0]]

Mapping generators: ~2, ~26/15, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1199.7990 ¢, ~26/15 = 950.9130 ¢, ~5/4 = 386.8478 ¢, ~7/4 = 970.1529 ¢
error map: ⟨-0.201 -0.129 +0.132 +0.925 +0.702 -1.142]
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.9953 ¢, ~5/4 = 386.7094 ¢, ~7/4 = 970.1192 ¢
error map: ⟨0.000 +0.036 +0.396 +1.293 +1.125 -0.832]

Optimal ET sequence: 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de

Badness (Sintel): 0.802

Pentacircle (896/891)

Subgroup: 2.3.5.7.11

Comma list: 896/891

Mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 -4], ⟨0 0 1 0 0], ⟨0 0 0 1 1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1199.3774 ¢, ~3/2 = 703.4693 ¢, ~5/4 = 387.5573 ¢, ~7/4 = 969.3690 ¢
error map: ⟨-0.623 +0.892 -0.002 -0.702 +1.061]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7426 ¢, ~5/4 = 387.2445 ¢, ~7/4 = 969.0476 ¢
error map: ⟨0.000 +1.788 +0.931 +0.222 +2.759]

Optimal ET sequence: 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee

Badness (Sintel): 1.20

Tridecimal pentacircle (gentle)

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 -4 -7], ⟨0 0 1 0 0 0], ⟨0 0 0 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3706 ¢, ~3/2 = 703.4872 ¢, ~5/4 = 387.5708 ¢, ~7/4 = 969.3987 ¢
error map: ⟨-0.629 +0.903 -0.002 -0.686 +0.985 +0.055]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.8328 ¢, ~5/4 = 387.2881 ¢, ~7/4 = 969.1612 ¢
error map: ⟨0.000 +1.878 +0.974 +0.335 +2.512 +1.804]

Optimal ET sequence: 12f, 17c, 22, 29, 34d, 41, 46, 58, 80, 87, 121, 167, 179ef, 208, 266ef, 433bceef, 641bbceeeff, 699bbcdeeeff

Badness (Sintel): 0.885

Topsy (847/845, 1001/1000)

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000

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

Mapping generators: ~2, ~3, ~5, ~20/13

Optimal tunings:

  • WE: ~2 = 1200.0343 ¢, ~3/2 = 701.9205 ¢, ~5/4 = 386.6098 ¢, ~20/13 = 745.4658 ¢
error map: ⟨+0.034 -0.000 +0.365 -0.353 -0.934 +0.753]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9261 ¢, ~5/4 = 386.6458 ¢, ~20/13 = 745.4431 ¢
error map: ⟨0.000 -0.029 +0.332 -0.408 -1.001 +0.675]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 701.955 ¢, ~5/4 = 386.699 ¢, ~13/10 = 454.558 ¢

Optimal ET sequence: 16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e

Badness (Sintel): 1.04

Lehmerismic (3025/3024)

Subgroup: 2.3.5.7.11

Comma list: 3025/3024

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

Mapping generators: ~2, ~3, ~5, ~55/36

Optimal tuning:

  • WE: ~2 = 1200.0197 ¢, ~3/2 = 701.9725 ¢, ~5/4 = 386.2210 ¢, ~55/36 = 733.4363 ¢
error map: ⟨+0.020 +0.037 -0.053 +0.039 -0.118]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9782 ¢, ~5/4 = 386.2319 ¢, ~55/36 = 733.4299 ¢
error map: ⟨0.000 +0.023 -0.082 +0.012 -0.164]

Optimal ET sequence: 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee

Badness (Sintel): 0.0802

Commas 2080/2079, 3025/3024

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3025/3024

Mapping: [⟨1 0 0 0 2 -3], ⟨0 1 0 1 2 6], ⟨0 0 1 0 -1 -2], ⟨0 0 0 2 1 3]]

Optimal tunings:

  • WE: ~2 = 1199.9717 ¢, ~3/2 = 702.0958 ¢, ~5/4 = 386.2618 ¢, ~55/36 = 733.4380 ¢
error map: ⟨-0.028 +0.113 -0.108 +0.118 -0.007 -0.134]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0990 ¢, ~5/4 = 386.2477 ¢, ~55/36 = 733.4490 ¢
error map: ⟨0.000 +0.144 -0.066 +0.171 +0.081 -0.082]

