Kirnberger's atom - Revision history

Overthink: Undo revision 230155 by Domin (talk) not a valid reason

2026-05-13T00:11:34Z

Undo revision 230155 by Domin (talk) not a valid reason

← Older revision		Revision as of 00:11, 13 May 2026
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	~~{{delete\|Gene wrote snot}}~~

	{{Infobox Interval		{{Infobox Interval
	\| Monzo = 161 -84 -12		\| Monzo = 161 -84 -12

Domin at 23:58, 12 May 2026

2026-05-12T23:58:59Z

← Older revision		Revision as of 23:58, 12 May 2026
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			{{delete\|Gene wrote snot}}

	{{Infobox Interval		{{Infobox Interval
	\| Monzo = 161 -84 -12		\| Monzo = 161 -84 -12

FloraC: Restore some lost details. These lists aren't clearer than prose. - essentially duplicate edo list from the optimal GPV sequence.

2026-04-24T11:53:00Z

Restore some lost details. These lists aren't clearer than prose. - essentially duplicate edo list from the optimal GPV sequence.

← Older revision		Revision as of 11:53, 24 April 2026
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	\| Comma = yes		\| Comma = yes
	}}		}}
	'''Kirnberger's atom''' ({{monzo\|legend=1\| 161 -84 -12 }}) (or ~~systematically~~ simply the '''atom'''), is an [[unnoticeable comma\|unnoticeable]] [[5-limit]] [[comma]], 0.01536093 [[cent]]s in size. It is the difference between:		'''Kirnberger's atom''' ({{monzo\|legend=1\| 161 -84 -12 }}), or simply the '''atom''', is an [[unnoticeable comma\|unnoticeable]] [[5-limit]] [[comma]], 0.01536093 [[cent]]s in size. It is the difference between a [[syntonic comma]] and a stack of eleven [[schisma]]s, between the [[Pythagorean comma]] and a stack of twelve schismas, or equivalently, between twelve syntonic commas and eleven Pythagorean commas.

	* [[81/80]] ~~and 11~~ [[~~32805~~/~~32768\|schismas~~]],		[[16384/10935\|Kirnberger's fifth]], which is the perfect fifth of [[3/2]] flattened by a [[schisma]], is practically identical to seven steps of [[12edo]], which realizes a rational intonation version of the equal temperament. Kirnberger's atom arises as the tiny interval by which twelve of Kirnberger's fifths exceed seven [[octave]]s, (16384/10935)<sup>12</sup>/2<sup>7</sup>.
	* A [[~~pythagorean comma~~]] ~~and 12 schismas~~
	* 12 81/80's and 11 pythagorean commas
	* The [[~~raider~~]] ~~and~~ the [[~~pirate~~]] ~~comma~~.

	~~[[16384/10935\|Kirnberger's fifth]], which is~~ the ~~perfect fifth of [[3/2]] flattened by a [[schisma]], is practically identical to seven steps of~~ [[~~12edo~~]]~~, which realizes a rational intonation version of the equal temperament. Kirnberger's atom arises as~~ the ~~tiny interval by which twelve of Kirnberger's fifths exceed seven~~ [[~~octave~~]]~~s, (16384/10935)<sup>12</sup>/2<sup>7</sup>~~.		It may also be expressed as the difference between the [[raider comma]] and the [[pirate comma]].

	== Temperaments ==		== Temperaments ==
	[[Tempering out]] Kirnberger's atom leads to the [[~~Very high accuracy temperaments #Atomic\|~~atomic]] temperament, in which eleven schismas make up a syntonic comma, and twelve schismas make up a [[Pythagorean comma]]. Many notable [[~~EDO\|edos~~]] temper out Kirnberger's atom, such as ~~{{EDOs\| 12, 612, 624, 1236, 1848, 2460, 3072, 3684, 4296, 7980, 12276, 16572, 20868, 25164, and 46032 }}~~. Any tuning system (such as [[41edo]]) for which the number of divisions of the octave is not divisible by 12 cannot temper out Kirnberger's atom.		[[Tempering out]] Kirnberger's atom leads to the 5-limit version of [[atomic]] temperament, in which eleven schismas make up a syntonic comma, and twelve schismas make up a Pythagorean comma. Many notable [[edo]]s temper out Kirnberger's atom, such as [[612edo]]. Any tuning system (such as [[41edo]]) for which the number of divisions of the octave is not divisible by 12 cannot temper out Kirnberger's atom.

