15th-octave temperaments

Fractional-octave temperaments

---

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

15edo has some close approximations including 22/21 to one step and 77/64 to four steps.

Temperaments discussed elsewhere include pentadecal.

Pentadecoid

Subgroup: 2.3.7.11

Comma list: 1362944/1361367, 1771561/1769472

Mapping: [⟨15 108 0 94], ⟨0 -2 1 -1]]

mapping generators: ~22/21, ~7

Optimal tuning (CTE): ~22/21 = 80.0000, ~8/7 = 231.0071

Optimal ET sequence: 120, 135, 660, 795, 930, 1065, 1200

Badness: 0.025231

Phosphorus

Subgroup: 2.3.5.7

Comma list: 250047/250000, [60 -96 1 32⟩

Mapping: [⟨15 8 -4 -4], ⟨0 13 32 38]]

mapping generators: ~[-21 34 4 -15⟩, ~210827008/199290375

Optimal tuning (CTE): ~[-21 34 4 -15⟩ = 80.0000, ~28/27 = 62.9267

Optimal ET sequence: 1125, 1335, 2460

Badness: 0.381587

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: 0.028944