4296edo - Revision history

Overthink: Undo revision 211363 by Overthink (talk) not very notable (also easily seen from table)

2026-03-06T23:34:29Z

Undo revision 211363 by Overthink (talk) not very notable (also easily seen from table)

← Older revision		Revision as of 23:34, 6 March 2026
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	4296 = 12 × 358, and is potentially of use as a device for constructing 5-limit 12-note circulating temperaments, and		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.		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.

	~~It is also a strong system in the 2.3.5.13.17.19.23.29.31 subgroup, being consistent to the 31-odd-limit in that subgroup.~~

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|4296\|prec=4}}		{{Harmonics in equal\|4296\|prec=4}}

Overthink: + brief description of higher limits

2025-09-28T03:45:05Z

+ brief description of higher limits

← Older revision		Revision as of 03:45, 28 September 2025
Line 6:		Line 6:
	4296 = 12 × 358, and is potentially of use as a device for constructing 5-limit 12-note circulating temperaments, and		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.		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.

			It is also a strong system in the 2.3.5.13.17.19.23.29.31 subgroup, being consistent to the 31-odd-limit in that subgroup.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|4296\|prec=4}}		{{Harmonics in equal\|4296\|prec=4}}

ArrowHead294 at 14:55, 20 February 2025

2025-02-20T14:55:43Z

← Older revision		Revision as of 14:55, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|4296~~}}		{{ED intro}}

	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.		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.

BudjarnLambeth at 07:07, 9 July 2023

2023-07-09T07:07:12Z

← Older revision		Revision as of 07:07, 9 July 2023
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	~~{{novelty}}{{stub}}~~{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|4296}}		{{EDO intro\|4296}}

BudjarnLambeth at 04:25, 9 July 2023

2023-07-09T04:25:52Z

← Older revision		Revision as of 04:25, 9 July 2023
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	{{Infobox ET}}		{{novelty}}{{stub}}{{Infobox ET}}
	{{EDO intro\|4296}}		{{EDO intro\|4296}}

Eliora at 12:37, 7 May 2023

2023-05-07T12:37:05Z

← Older revision		Revision as of 12:37, 7 May 2023
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	{{Infobox ET}}		{{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.		{{EDO intro\|4296}}

	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 ===		=== Prime harmonics ===
	{{Harmonics in equal\|4296\|prec=4}}		{{Harmonics in equal\|4296\|prec=4}}

FloraC: Cleanup

2023-02-20T09:14:38Z

Cleanup

← Older revision		Revision as of 09:14, 20 February 2023
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	{{Infobox ET}}		{{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~~>~~, 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&~~amp;~~1848 temperament.		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, \|-17 62 -35~~>~~, fortune, \|-107 47 14~~>~~ and the monzisma, \|54 -37 2~~>~~, are all one step of 4296et.		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.

	{{~~Primes~~ in ~~edo~~\|4296\|prec=4}}		=== Prime harmonics ===
			{{Harmonics in equal\|4296\|prec=4}}
	~~[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->~~

Plumtree: Infobox ET added

2022-10-04T22:18:28Z

Infobox ET added

← Older revision		Revision as of 22:18, 4 October 2022
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>, 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.		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.

Fredg999 category edits: Categories

2022-07-04T01:30:13Z

Cmloegcmluin: link to new page for Supports

2022-01-25T18:39:10Z

link to new page for Supports

← Older revision		Revision as of 18:39, 25 January 2022
Line 1:		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 ~~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.		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.

	{{Primes in edo\|4296\|prec=4}}		{{Primes in edo\|4296\|prec=4}}

← Older revision		Revision as of 01:30, 4 July 2022
Line 1:		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.		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.		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.

	{{Primes in edo\|4296\|prec=4}}		{{Primes in edo\|4296\|prec=4}}

			[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->

← Older revision		Revision as of 18:39, 25 January 2022
Line 1:		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 ~~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.		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.

	{{Primes in edo\|4296\|prec=4}}		{{Primes in edo\|4296\|prec=4}}