11th-octave temperaments

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.

Fractional-octave temperaments

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← 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