Simple map - Revision history

FloraC: Changed redirect target from Uniform map#Simple map to Uniform map#Integer uniform map, or simple map

2025-03-31T08:38:50Z

Changed redirect target from Uniform map#Simple map to Uniform map#Integer uniform map, or simple map

← Older revision		Revision as of 08:38, 31 March 2025
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	#REDIRECT [[~~Uniform_map~~#~~Simple_map~~]]		#REDIRECT [[Uniform map #Integer uniform map, or simple map]]

Cmloegcmluin: merge "simple map" back in to the "uniform map" page, to avoid unnecessary fracturing of information

2021-12-29T16:20:31Z

merge "simple map" back in to the "uniform map" page, to avoid unnecessary fracturing of information

← Older revision		Revision as of 16:20, 29 December 2021
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	~~A '''simple map''' is a specific type of~~ [[map]] used for [[EDO]]s. Every EDO has one simple map. The simple map for <math>n</math>-EDO is found by taking the [[just intonation point]] (JIP) for some prime limit <math>p</math>, {{map\|log₂2 log₂3 log₂5 ... log₂<math>p</math>}}, multiplying it by <math>n</math>, then individually rounding entries to the nearest integer. For example, the simple map for 7-limit 19-EDO is 19·{{map\|log₂2 log₂3 log₂5 log₂7}} = {{map\|19 30.115 44.117 53.340}} which rounds to {{map\|19 30 44 53}}.		#REDIRECT [[Uniform_map#Simple_map]]

	The simple map is not necessarily the ''best'' map for its EDO in terms of overall tuning accuracy, but it is the simplest map to calculate. The classic example of a simple map which is not the best map is 17p in the 5-limit, 17·{{map\|log₂2 log₂3 log₂5}} = {{map\|17 26.944 39.473}} which rounds to {{map\|17 27 39}}. The approximation of prime 5 is really bad here; it's about exactly halfway between 39 and 40 steps, but slightly below, which is why it rounds down. But it turns out that if we round up instead, using 40 steps to approximate prime 5, then the absolute errors in the primes remain about the same. However, the error in 5/3 is much less, because the error in 5 and the error in 3 are now in the same direction, canceling each other out, and so overall {{map\|17 27 40}} has less error than {{map\|17 27 39}}.

	~~== Vs. integer uniform maps ==~~

	~~Another name for a simple map is an [[integer uniform map]]. The two different terms provide two different ways of presenting the same object, which can be helpful in different contexts:~~
	* In contexts pertaining to tuning accuracy, "simple map" works well. This is probably the more common context.
	* In contexts pertaining to other uniform maps, "integer uniform map" works well.

	~~[[Category:Regular temperament theory]]~~
	~~[[Category:Terms]]~~
	~~[[Category:Math~~]]

Cmloegcmluin: consolidate categories

2021-12-25T13:23:04Z

consolidate categories

← Older revision		Revision as of 13:23, 25 December 2021
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	* In contexts pertaining to other uniform maps, "integer uniform map" works well.		* In contexts pertaining to other uniform maps, "integer uniform map" works well.

	~~[[Category:Temperament]]~~
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	~~[[Category:Theory]]~~
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Math]]		[[Category:Math]]

Cmloegcmluin: incorporate Dave's feedback

2021-12-15T18:49:05Z

incorporate Dave's feedback

← Older revision		Revision as of 18:49, 15 December 2021
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	A '''simple map''' is a specific type of [[map]] used for [[EDO]]s. Every EDO has one simple map. The simple map for <math>n</math>-EDO is found by ~~multiplying~~ the [[just intonation point]] (JIP) {{map\|log₂2 log₂3 log₂5 ...}} by <math>n</math>, then individually rounding entries to the nearest integer. For example, the simple map for 7-limit 19-EDO is 19·{{map\|log₂2 log₂3 log₂5 log₂7}} = {{map\|19 30.115 44.117 53.340}} which rounds to {{map\|19 30 44 53}}.		A '''simple map''' is a specific type of [[map]] used for [[EDO]]s. Every EDO has one simple map. The simple map for <math>n</math>-EDO is found by taking the [[just intonation point]] (JIP) for some prime limit <math>p</math>, {{map\|log₂2 log₂3 log₂5 ... log₂<math>p</math>}}, multiplying it by <math>n</math>, then individually rounding entries to the nearest integer. For example, the simple map for 7-limit 19-EDO is 19·{{map\|log₂2 log₂3 log₂5 log₂7}} = {{map\|19 30.115 44.117 53.340}} which rounds to {{map\|19 30 44 53}}.

