Uniform map - Revision history

FloraC: /* Integer uniform map, or simple map */ capitalization

2025-04-01T11:07:23Z

Integer uniform map, or simple map: capitalization

← Older revision		Revision as of 11:07, 1 April 2025
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	== Integer uniform map, or simple map ==		== Integer uniform map, or simple map ==
	{{Main\| ~~patent~~ val }}		{{Main\| Patent val }}

	A uniform map whose multiplier is an integer is called an '''integer uniform map'''. For example, we could use the integer 17 itself directly. So {{nowrap\|17{{dot}}{{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 }} {{=}} {{map\| 17 26.944 39.473 }}}}, which also rounds to {{map\| 17 27 39 }}. Every edo has one integer uniform map.		A uniform map whose multiplier is an integer is called an '''integer uniform map'''. For example, we could use the integer 17 itself directly. So {{nowrap\|17{{dot}}{{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 }} {{=}} {{map\| 17 26.944 39.473 }}}}, which also rounds to {{map\| 17 27 39 }}. Every edo has one integer uniform map.

ArrowHead294 at 13:20, 31 March 2025

2025-03-31T13:20:25Z

← Older revision		Revision as of 13:20, 31 March 2025
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	Every [[edo]] has a finite number of possible uniform maps within a given prime limit, but an infinite number of uniform maps if no prime limit is given.		Every [[edo]] has a finite number of possible uniform maps within a given prime limit, but an infinite number of uniform maps if no prime limit is given.

	To find a uniform map for some ''n''-edo, choose any multiplier that rounds to ''n'' (anywhere from {{nowrap\|''n'' − 0.5}} to {{nowrap\|''n'' + 0.5}}). For example, 16.9 rounds to 17, so we could use that to find another example for 17edo. Uniformly multiplying 16.9 · {{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 }} = {{map\| 16.9 26.786 39.241 }}, which rounds to {{map\| 17 27 39 }}.		To find a uniform map for some ''n''-edo, choose any multiplier that rounds to ''n'' (anywhere from {{nowrap\|''n'' − 0.5}} to {{nowrap\|''n'' + 0.5}}). For example, 16.9 rounds to 17, so we could use that to find another example for 17edo. Uniformly multiplying {{nowrap\|16.9{{dot}}{{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 }} {{=}} {{map\| 16.9 26.786 39.241 }}}}, which rounds to {{map\| 17 27 39 }}.

	Subgroup uniform maps are also possible. For example, a uniform map on the 2.5.9/7 subgroup would be found by uniformly multiplying {{map\|log<sub>2</sub>2 log<sub>2</sub>5 log<sub>2</sub>(9/7)}}.		Subgroup uniform maps are also possible. For example, a uniform map on the 2.5.9/7 subgroup would be found by uniformly multiplying {{map\|log<sub>2</sub>2 log<sub>2</sub>5 log<sub>2</sub>(9/7)}}.

ArrowHead294 at 13:19, 31 March 2025

2025-03-31T13:19:53Z

← Older revision		Revision as of 13:19, 31 March 2025
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			[[File:Uniform maps.png\|thumb\|right\|500px\|This visualizes edo maps. We could call it the uniform map continuum. Generator size decreases to the right, so the edo number goes up. Each colored cell is the closest approximation of steps to a prime for that generator size. We can see uniform maps as any set of step counts for primes found as perfectly vertical lines drawn through such a diagram. And integer uniform maps would be any such vertical line that is also drawn straight through a number which appears in the row for ''n''-edo.]]
	A '''uniform map''' is any [[val\|map]] found by ''uniformly'' multiplying every entry of the [[just intonation point]] (JIP) for some prime limit ''p'', {{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, by the same positive real number before rounding them to integers.		A '''uniform map''' is any [[val\|map]] found by ''uniformly'' multiplying every entry of the [[just intonation point]] (JIP) for some prime limit ''p'', {{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, by the same positive real number before rounding them to integers.

	[[File:Uniform maps.png\|thumb\|right\|500px\|This visualizes edo maps. We could call it the uniform map continuum. Generator size decreases to the right, so the edo number goes up. Each colored cell is the closest approximation of steps to a prime for that generator size. We can see uniform maps as any set of step counts for primes found as perfectly vertical lines drawn through such a diagram. And integer uniform maps would be any such vertical line that is also drawn straight through a number which appears in the row for ''n''-edo.]]

	What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found in [[Patent val #Generalized patent val]]). The map with the overall best tuning accuracy for an edo will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ET sequence]]s given for many [[regular temperament]]s across the wiki.		What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found in [[Patent val #Generalized patent val]]). The map with the overall best tuning accuracy for an edo will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ET sequence]]s given for many [[regular temperament]]s across the wiki.

	For an example, choosing 17.1 as our uniform multiplier, we find the map 17.1 · {{map\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5}} = {{map\|17.1 27.103 39.705}} which rounds to {{map\| 17 27 40 }}. This is one of the many uniform maps for 17edo.		For an example, choosing 17.1 as our uniform multiplier, we find the map {{nowrap\|17.1{{dot}}{{map\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5}} {{=}} {{map\|17.1 27.103 39.705}}}} which rounds to {{map\| 17 27 40 }}. This is one of the many uniform maps for 17edo.

