11th-octave temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings
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{{Mapping|legend=1| 11 0 43 | 0 1 -1 }}
{{Mapping|legend=1| 11 0 43 | 0 1 -1 }}
: Mapping generators: ~16/15, ~3
: Mapping generators: ~16/15, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~16/15 = 109.091¢, ~3/2 = 702.794¢
* [[WE]]: ~16/15 = 109.0904{{c}}, ~3/2 = 702.6861{{c}}
* [[CWE]]: ~16/15 = 109.091¢, ~3/2 = 702.908¢
: [[error map]]: {{val| -0.445 +0.286 +0.613 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 702.9082{{c}}
: error map: {{val| 0.000 +0.953 +1.687 }}


{{Optimal ET sequence|legend=1| 22, 55, 77, 99 }}
{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 869bcc, 968bcc, 1067bccc, 1166bccc }}


[[Badness]] (Sintel): 10.652
[[Badness]] (Sintel): 10.7


=== Undekapent ===
=== Undekapent ===
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{{Mapping|legend=1| 11 0 8 | 0 1 1 }}
{{Mapping|legend=1| 11 0 8 | 0 1 1 }}
: Mapping generators: ~648/625, ~3
: Mapping generators: ~648/625, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~648/625 = 109.091¢, ~3/2 = 705.652¢
* [[WE]]: ~648/625 = 109.1809{{c}}, ~3/2 = 704.4091{{c}}
* [[CWE]]: ~648/625 = 109.091¢, ~3/2 = 704.698¢
: [[error map]]: {{val| +0.990 +3.444 -7.468 }}
* [[CWE]]: ~648/625 = 109.0909{{c}}, ~3/2 = 704.6983{{c}}
: error map: {{val| 0.000 +2.743 -8.888 }}


{{Optimal ET sequence|legend=1| 22, 77c, 99c, 121c }}
{{Optimal ET sequence|legend=1| 22, 77c, 99c }}


[[Badness]] (Sintel): 17.067
[[Badness]] (Sintel): 17.1


=== Elven ===
=== Elven ===
{{see also| Sensibeta temperaments #Elven }}
{{See also| Sensibeta temperaments #Elven }}
 
This temperament has a period of 1/11 octave, which represents [[3125/2916]] in the 5-limit.
This temperament has a period of 1/11 octave, which represents [[3125/2916]] in the 5-limit.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo|-23 -66 55}}
[[Comma list]]: {{monzo| -23 -66 55 }}


{{Mapping|legend=1| 11 2 7 | 0 5 6 }}
{{Mapping|legend=1| 11 2 7 | 0 5 6 }}
 
: mapping generators: ~3125/2916, ~{{monzo| 10 29 -24 }}
: Mapping generators: ~3125/2916, ~{{monzo| 10 29 -24 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~3125/2916 = 109.091¢, ~5/3 = 882.353¢
* [[WE]]: ~3125/2916 = 109.0998{{c}}, ~{{monzo| 10 29 -24 }} = 336.8921{{c}}
* [[CWE]]: ~3125/2916 = 109.091¢, ~5/3 = 882.336¢
: [[error map]]: {{val| +0.097 +0.705 -1.263 }}
* [[CWE]]: ~3125/2916 = 109.0909{{c}}, ~{{monzo| 10 29 -24 }} = 336.8814{{c}}
: error map: {{val| 0.000 +0.634 -1.389 }}


{{Optimal ET sequence|legend=1| 121, 253, 374 }}
{{Optimal ET sequence|legend=1| 121, 253, 374, 627c }}


[[Badness]] (Sintel): 429.4
[[Badness]] (Sintel): 429


== Hendeca ==
== Hendeca ==
Line 72: Line 76:


{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


: Mapping generators: ~16/15, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
[[Optimal tuning]] ([[POTE]]): ~16/15 = 109.091, ~3/2 = 703.313
: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}


{{Optimal ET sequence|legend=1| 11, 22, 55d, 77d, 99dd }}
{{Optimal ET sequence|legend=1| 11, 22, 55d, 77d, 99dd }}