	The simple map is not necessarily the ''best'' map for its EDO in terms of overall tuning accuracy, but it is the simplest map to calculate. The classic example of a simple map which is not the best map is 17p in the 5-limit, 17·{{map\|log₂2 log₂3 log₂5}} = {{map\|17 26.944 39.473}} which rounds to {{map\|17 27 39}}. ~~We can see that the~~ approximation of prime 5 is really bad here; it's about exactly halfway between 39 and 40 steps, but slightly below, which is why it ~~rounded~~ down. It turns out that if we ~~shrink our steps slightly so that it takes~~ 40 steps to approximate 5 ~~instead~~, then the ~~approximations of~~ primes 2 and 3 ~~remain pretty good~~, and so overall {{map\|17 27 40}} has less error than {{map\|17 27 39}}~~. For more information on this issue, see: [[Douglas_Blumeyer's_RTT_How-To#A_multitude_of_maps]]~~.		The simple map is not necessarily the ''best'' map for its EDO in terms of overall tuning accuracy, but it is the simplest map to calculate. The classic example of a simple map which is not the best map is 17p in the 5-limit, 17·{{map\|log₂2 log₂3 log₂5}} = {{map\|17 26.944 39.473}} which rounds to {{map\|17 27 39}}. The approximation of prime 5 is really bad here; it's about exactly halfway between 39 and 40 steps, but slightly below, which is why it rounds down. But it turns out that if we round up instead, using 40 steps to approximate prime 5, then the absolute errors in the primes remain about the same. However, the error in 5/3 is much less, because the error in 5 and the error in 3 are now in the same direction, canceling each other out, and so overall {{map\|17 27 40}} has less error than {{map\|17 27 39}}.

	== Vs. integer uniform maps ==		== Vs. integer uniform maps ==

	~~Every~~ simple map is ~~also a~~ [[integer uniform map]]~~, and vice versa~~. ~~They are identical objects, but the~~ two different terms provide two different ways of presenting ~~about~~ the same object, which can be helpful in different contexts.		Another name for a simple map is an [[integer uniform map]]. The two different terms provide two different ways of presenting the same object, which can be helpful in different contexts:
			* In contexts pertaining to tuning accuracy, "simple map" works well. This is probably the more common context.
	* In contexts pertaining to tuning accuracy, "simple map" works ~~great~~. This is probably the more common context.		* In contexts pertaining to other uniform maps, "integer uniform map" works well.
	* In contexts pertaining to other uniform maps, "integer uniform map" works ~~great~~.

	[[Category:Temperament]]		[[Category:Temperament]]

Cmloegcmluin: extract from uniform map page

2021-12-12T21:36:36Z

extract from uniform map page

← Older revision		Revision as of 21:36, 12 December 2021
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	#~~REDIRECT~~ [[~~Uniform~~ map~~#Simple~~ map]]		A '''simple map''' is a specific type of [[map]] used for [[EDO]]s. Every EDO has one simple map. The simple map for <math>n</math>-EDO is found by multiplying the [[just intonation point]] (JIP) {{map\|log₂2 log₂3 log₂5 ...}} by <math>n</math>, then individually rounding entries to the nearest integer. For example, the simple map for 7-limit 19-EDO is 19·{{map\|log₂2 log₂3 log₂5 log₂7}} = {{map\|19 30.115 44.117 53.340}} which rounds to {{map\|19 30 44 53}}.

			The simple map is not necessarily the ''best'' map for its EDO in terms of overall tuning accuracy, but it is the simplest map to calculate. The classic example of a simple map which is not the best map is 17p in the 5-limit, 17·{{map\|log₂2 log₂3 log₂5}} = {{map\|17 26.944 39.473}} which rounds to {{map\|17 27 39}}. We can see that the approximation of prime 5 is really bad here; it's about exactly halfway between 39 and 40 steps, but slightly below, which is why it rounded down. It turns out that if we shrink our steps slightly so that it takes 40 steps to approximate 5 instead, then the approximations of primes 2 and 3 remain pretty good, and so overall {{map\|17 27 40}} has less error than {{map\|17 27 39}}. For more information on this issue, see: [[Douglas_Blumeyer's_RTT_How-To#A_multitude_of_maps]].

			== Vs. integer uniform maps ==

			Every simple map is also a [[integer uniform map]], and vice versa. They are identical objects, but the two different terms provide two different ways of presenting about the same object, which can be helpful in different contexts.

			* In contexts pertaining to tuning accuracy, "simple map" works great. This is probably the more common context.
			* In contexts pertaining to other uniform maps, "integer uniform map" works great.

			[[Category:Temperament]]
			[[Category:Regular temperament theory]]
			[[Category:Theory]]
			[[Category:Terms]]
			[[Category:Math]]

Cmloegcmluin: Redirected page to Uniform map#Simple map

2021-12-12T00:24:07Z

Redirected page to Uniform map#Simple map

New page

#REDIRECT [[Uniform map#Simple map]]