87th-octave temperaments

Fractional-octave temperaments

---

← 86th-octave • 87th-octave • 88th-octave →

One step of 87edo is in direct proximity to starling comma, 126/125.

87-126/125-commatic

An extremely precise microtemperament provided by tempering the comma which is the difference between a stack of 87 126/125s and the octave. 87edo itself, notably, does the mapping consistently.

Subgroup: 2.3.5.7

Comma list: [86 174 -261 87⟩

Mapping: [⟨87 87 174 262], ⟨0 1 0 -2], ⟨0 0 1 3]]

Optimal tunings:

• WE: ~126/125 = 1\87, ~3/2 = 701.955 ¢, ~5/4 = 386.314

error map: ⟨-0.000, -0.000, 0.000, -0.000]

• CWE: ~126/125 = 1\87, ~3/2 = 701.955 ¢, ~5/4 = 386.314

error map: ⟨-0.000, -0.000, 0.000, -0.000]

Optimal ET sequence: 87, 609, 696, 783, 1479, 2262, 3045, 7482, 8178, 8961, 9744, 10527, 11223, 12006, 20967, 21750, 32973, 33756, 44979, 111708, 156687, 268395, 425082

Badness: 2.0954 × 10³

Francium

Francium tempers out the euzenius comma and the 87-126/125 comma in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 78125000/78121827, [152 -112 -41 43⟩

Mapping: [⟨87 1 -106 -406], ⟨0 4 9 19]]

mapping generators: ~126/125, ~[37 -25 -12 11⟩

Optimal tuning (CTE): ~126/125 = 1\87, ~[37 -25 -12 11⟩ = 472.043

Optimal ET sequence: 783, …, 8178, 8961, 9744, 10527, 20271, 30798

Badness: 0.762

11-limit

Subgroup: 2.3.5.7.11

Comma list: 78125000/78121827, 199297406/199290375, [74 -17 -5 -4 -7⟩

Mapping: [⟨87 1 -106 -406 1225], ⟨0 4 9 19 -27]]

mapping generators: ~126/125, ~[35 1 -6 -3 -4⟩

Optimal tuning (CTE): ~126/125 = 1\87, ~[35 1 -6 -3 -4⟩ = 472.043

Optimal ET sequence: 783, 8178, 8961, 9744, 28449de, 38193bcde, 47937bcdee

Badness: 0.352