	== Approximation ==		== Approximation ==

Overthink: fix typo

2026-03-22T03:22:06Z

fix typo

← Older revision		Revision as of 03:22, 22 March 2026
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	\| Comma = yes		\| Comma = yes
	}}		}}
	'''Kirnberger's atom''' ({{monzo\|~~legned~~=1\| 161 -84 -12 }}) (or systematically simply the '''atom'''), is an [[unnoticeable comma\|unnoticeable]] [[5-limit]] [[comma]], 0.01536093 [[cent]]s in size. It is the difference between:		'''Kirnberger's atom''' ({{monzo\|legend=1\| 161 -84 -12 }}) (or systematically simply the '''atom'''), is an [[unnoticeable comma\|unnoticeable]] [[5-limit]] [[comma]], 0.01536093 [[cent]]s in size. It is the difference between:

	* [[81/80]] and 11 [[32805/32768\|schismas]],		* [[81/80]] and 11 [[32805/32768\|schismas]],

VectorGraphics at 00:57, 22 March 2026

2026-03-22T00:57:07Z

← Older revision		Revision as of 00:57, 22 March 2026
Line 5:		Line 5:
	\| Comma = yes		\| Comma = yes
	}}		}}
	'''Kirnberger's atom''' ({{monzo\|legned=1\| 161 -84 -12 }}), is an [[unnoticeable comma\|unnoticeable]] [[5-limit]] [[comma]], 0.01536093 [[cent]]s in size. It is the difference between:		'''Kirnberger's atom''' ({{monzo\|legned=1\| 161 -84 -12 }}) (or systematically simply the '''atom'''), is an [[unnoticeable comma\|unnoticeable]] [[5-limit]] [[comma]], 0.01536093 [[cent]]s in size. It is the difference between:

	* [[81/80]] and 11 [[32805/32768\|schismas]],		* [[81/80]] and 11 [[32805/32768\|schismas]],

Overthink: /* Approximation */ use the actual closest edo

2026-03-16T21:33:16Z

Approximation: use the actual closest edo

← Older revision		Revision as of 21:33, 16 March 2026
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	== Approximation ==		== Approximation ==
	However, if one wants to accurately represent the interval without tempering it out, there are very large edos that do this. [[78005edo]] not only has a step size that is very close to Kirnberger's atom and consistently represents it, but it is also one of, if not the most accurate 5-limit edo for its size. [[~~78123edo~~]]~~'s step size is even closer~~, but ~~Kirnberger's atom~~ is not consistently represented (1 step via [[direct approximation]] and 3 steps by [[patent val]]).		However, if one wants to accurately represent the interval without tempering it out, there are very large edos that do this. [[78005edo]] not only has a step size that is very close to Kirnberger's atom and consistently represents it, but it is also one of, if not the most accurate 5-limit edo for its size. The edo with the closest step to Kirnberger's atom is [[78120edo]], but it is not consistently represented (1 step via [[direct approximation]] and 24 steps by [[patent val]]).

	[[Category:Atomic]]		[[Category:Atomic]]

Overthink: /* Temperaments */ trim list; nvm about the optimal ET sequence (5-limit omits some of them, included in later limits but still not the best reference?)

2026-03-16T21:30:06Z

Temperaments: trim list; nvm about the optimal ET sequence (5-limit omits some of them, included in later limits but still not the best reference?)

← Older revision		Revision as of 21:30, 16 March 2026
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	== Temperaments ==		== Temperaments ==
	[[Tempering out]] Kirnberger's atom leads to the [[Very high accuracy temperaments #Atomic\|atomic]] temperament, in which eleven schismas make up a syntonic comma, and twelve schismas make up a [[Pythagorean comma]]. Many notable [[EDO\|edos]] temper out Kirnberger's atom, such as {{EDOs\| 12, 612, 624, 1236, 1848, 2460, 3072~~, 3084~~, 3684, 4296~~, 4308, 4908~~, 7980, 12276, 16572, 20868, 25164~~, 29460, 33756~~, and 46032 }}~~, can be seen by the [[optimal ET sequence]] at [[Very high accuracy temperaments #Atomic]]~~. Any tuning system (such as [[41edo]]) for which the number of divisions of the octave is not divisible by 12 cannot temper out Kirnberger's atom.		[[Tempering out]] Kirnberger's atom leads to the [[Very high accuracy temperaments #Atomic\|atomic]] temperament, in which eleven schismas make up a syntonic comma, and twelve schismas make up a [[Pythagorean comma]]. Many notable [[EDO\|edos]] temper out Kirnberger's atom, such as {{EDOs\| 12, 612, 624, 1236, 1848, 2460, 3072, 3684, 4296, 7980, 12276, 16572, 20868, 25164, and 46032 }}. Any tuning system (such as [[41edo]]) for which the number of divisions of the octave is not divisible by 12 cannot temper out Kirnberger's atom.