Optimal ET sequence: 15, 24d, 26, 31f, 39df, 41, 46, 65d, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d

Badness (Sintel): 0.272

Trimitone (8019/8000)

Subgroup: 2.3.5.7.11

Comma list: 8019/8000

Mapping: [⟨1 0 0 0 6], ⟨0 1 0 0 -6], ⟨0 0 1 0 3], ⟨0 0 0 1 0]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1200.1317 ¢, ~3/2 = 701.4920 ¢, ~5/4 = 386.4054 ¢, ~7/4 = 968.5622 ¢
error map: ⟨+0.132 -0.331 +0.355 -0.000 -0.263]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4730 ¢, ~5/4 = 386.5375 ¢, ~7/4 = 968.6211 ¢
error map: ⟨0.000 -0.482 +0.224 -0.205 -0.543]

Optimal ET sequence: 12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be

Badness (Sintel): 1.50

Commas 729/728, 1001/1000

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1001/1000

Mapping: [⟨1 0 0 0 6 -3], ⟨0 1 0 0 -6 6], ⟨0 0 1 0 3 0], ⟨0 0 0 1 0 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1200.1318 ¢, ~3/2 = 701.4926 ¢, ~5/4 = 386.4063 ¢, ~7/4 = 968.5613 ¢
error map: ⟨+0.132 -0.331 +0.356 -0.001 -0.264 -0.002]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4770 ¢, ~5/4 = 386.5437 ¢, ~7/4 = 968.6150 ¢
error map: ⟨0.000 -0.478 +0.230 -0.211 -0.549 -0.280]

Optimal ET sequence: 53, 58, 72, 111, 130, 183, 190, 243e, 248, 301, 373, 804, 1177be

Badness (Sintel): 0.875

Kalismic (9801/9800)

Subgroup: 2.3.5.7.11

Comma list: 9801/9800

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

Mapping generators: ~99/70, ~3, ~5, ~7

Optimal tuning:

  • WE: ~99/70 = 600.0018 ¢, ~3/2 = 701.9397 ¢, ~5/4 = 386.3193 ¢, ~7/4 = 968.8374 ¢
error map: ⟨+0.004 -0.012 +0.013 +0.019 -0.028]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.9399 ¢, ~5/4 = 386.3226 ¢, ~7/4 = 968.8399 ¢
error map: ⟨0.000 -0.015 +0.009 +0.014 -0.035]

Optimal ET sequence: 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e

Badness (Sintel): 0.0415

Commas 1716/1715, 2080/2079

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079

Mapping: [⟨2 0 0 0 3 -7], ⟨0 1 0 0 -2 1], ⟨0 0 1 0 1 0], ⟨0 0 0 1 1 2]]

Lattice basis:

3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299

Optimal tuning:

  • WE: ~99/70 = 599.9732 ¢, ~3/2 = 702.0465 ¢, ~5/4 = 386.4214 ¢, ~7/4 = 969.1757 ¢
error map: ⟨-0.054 +0.038 +0.001 +0.243 -0.001 -0.210]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.0554 ¢, ~5/4 = 386.3788 ¢, ~7/4 = 969.1756 ¢
error map: ⟨0.000 +0.100 +0.065 +0.350 +0.126 -0.121]

Minimax tuning:

  • 13- and 15-odd-limit: ~99/70 = [1/2 0 0 0 0 0⟩, ~3 = [7/10 4/5 0 -2/5 0 1/5⟩, ~5 = [7/10 -1/5 1 -2/5 0 1/5⟩, ~7 = [7/5 -2/5 0 1/5 0 2/5⟩
eigenmonzo (unchanged-intervals) basis: 2.5/3.9/7.13

Optimal ET sequence: 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d

Badness (Sintel): 0.256

Unisquary (12005/11979)

Subgroup: 2.3.5.7.11

Comma list: 12005/11979

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

Mapping generators: ~2, ~3, ~5, ~11/7

Optimal tunings:

