8th-octave temperaments

Fractional-octave temperaments

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← 7th-octave • 8th-octave • 9th-octave →

Discussed elsewhere are:

• Octant, → Schismatic family
• Octoid, → Ragismic microtemperaments

Octatonic

12/11 is very close to 1 step of 8edo, and hence this temperament tempers out the octatonic comma, the difference between a stack of 8 12/11's and the octave. The octatonic temperament makes a consistent circle.

Subgroup: 2.3.11

Comma list: [15 8 0 0 -8⟩

Mapping: [⟨8 0 15], ⟨0 1 1]]

mapping generators: ~12/11 = 1\8, ~3

Supporting ETs: 16, 24, 32, 48, 56, 80, 72, 120, 128, 152, ...

Octium

Octium temperament is named after an "extended provisional" name for oxygen, the 8th element. The term oxygen itself is already the name of an exotemperament. It reaches 24/23 in one generator and 19/16 in two.

Subgroup: 2.3.19.23.29.31

Comma list: 94221/94208, 419957/419904, 219501/219488, 7997934975003/7996960669696

Sval mapping: [⟨8 2 35 25 47 2], ⟨0 21 -2 22 -16 74]]

mapping generators: ~10051/9216 = 1\8, ~24/23 = 76.284

Optimal tuning (CTE): ~24/23 = 76.284

Supporting ETs: 472, 1400, 1872, 3272, 5144

Octoid (5-limit)

For the 7-limit temperament, see Ragismic microtemperaments#Octoid.

Subgroup: 2.3.5

Comma list: 59604644775390625/59296646043258912

Sval mapping: [⟨8 1 3], ⟨0 3 4]]

Optimal tuning (CWE): ~2125764/1953125 = 1\8, ~4374/3125 = 584.025

Supporting ETs: 8, 152, 224, 72, 376, 80, 600, 528, 296, 64c, 232, 88bc, 824, 680