Octaphore

Interval information

Ratio	94450499584/94143178827
Factorization	214 × 3-23 × 78
Monzo	[14 -23 0 8⟩
Size in cents	5.642232¢
Names	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
Harmonic entropy (Shannon, [math]\displaystyle{ \sqrt{nd} }[/math])	~1.49165 bits
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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]]

mapping generators: ~2, ~28/27, ~5

Optimal tuning (POTE): ~2 = 1200.000, ~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: 200704/200475, 94450499584/94143178827

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

Optimal tuning (POTE): ~2 = 1200.000, ~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 = 1200.000, ~27/26 = 62.281, ~5/4 = 386.512

Unicorn (2.3.7 subgroup)

See also: Unicorn

If we temper out the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of unicorn, where it finds prime 5 by interpreting five gens as a flat ~6/5 by tempering out 126/125.

Subgroup: 2.3.7

Comma list: 94450499584/94143178827

Mapping: [⟨1 2 4], ⟨0 -8 -23]]

mapping generators: ~2, ~28/27

Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 62.233

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

See also

• Unicorn family
• Unicorn comma