  • CTE and CWE: ~2 = 1200.000 ¢, ~3/2 = 702.031 ¢, ~5/4 = 386.233 ¢, ~11/7 = 783.508 ¢

Optimal ET sequence: 12, 21, 23de, 26, 37, 46, 58, 72, 118, 130, 190, 239, 311, 441, 559, 752e, 870

Hensquary

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 1716/1715

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

Mapping generators: ~2, ~3, ~5, ~11/7

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.072 ¢, ~5/4 = 386.528 ¢, ~11/7 = 783.823 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.253 ¢, ~5/4 = 386.776 ¢, ~11/7 = 783.875 ¢

Optimal ET sequence: 26, 37, 46, 58, 72, 121, 130, 193, 239, 251e, 323e, 369

Ekasquary

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 4459/4455

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

Mapping generators: ~2, ~3, ~5, ~11/7

Optimal tunings:

  • CTE and CWE: ~2 = 1200.000 ¢, ~3/2 = 701.952 ¢, ~5/4 = 386.139 ¢, ~11/7 = 783.548 ¢

Optimal ET sequence: 46f, 49f, 58, 72, 118, 121, 130, 190, 193, 248, 311, 441, 752e, 1072

Semicanousmic (14641/14580)

Subgroup: 2.3.5.7.11

Comma list: 14641/14580

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

Mapping generators: ~2, ~3, ~18/11, ~7

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.334 ¢, ~18/11 = 854.555 ¢, ~7/4 = 968.826 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.292 ¢, ~18/11 = 854.548 ¢, ~7/4 = 968.756 ¢

Optimal ET sequence: 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e

Badness (Smith): 0.351 × 10-6

Tridecimal semicanousmic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 14641/14580

Mapping: [⟨1 0 2 0 1 -6], ⟨0 1 2 0 2 3], ⟨0 0 -4 0 -1 3], ⟨0 0 0 1 0 1]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.469 ¢, ~18/11 = 854.644 ¢, ~7/4 = 968.959 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.483 ¢, ~18/11 = 854.642 ¢, ~7/4 = 968.989 ¢

Optimal ET sequence: 87, 111, 181, 198, 323, 410

Badness (Smith): 17.1 × 10-6

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 1089/1088, 14641/14580

Mapping: [⟨1 0 2 0 1 -6 -4], ⟨0 1 2 0 2 3 6], ⟨0 0 -4 0 -1 3 -2], ⟨0 0 0 1 0 1 0]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.413 ¢, ~18/11 = 854.633 ¢, ~7/4 = 969.032 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.411 ¢, ~18/11 = 854.634 ¢, ~7/4 = 969.026 ¢

Optimal ET sequence: 87, 94, 111, 181, 198g, 212g, 292, 323, 410

Badness (Smith): 34.0 × 10-6

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 1089/1088, 1216/1215, 1445/1444

Mapping: [⟨1 0 2 0 1 -6 -4 -4], ⟨0 1 2 0 2 3 6 7], ⟨0 0 -4 0 -1 3 -2 -4], ⟨0 0 0 1 0 1 0 0]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.368 ¢, ~18/11 = 854.639 ¢, ~7/4 = 969.075 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.351 ¢, ~18/11 = 854.644 ¢, ~7/4 = 969.010 ¢

Optimal ET sequence: 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef

Badness (Smith): 41.9 × 10-6

Semiporwellismic (16384/16335)

Subgroup: 2.3.5.7.11

Comma list: 16384/16335

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

Mapping generators: ~2, ~3, ~128/99, ~7

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.470 ¢, ~128/99 = 442.106 ¢, ~7/4 = 968.826 ¢

Optimal ET sequence: 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee

Badness (Smith): 0.219 × 10-6

Symbiotic (19712/19683)

Subgroup: 2.3.5.7.11

Comma list: 19712/19683

Mapping: [⟨1 0 0 0 -8], ⟨0 1 0 0 9], ⟨0 0 1 0 0], ⟨0 0 0 1 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.213 ¢, ~5/4 = 386.314 ¢, ~7/4 = 968.736 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.244 ¢, ~5/4 = 386.406 ¢, ~7/4 = 968.859 ¢

