4296edo: Difference between revisions

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The 4296 equal division divides the octave into 4296 steps of 0.2793 cents each, which means that one cent is exactly 3.58 steps of 4296 edo. It is an extraordinarily strong 5-limit system, tempering out raider, |71 -99 37>, pirate, |-90 -15 49> and the Kirnberger atom, |161 -84 -12>. Not until [[73709edo|73709]] do we reach a division with a lower 5-limit relative error, and not until [[6796263edo|6796263]] do we find a lower logflat badness. It is uniquely consistent through the 9 odd limit, and in the 7-limit, it tempers out the landscape comma, 250047/250000, and so ~~supports~~ the 7-limit version of the 612&1848 temperament.

The 4296 equal division divides the octave into 4296 steps of 0.2793 cents each, which means that one cent is exactly 3.58 steps of 4296 edo. It is an extraordinarily strong 5-limit system, tempering out raider, |71 -99 37>, pirate, |-90 -15 49> and the Kirnberger atom, |161 -84 -12>. Not until [[73709edo|73709]] do we reach a division with a lower 5-limit relative error, and not until [[6796263edo|6796263]] do we find a lower logflat badness. It is uniquely consistent through the 9 odd limit, and in the 7-limit, it tempers out the landscape comma, 250047/250000, and so [[support]]s the 7-limit version of the 612&1848 temperament.

4296 = 12 * 358, and is potentially of use as a device for constructing 5-limit 12-note circulating temperaments. From that point of view, one might note that 81/80 is 77 steps, 531441/524288, the Pythagorean comma, 84 steps, and 32805/32768, the schisma, 7 steps, making it exactly 1/12 of a Pythagorean comma and 1/11 of a syntonic comma, useful approximations when dealing with this problem. Senior, |-17 62 -35>, fortune, |-107 47 14> and the monzisma, |54 -37 2>, are all one step of 4296et.

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-The 4296 equal division divides the octave into 4296 steps of 0.2793 cents each, which means that one cent is exactly 3.58 steps of 4296 edo. It is an extraordinarily strong 5-limit system, tempering out raider, |71 -99 37&gt;, pirate, |-90 -15 49&gt; and the Kirnberger atom, |161 -84 -12&gt;. Not until [[73709edo|73709]] do we reach a division with a lower 5-limit relative error, and not until [[6796263edo|6796263]] do we find a lower logflat badness. It is uniquely consistent through the 9 odd limit, and in the 7-limit, it tempers out the landscape comma, 250047/250000, and so supports the 7-limit version of the 612&amp;1848 temperament.
+The 4296 equal division divides the octave into 4296 steps of 0.2793 cents each, which means that one cent is exactly 3.58 steps of 4296 edo. It is an extraordinarily strong 5-limit system, tempering out raider, |71 -99 37&gt;, pirate, |-90 -15 49&gt; and the Kirnberger atom, |161 -84 -12&gt;. Not until [[73709edo|73709]] do we reach a division with a lower 5-limit relative error, and not until [[6796263edo|6796263]] do we find a lower logflat badness. It is uniquely consistent through the 9 odd limit, and in the 7-limit, it tempers out the landscape comma, 250047/250000, and so [[support]]s the 7-limit version of the 612&amp;1848 temperament.
 = 12 * 358, and is potentially of use as a device for constructing 5-limit 12-note circulating temperaments. From that point of view, one might note that 81/80 is 77 steps, 531441/524288, the Pythagorean comma, 84 steps, and 32805/32768, the schisma, 7 steps, making it exactly 1/12 of a Pythagorean comma and 1/11 of a syntonic comma, useful approximations when dealing with this problem. Senior, |-17 62 -35&gt;, fortune, |-107 47 14&gt; and the monzisma, |54 -37 2&gt;, are all one step of 4296et.
 {{Primes in edo|4296|prec=4}}