6691edo: Difference between revisions

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6691edo is a very strong [[11-limit]] system, with a lower 11-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] than any smaller division until [[40006edo|40006]]. It is also strong in the [[7-limit]], where only [[3125edo|3125]] is both smaller and with a lesser relative error.

We may note it is a [[euzenius]] and [[parimic]] system. A basis for the 11-limit commas is {[[1771561/1771470]], 3294225/3294172, 67110351/67108864, [[78125000/78121827]]} and for the 7-limit commas, {78125000/78121827, ~~281484423828125/281474976710656~~, ~~8936733825332544112/8936247052719140625~~}.

We may note it is a [[euzenius]] and [[parimic]] system. A basis for the 11-limit commas is {[[1771561/1771470]], 3294225/3294172, 67110351/67108864, [[78125000/78121827]]} and for the 7-limit commas, {78125000/78121827, {{monzo| -48 0 11 8 }}, {{monzo| 4 -28 -8 21 }}}.

The approximation to [[harmonic]] [[13/1|13]] is weaker, though it is still [[consistent]] to the [[15-odd-limit]]. In fact, it is consistent to the no-13 or no-17 no-23 [[29-odd-limit]]. In the 13-limit we may note it [[tempering out|tempers out]] [[10648/10647]], and is a good tuning for the corresponding rank-5 temperament. It also tempers out [[140625/140608]]. In the 17-limit, [[194481/194480]]; in the 19-limit, 14080/14079, 23409/23408, 43681/43680, 89376/89375, 165376/165375; and in the 23-limit, 21505/21504, 23276/23275, 25921/25920, 52326/52325, 76545/76544 among others.

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 edo is a very strong [[11-limit]] system, with a lower 11-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] than any smaller division until [[40006edo|40006]]. It is also strong in the [[7-limit]], where only [[3125edo|3125]] is both smaller and with a lesser relative error.
-We may note it is a [[euzenius]] and [[parimic]] system. A basis for the 11-limit commas is {[[1771561/1771470]], 3294225/3294172, 67110351/67108864, [[78125000/78121827]]} and for the 7-limit commas, {78125000/78121827, 281484423828125/281474976710656, 8936733825332544112/8936247052719140625}.
+We may note it is a [[euzenius]] and [[parimic]] system. A basis for the 11-limit commas is {[[1771561/1771470]], 3294225/3294172, 67110351/67108864, [[78125000/78121827]]} and for the 7-limit commas, {78125000/78121827, {{monzo| -48 0 11 8 }}, {{monzo| 4 -28 -8 21 }}}.
 The approximation to [[harmonic]] [[13/1|13]] is weaker, though it is still [[consistent]] to the [[15-odd-limit]]. In fact, it is consistent to the no-13 or no-17 no-23 [[29-odd-limit]]. In the 13-limit we may note it [[tempering out|tempers out]] [[10648/10647]], and is a good tuning for the corresponding rank-5 temperament. It also tempers out [[140625/140608]]. In the 17-limit, [[194481/194480]]; in the 19-limit, 14080/14079, 23409/23408, 43681/43680, 89376/89375, 165376/165375; and in the 23-limit, 21505/21504, 23276/23275, 25921/25920, 52326/52325, 76545/76544 among others.