4296edo: Difference between revisions

(5 intermediate revisions by 4 users not shown)

Line 1:

The '''4296 equal divisions of the octave''' ('''4296edo'''), or '''4296-tone equal temperament''' ('''4296tet'''), '''4296 equal temperament''' ('''4296et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into 4296 [[equal]] parts of about of 0.2793 [[cent]]s each, which means that one cent is exactly 3.58 steps of 4296edo. It 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. 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.

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.

=== Prime harmonics ===

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''4296 equal divisions of the octave''' ('''4296edo'''), or '''4296-tone equal temperament''' ('''4296tet'''), '''4296 equal temperament''' ('''4296et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into 4296 [[equal]] parts of about of 0.2793 [[cent]]s each, which means that one cent is exactly 3.58 steps of 4296edo. It 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.
+{{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, {{monzo| -17 62 -35 }}, fortune, {{monzo| -107 47 14 }} and the [[monzisma]], {{monzo| 54 -37 2 }}, 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.
+= 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.
 === Prime harmonics ===
 {{Harmonics in equal|4296|prec=4}}