Octaphore

Interval information

Ratio

94450499584/94143178827

Factorization

2¹⁴ × 3^-23 × 7⁸

Monzo

[14 -23 0 8⟩

Size in cents

5.642232¢

Names

the octaphore,
enneagari comma

FJS name

[math]\displaystyle{ \text{5d6}^{7,7,7,7,7,7,7,7} }[/math]

Special properties

reduced

Tenney height (log₂ nd)

72.913

Weil height (log₂ max(n, d))

72.9177

Wilson height (sopfr(nd))

153

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The octaphore, also known as the enneagari comma, is a small 7-limit (also 2.3.7-subgroup) comma measuring about 5.64 cents. It is so named because it is the amount by which eight 28/27 third-tones exceed the 4/3 perfect fourth. It can also be found as the amount by which seven 28/27 third-tones exceed the 9/7 supermajor third, or as the sum of the garischisma (33554432/33480783) and the septimal ennealimma (40353607/40310784).

Temperaments

Tempering out the octaphore comma in the full 7-limit leads to rank-3 octaphore temperament, and excluding prime 5 from the subgroup leads to the 2.3.7 subgroup rank-2 Unicorn temperament.

Octaphore

Subgroup: 2.3.5.7

Comma list: 94450499584/94143178827

Mapping: [⟨1 2 2 4], ⟨0 -8 0 -23], ⟨0 0 1 0]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.233, ~5/4 = 386.314

Optimal ET sequence: 19, 39d, 58, 77, 96d, 135

Undecimal Octaphore

By noticing that the interval at ⟨4 2 -2] is quite close to 11/8, we can add prime 11 to the mapping by tempering out the Reef comma.

Subgroup: 2.3.5.7.11

Comma list: 94450499584/94143178827, 200704/200475

Mapping: [⟨1 2 2 4 4], ⟨0 -8 0 -23 2], ⟨0 0 1 0 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.233, ~5/4 = 386.481

Tridecimal Octaphore

By noticing that two generators is extremely close to 14/13, we can add prime 13 to the mapping by tempering out the Squbema, or equivalently by tempering out the Tesseract Comma.

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 3584/3575, 660275/657072

Mapping: [⟨1 2 2 4 4 5], ⟨0 -8 0 -23 2 -25], ⟨0 0 1 0 -2 0]]

Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 62.281, ~5/4 = 386.512

2.3.7 Unicorn

See also: Unicorn

If we temper the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of unicorn:

Subgroup: 2.3.7

Comma list: 94450499584/94143178827

Mapping: [⟨1 2 4], ⟨0 -8 -23]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.233

Optimal ET sequence: 19, 20d, 39d, 58, 77, 96d, 135

Septimal Unicorn

By noticing that the interval at five generators is nearly 6/5, we can introduce prime 5 to the mapping by tempering out 126/125.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 126/125

Mapping: [⟨1 2 3 4], ⟨0 -8 -13 -23]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.324

Optimal ET sequence: 19, 39d, 58, 77, 135c, 212c