6691edo: Difference between revisions

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The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It is a very strong [[11-limit]] division, with a lower 11-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] than any 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.

~~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}~~.

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.

~~{{Harmonics in equal}}~~

~~{{stub}}~~

[[~~Category:Equal divisions of~~ the ~~octave~~|~~####~~]]

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.

=== Prime harmonics ===

=== Subsets and supersets ===

6691edo is the 863rd [[prime edo]].

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 {{Infobox ET}}
-The '''6691 division''' divides the octave into 6691 equal parts of 0.17935 cents each. It is a very strong [[11-limit]] division, with a lower 11-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] than any 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.
+{{ED intro}}
-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}.
+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.
-{{Harmonics in equal}}
-{{stub}}
-[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
+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.
+=== Prime harmonics ===
+{{Harmonics in equal|6691}}
+=== Subsets and supersets ===
+edo is the 863rd [[prime edo]].