15th-octave temperaments

Fractional-octave temperaments

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← 14th-octave • 15th-octave • 16th-octave →

Pentadecal

The pentadecal temperament tempers out the 15-5/3-comma (Quintriyo, pentadecatonic minor thirds comma), [-11 -15 15⟩. This temperament can be described as 15&60 temperament; in the 7-limit, it tempers out the cloudy comma, 16807/16384, and the keema, 875/864.

Subgroup: 2.3.5

Comma: 30517578125/29386561536

Mapping: [⟨15 0 11], ⟨0 1 1]]

POTE generator: ~3/2 = 702.652

Optimal ET sequence: 15, 45, 60, 75, 135, 210c, 345cc

Badness: 0.692042

7-limit

Subgroup: 2.3.5.7

Comma list: 875/864, 16807/16384

Mapping: [⟨15 0 11 42], ⟨0 1 1 0]]

mapping generators: ~21/20, ~3

Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 700.223 (~126/125 = 19.777)

Optimal ET sequence: 15, 45, 60

Badness: 0.114833

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 1372/1331

Mapping: [⟨15 0 11 42 52], ⟨0 1 1 0 0]]

mapping generators: ~21/20, ~3

Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 702.733 (~56/55 = 17.267)

Optimal ET sequence: 15, 45, 60, 75de, 135de

Badness: 0.077420

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 1372/1331

Mapping: [⟨15 0 11 42 52 8], ⟨0 1 1 0 0 2]]

mapping generators: ~21/20, ~3

Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 701.715 (~91/90 = 18.285)

Optimal ET sequence: 15, 45f, 60, 135de, 195cddee

Badness: 0.051740

Quindecal

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 875/864

Mapping: [⟨15 0 11 42 28], ⟨0 1 1 0 1]]

mapping generators: ~21/20, ~3

Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 700.318 (~126/125 = 19.682)

Optimal ET sequence: 15, 45e, 60e

Badness: 0.044405

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 196/195, 352/351, 875/864

Mapping: [⟨15 0 11 42 28 103], ⟨0 1 1 0 1 -2]]

mapping generators: ~21/20, ~3

Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 701.647 (~65/64 = 18.353)

Optimal ET sequence: 15f, 45e, 60e

Badness: 0.055361

Quindecic

Quindecic preserves the 11-limit structure of 15edo, with an independent generator for harmonic 13.

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 49/48, 55/54, 77/75

Mapping: [⟨15 24 35 42 52 0], ⟨0 0 0 0 0 1]]

mapping generators: ~22/21, ~13

Optimal tunings:

• CTE: ~22/21 = 80.000, ~13/8 = 840.528 (~40/39 = 39.472)
• POTE: ~22/21 = 80.000, ~13/8 = 852.924 (~40/39 = 27.076)

Optimal ET sequence: 15, 30

Badness (Smith): 0.028944