87th-octave temperaments: Difference between revisions
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Comma list: 78125000/78121827, {{monzo| 152 -112 -41 43 }} | Comma list: 78125000/78121827, {{monzo| 152 -112 -41 43 }} | ||
{{Mapping|legend=0| 87 1 -106 -406 | 0 4 9 19 }} | |||
:mapping generators: ~126/125, ~{{monzo| 37 -25 -12 11 }} | :mapping generators: ~126/125, ~{{monzo| 37 -25 -12 11 }} | ||
Optimal tuning (CTE): ~126/125 = 1\87, ~{{monzo| 37 -25 -12 11 }} = 472.043 | Optimal tuning (CTE): ~126/125 = 1\87, ~{{monzo| 37 -25 -12 11 }} = 472.043 | ||
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Comma list: 78125000/78121827, 199297406/199290375, {{monzo| 74 -17 -5 -4 -7 }} | Comma list: 78125000/78121827, 199297406/199290375, {{monzo| 74 -17 -5 -4 -7 }} | ||
{{Mapping|legend=0| 87 1 -106 -406 1225 | 0 4 9 19 -27 }} | |||
:mapping generators: ~126/125, ~{{monzo| 35 1 -6 -3 -4 }} | :mapping generators: ~126/125, ~{{monzo| 35 1 -6 -3 -4 }} | ||
Optimal tuning (CTE): ~126/125 = 1\87, ~{{monzo| 35 1 -6 -3 -4 }} = 472.043 | Optimal tuning (CTE): ~126/125 = 1\87, ~{{monzo| 35 1 -6 -3 -4 }} = 472.043 | ||
Revision as of 12:33, 24 September 2026
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]]
- 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]
Supporting ETs: 87, 609, 696, 783, 870, 1305, 1479, ...
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