	== Approximation ==		== Approximation ==

Hotcrystal0: adding back the list

2026-03-16T20:39:37Z

adding back the list

← Older revision		Revision as of 20:39, 16 March 2026
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	== Temperaments ==		== Temperaments ==
	[[Tempering out]] Kirnberger's atom leads to the [[Very high accuracy temperaments #Atomic\|atomic]] temperament, in which eleven schismas make up a syntonic comma, and twelve schismas make up a [[Pythagorean comma]]. Many notable [[EDO\|edos]] temper out Kirnberger's atom, as can be seen by the [[optimal ET sequence]] at [[Very high accuracy temperaments #Atomic]]. Any tuning system (such as [[41edo]]) for which the number of divisions of the octave is not divisible by 12 cannot temper out Kirnberger's atom.		[[Tempering out]] Kirnberger's atom leads to the [[Very high accuracy temperaments #Atomic\|atomic]] temperament, in which eleven schismas make up a syntonic comma, and twelve schismas make up a [[Pythagorean comma]]. Many notable [[EDO\|edos]] temper out Kirnberger's atom, such as {{EDOs\| 12, 612, 624, 1236, 1848, 2460, 3072, 3084, 3684, 4296, 4308, 4908, 7980, 12276, 16572, 20868, 25164, 29460, 33756, and 46032 }}, can be seen by the [[optimal ET sequence]] at [[Very high accuracy temperaments #Atomic]]. Any tuning system (such as [[41edo]]) for which the number of divisions of the octave is not divisible by 12 cannot temper out Kirnberger's atom.

	== Approximation ==		== Approximation ==

Overthink: /* Temperaments */ link

2026-03-16T19:42:05Z

Temperaments: link

← Older revision		Revision as of 19:42, 16 March 2026
Line 15:		Line 15:

	== Temperaments ==		== Temperaments ==
	[[Tempering out]] Kirnberger's atom leads to the [[Very high accuracy temperaments #Atomic\|atomic]] temperament, in which eleven schismas make up a syntonic comma, and twelve schismas make up a [[Pythagorean comma]]. Many notable edos temper out Kirnberger's atom, as can be seen by the [[optimal ET sequence]] at [[Very high accuracy temperaments #Atomic]]. Any tuning system (such as [[41edo]]) for which the number of divisions of the octave is not divisible by 12 cannot temper out Kirnberger's atom.		[[Tempering out]] Kirnberger's atom leads to the [[Very high accuracy temperaments #Atomic\|atomic]] temperament, in which eleven schismas make up a syntonic comma, and twelve schismas make up a [[Pythagorean comma]]. Many notable [[EDO\|edos]] temper out Kirnberger's atom, as can be seen by the [[optimal ET sequence]] at [[Very high accuracy temperaments #Atomic]]. Any tuning system (such as [[41edo]]) for which the number of divisions of the octave is not divisible by 12 cannot temper out Kirnberger's atom.

	== Approximation ==		== Approximation ==

Overthink: /* Temperaments */ link optimal ET sequence

2026-03-16T19:41:11Z

Temperaments: link optimal ET sequence

← Older revision		Revision as of 19:41, 16 March 2026
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	== Temperaments ==		== Temperaments ==
	~~Kirnberger's atom is~~ [[~~tempering out\|tempered~~ out]] ~~in notable edos such as {{EDOs\| 12, 612, 624, 1236, 1848, 2460, 3072, 3084, 3684, 4296, 4308, 4908, 7980, 12276, 16572, 20868, 25164, 29460, 33756, and 46032 }}, leading~~ to the [[Very high accuracy temperaments #Atomic\|atomic ~~temperament~~]], in which eleven schismas make up a syntonic comma and twelve schismas make up a [[Pythagorean comma]]~~; any~~ tuning system ([[41edo]], for ~~example)~~ which the number of divisions of the octave is not ~~a multiple of~~ 12 cannot temper out Kirnberger's atom.		[[Tempering out]] Kirnberger's atom leads to the [[Very high accuracy temperaments #Atomic\|atomic]] temperament, in which eleven schismas make up a syntonic comma, and twelve schismas make up a [[Pythagorean comma]]. Many notable edos temper out Kirnberger's atom, as can be seen by the [[optimal ET sequence]] at [[Very high accuracy temperaments #Atomic]]. Any tuning system (such as [[41edo]]) for which the number of divisions of the octave is not divisible by 12 cannot temper out Kirnberger's atom.

	== Approximation ==		== Approximation ==