4296edo: Difference between revisions

(6 intermediate revisions by 5 users not shown)

Line 1:

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~~&~~gt;, are all one step of 4296et~~.

4296edo is an extraordinarily strong 5-limit system, tempering out raider, {{monzo| 71 -99 37 }}, pirate, {{monzo| -90 -15 49 }} and the [[Kirnberger's atom]], {{monzo| 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 septimal [[atomic]], the 612 & 1848 temperament.

4296 = 12 × 358, and is potentially of use as a device for constructing 5-limit 12-note circulating temperaments, and

which means that one cent is exactly 3.58 steps of 4296edo. 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, {{monzo| -17 62 -35 }}, fortune, {{monzo| -107 47 14 }} and the [[monzisma]], {{monzo| 54 -37 2 }}, are all one step of 4296et.

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

=== Prime harmonics ===

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-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.
+{{ED intro}}
-= 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.
+edo is an extraordinarily strong 5-limit system, tempering out raider, {{monzo| 71 -99 37 }}, pirate, {{monzo| -90 -15 49 }} and the [[Kirnberger's atom]], {{monzo| 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 septimal [[atomic]], the 612 & 1848 temperament.
-{{Primes in edo|4296|prec=4}}
+= 12 × 358, and is potentially of use as a device for constructing 5-limit 12-note circulating temperaments, and
+which means that one cent is exactly 3.58 steps of 4296edo. 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, {{monzo| -17 62 -35 }}, fortune, {{monzo| -107 47 14 }} and the [[monzisma]], {{monzo| 54 -37 2 }}, are all one step of 4296et.
-[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
+=== Prime harmonics ===
+{{Harmonics in equal|4296|prec=4}}