[[Badness]] (Sintel): 4.371
[[Badness]] (Sintel): 4.37


=== 11-limit ===
=== 11-limit ===
Line 88: Line 95:
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}


Optimal tuning (POTE): ~16/15 = 109.091, ~3/2 = 703.090
Optimal tunings:
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


{{Optimal ET sequence|legend=0| 11, 22, 55d, 77d, 99dd }}
{{Optimal ET sequence|legend=0| 11, 22, 55d }}


Badness (Sintel): 2.464
Badness (Sintel): 2.46


== Undeka ==
== Undeka ==
Line 100: Line 109:


{{Mapping|legend=1| 11 0 8 31 | 0 1 1 0 }}
{{Mapping|legend=1| 11 0 8 31 | 0 1 1 0 }}
: mapping generators: ~21/20, ~3


: Mapping generators: ~21/20, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~21/20 = 108.9318{{c}}, ~3/2 = 707.7579{{c}}
[[Optimal tuning]] ([[POTE]]): ~21/20 = 109.091, ~3/2 = 708.792
: [[error map]]: {{val| -1.750 +4.053 -8.852 +8.059 }}
* [[CWE]]: ~21/20 = 109.0909{{c}}, ~3/2 = 707.7526{{c}}
: error map: {{val| 0.000 +5.798 -5.834 +12.992 }}


{{Optimal ET sequence|legend=1| 11c, 22 }}
{{Optimal ET sequence|legend=1| 11b, 22 }}


[[Badness]] (Sintel): 3.588
[[Badness]] (Sintel): 3.59


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 11 0 8 31 38 | 0 1 1 0 0 }}
Mapping: {{mapping| 11 0 8 31 38 | 0 1 1 0 0 }}


Optimal tuning (POTE): ~21/20 = 109.091, ~3/2 = 706.768
Optimal tunings:
* WE: ~21/20 = 109.0321{{c}}, ~3/2 = 706.3870{{c}}
* CWE: ~21/20 = 109.0909{{c}}, ~3/2 = 706.4785{{c}}


{{Optimal ET sequence|legend=0| 11c, 22 }}
{{Optimal ET sequence|legend=0| 11c, 22 }}


Badness (Sintel): 2.270
Badness (Sintel): 2.27


=== 13-limit ===
=== 13-limit ===
Line 129: Line 143:
Mapping: {{mapping| 11 0 8 31 38 23 | 0 1 1 0 0 1 }}
Mapping: {{mapping| 11 0 8 31 38 23 | 0 1 1 0 0 1 }}


Optimal tuning (POTE): ~13/12 = 109.091, ~3/2 = 707.764
Optimal tunings:
* WE: ~13/12 = 109.1184{{c}}, ~3/2 = 707.9422{{c}}
* CWE: ~13/12 = 109.0909{{c}}, ~3/2 = 707.9065{{c}}


{{Optimal ET sequence|legend=0| 11cf, 22 }}
{{Optimal ET sequence|legend=0| 11cf, 22 }}


Badness (Sintel): 2.336
Badness (Sintel): 2.34


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}

Revision as of 10:27, 5 July 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.

← 10th-octave • 11th-octave • 12th-octave →

An 11th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 11. Although 11edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 11 are.

5-limit temperaments

Hendecapent

This temperament has a period of 1/11 octave, which represents 16/15 in the 5-limit. There are some 7-limit extensions include hendecatonic (22 & 77) and hendeca (22 & 33).

Subgroup: 2.3.5

Comma list: 8796093022208/8649755859375

Mapping: [⟨11 0 43], ⟨0 1 -1]]

Mapping generators: ~16/15, ~3

Optimal tunings:

  • WE: ~16/15 = 109.0904 ¢, ~3/2 = 702.6861 ¢
error map: ⟨-0.445 +0.286 +0.613]
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.9082 ¢
error map: ⟨0.000 +0.953 +1.687]

Optimal ET sequence: 22, 55, 77, 99, 869bcc, 968bcc, 1067bccc, 1166bccc

Badness (Sintel): 10.7

Undekapent

This temperament has a period of 1/11 octave, three of them represent 6/5 in the 5-limit.