Optimal ET sequence: 17c, 19e, 24, 34d, 41, 53, 58, 94, 99e, 118, 152, 270, 581, 733, 851, 1003, 1273, 1854, 2124b

Badness (Smith): 0.120 × 10-6

Tridecimal symbiotic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 19712/19683

Mapping: [⟨1 0 0 0 -8 -13], ⟨0 1 0 0 9 12], ⟨0 0 1 0 0 -1], ⟨0 0 0 1 -1 0]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.228 ¢, ~5/4 = 386.284 ¢, ~7/4 = 968.789 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.254 ¢, ~5/4 = 386.401 ¢, ~7/4 = 968.898 ¢

Optimal ET sequence: 17c, 34dff, 36ce, 41, 53, 58, 94, 111, 152f, 212, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bf

Badness (Smith): 3.31 × 10-6

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 2080/2079

Mapping: [⟨1 0 0 0 -8 -13 -6], ⟨0 1 0 0 9 12 5], ⟨0 0 1 0 0 -1 1], ⟨0 0 0 1 -1 0 0]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.230 ¢, ~5/4 = 386.304 ¢, ~7/4 = 968.797 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.255 ¢, ~5/4 = 386.408 ¢, ~7/4 = 968.902 ¢

Optimal ET sequence: 17ch, 24f, 34dffh, 36ceh, 41, 53, 58h, 94, 111, 152f, 212h, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bfh

Badness (Smith): 4.37 × 10-6

Olympic (131072/130977)

Subgroup: 2.3.5.7.11

Comma list: 131072/130977

Mapping: [⟨1 0 0 0 17], ⟨0 1 0 0 -5], ⟨0 0 1 0 0], ⟨0 0 0 1 -2]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.103 ¢, ~5/4 = 386.314 ¢, ~7/4 = 969.012 ¢

Optimal ET sequence: 41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee

Tridecimal olympic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095

Mapping: [⟨1 0 0 0 17 12], ⟨0 1 0 0 -5 -2], ⟨0 0 1 0 0 -1], ⟨0 0 0 1 -2 -1]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.103 ¢, ~5/4 = 386.298 ¢, ~7/4 = 969.007 ¢

Optimal ET sequence: 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e *

* optimal patent val: 3044

Seascape (160083/160000)

Subgroup: 2.3.5.7.11

Comma list: 160083/160000

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

Mapping generators: ~2, ~400/231, ~5, ~7

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~400/231 = 950.960 ¢, ~5/4 = 386.417 ¢, ~7/4 = 968.751 ¢

Optimal ET sequence: 15, 19, 29, 43, 53, 58, 68, 72

Fifthchromismic (1610510/1594323)

Subgroup: 2.3.5.7.11

Comma list: 1610510/1594323

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

Mapping generators: ~2, ~3, ~22/9, ~7

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.738 ¢, ~11/9 = 344.406 ¢, ~7/4 = 968.826 ¢

Optimal ET sequence: 77, 80, 87, 94, 167

Fifthchroma

See also: Fifth-chroma temperaments

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 951665/944784

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

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.776 ¢, ~11/9 = 344.410 ¢, ~7/4 = 968.826 ¢

Optimal ET sequence: 77, 80, 87, 94, 167

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 715/714, 155771/155520

Mapping: [⟨1 0 4 0 -1 4 6], ⟨0 1 3 0 2 -1 3], ⟨0 0 -5 0 1 1 -3], ⟨0 0 0 1 0 0 -1]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.713 ¢, ~11/9 = 344.403 ¢, ~7/4 = 969.193 ¢

Optimal ET sequence: 77, 80, 87, 94, 167

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 352/351, 715/714, 1463/1458, 2057/2052

Mapping: [⟨1 0 4 0 -1 4 6 2], ⟨0 1 3 0 2 -1 3 4], ⟨0 0 -5 0 1 1 -3 -1], ⟨0 0 0 1 0 0 -1 -1]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.736 ¢, ~11/9 = 344.417 ¢, ~7/4 = 969.159 ¢

Optimal ET sequence: 77, 80, 87, 94, 167h

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