Subgroup: 2.3.5

Comma list: 48828125/45349632

Mapping: [⟨11 0 8], ⟨0 1 1]]

Mapping generators: ~648/625, ~3

Optimal tunings:

  • WE: ~648/625 = 109.1809 ¢, ~3/2 = 704.4091 ¢
error map: ⟨+0.990 +3.444 -7.468]
  • CWE: ~648/625 = 109.0909 ¢, ~3/2 = 704.6983 ¢
error map: ⟨0.000 +2.743 -8.888]

Optimal ET sequence: 22, 77c, 99c

Badness (Sintel): 17.1

Elven

See also: Sensibeta temperaments § Elven

This temperament has a period of 1/11 octave, which represents 3125/2916 in the 5-limit.

Subgroup: 2.3.5

Comma list: [-23 -66 55⟩

Mapping: [⟨11 2 7], ⟨0 5 6]]

mapping generators: ~3125/2916, ~[10 29 -24⟩

Optimal tunings:

  • WE: ~3125/2916 = 109.0998 ¢, ~[10 29 -24⟩ = 336.8921 ¢
error map: ⟨+0.097 +0.705 -1.263]
  • CWE: ~3125/2916 = 109.0909 ¢, ~[10 29 -24⟩ = 336.8814 ¢
error map: ⟨0.000 +0.634 -1.389]

Optimal ET sequence: 121, 253, 374, 627c

Badness (Sintel): 429

Hendeca

Not to be confused with Hendec.

This temperament has a period of 1/11 octave, nine of them represent 7/4 in the 7-limit, as well as undeka.

Subgroup: 2.3.5.7

Comma list: 225/224, 122880/117649

Mapping: [⟨11 0 43 31], ⟨0 1 -1 0]]

mapping generators: ~16/15, ~3

Optimal tunings:

  • WE: ~16/15 = 108.9526 ¢, ~3/2 = 702.4215 ¢
error map: ⟨-1.521 -1.055 -2.252 +8.705]
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.2071 ¢
error map: ⟨0.000 +1.252 +1.388 +12.992]

Optimal ET sequence: 11, 22, 55d, 77d, 99dd

Badness (Sintel): 4.37

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 352/343

Mapping: [⟨11 0 43 31 38], ⟨0 1 -1 0 0]]

Optimal tunings:

  • WE: ~16/15 = 109.0109 ¢, ~3/2 = 702.5746 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.0395 ¢

Optimal ET sequence: 11, 22, 55d

Badness (Sintel): 2.46

Undeka

Subgroup: 2.3.5.7

Comma list: 875/864, 3200/3087

Mapping: [⟨11 0 8 31], ⟨0 1 1 0]]

mapping generators: ~21/20, ~3

Optimal tunings:

  • WE: ~21/20 = 108.9318 ¢, ~3/2 = 707.7579 ¢
error map: ⟨-1.750 +4.053 -8.852 +8.059]
  • CWE: ~21/20 = 109.0909 ¢, ~3/2 = 707.7526 ¢
error map: ⟨0.000 +5.798 -5.834 +12.992]

Optimal ET sequence: 11b, 22

Badness (Sintel): 3.59

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 352/343, 385/384

Mapping: [⟨11 0 8 31 38], ⟨0 1 1 0 0]]

Optimal tunings:

  • WE: ~21/20 = 109.0321 ¢, ~3/2 = 706.3870 ¢
  • CWE: ~21/20 = 109.0909 ¢, ~3/2 = 706.4785 ¢

Optimal ET sequence: 11c, 22

Badness (Sintel): 2.27

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 65/63, 100/99, 169/165, 352/343

Mapping: [⟨11 0 8 31 38 23], ⟨0 1 1 0 0 1]]

Optimal tunings:

  • WE: ~13/12 = 109.1184 ¢, ~3/2 = 707.9422 ¢
  • CWE: ~13/12 = 109.0909 ¢, ~3/2 = 707.9065 ¢

Optimal ET sequence: 11cf, 22

Badness (Sintel): 2.34

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