	Every [[edo]] has a finite number of possible uniform maps within a given prime limit, but an infinite number of uniform maps if no prime limit is given.		Every [[edo]] has a finite number of possible uniform maps within a given prime limit, but an infinite number of uniform maps if no prime limit is given.
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	{{Main\| patent val }}		{{Main\| patent val }}

	A uniform map whose multiplier is an integer is called an '''integer uniform map'''. For example, we could use the integer 17 itself directly. So 17 · {{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 }} = {{map\| 17 26.944 39.473 }}, which also rounds to {{map\| 17 27 39 }}. Every edo has one integer uniform map.		A uniform map whose multiplier is an integer is called an '''integer uniform map'''. For example, we could use the integer 17 itself directly. So {{nowrap\|17{{dot}}{{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 }} {{=}} {{map\| 17 26.944 39.473 }}}}, which also rounds to {{map\| 17 27 39 }}. Every edo has one integer uniform map.

	Another name for an integer uniform map is a '''simple map'''. The two different terms provide two different ways of presenting the same object, which can be helpful in different contexts:		Another name for an integer uniform map is a '''simple map'''. The two different terms provide two different ways of presenting the same object, which can be helpful in different contexts:
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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.

	To define ''simple map'' independently of ''integer uniform map'', it is a specific type of map used for [[edo]]s. Every edo has one simple map. The simple map for ''n''-edo is found by taking the [[just intonation point]] (JIP) for some prime limit ''p'', {{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, multiplying it by ''n'', then individually rounding entries to the nearest integer. For example, the simple map for 7-limit 19edo is ~~19·~~{{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 log<sub>2</sub>7 }} = {{map\| 19 30.115 44.117 53.340 }} which rounds to {{map\| 19 30 44 53 }}.		To define ''simple map'' independently of ''integer uniform map'', it is a specific type of map used for [[edo]]s. Every edo has one simple map. The simple map for ''n''-edo is found by taking the [[just intonation point]] (JIP) for some prime limit ''p'', {{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, multiplying it by ''n'', then individually rounding entries to the nearest integer. For example, the simple map for 7-limit 19edo is {{nowrap\|19{{dot}}{{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 log<sub>2</sub>7 }} {{=}} {{map\| 19 30.115 44.117 53.340 }}}} which rounds to {{map\| 19 30 44 53 }}.

	So 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\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 }} = {{map\| 17 26.944 39.473 }} which rounds to {{map\| 17 27 39 }}. The approximation of prime 5 is really bad here; it is 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 }}.		So 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, {{nowrap\|17{{dot}}{{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 }} {{=}} {{map\| 17 26.944 39.473 }}}} which rounds to {{map\| 17 27 39 }}. The approximation of prime 5 is really bad here; it is 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 }}.

	== Terminology ==		== Terminology ==

FloraC: Consolidate sections; misc. cleanup and style improvements

2025-03-31T08:38:24Z

Consolidate sections; misc. cleanup and style improvements

← Older revision		Revision as of 08:38, 31 March 2025
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	A '''uniform map''' is any [[map]] found by ''uniformly'' multiplying every entry of the [[just intonation point]] (JIP) for some prime limit ~~<math>~~p~~</math>~~, {{map\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 ~~...~~ log<sub>2</sub>~~<math>~~p~~</math>~~}}, by the same positive real number before rounding them to integers.		A '''uniform map''' is any [[val\|map]] found by ''uniformly'' multiplying every entry of the [[just intonation point]] (JIP) for some prime limit ''p'', {{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, by the same positive real number before rounding them to integers.

	[[File:Uniform maps.png\|thumb\|right\|500px\|This visualizes ~~EDO~~ maps. We could call it the uniform map continuum. Generator size decreases to the right, so the ~~EDO~~ number goes up. Each colored cell is the closest approximation of steps to a prime for that generator size. We can see uniform maps as any set of step counts for primes found as perfectly vertical lines drawn through such a diagram. And integer uniform maps would be any such vertical line that is also drawn straight through a number which appears in the row for n-~~ED2~~.]]		[[File:Uniform maps.png\|thumb\|right\|500px\|This visualizes edo maps. We could call it the uniform map continuum. Generator size decreases to the right, so the edo number goes up. Each colored cell is the closest approximation of steps to a prime for that generator size. We can see uniform maps as any set of step counts for primes found as perfectly vertical lines drawn through such a diagram. And integer uniform maps would be any such vertical line that is also drawn straight through a number which appears in the row for ''n''-edo.]]

	What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found [[~~Patent_val~~#~~Generalized_patent_val\|here~~]]). The map with the overall best tuning accuracy for an ~~EDO~~ will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ET sequence]]s given for many [[regular temperament]]s across the wiki.		What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found in [[Patent val #Generalized patent val]]). The map with the overall best tuning accuracy for an edo will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ET sequence]]s given for many [[regular temperament]]s across the wiki.

	For an example, choosing 17.1 as our uniform multiplier, we find the map ~~{{nowrap\|~~17.1 · }}{{map\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5}} = {{map\|17.1 27.103 39.705}} which rounds to {{map\|17 27 40}}. This is one of the many uniform maps for ~~17-EDO~~.		For an example, choosing 17.1 as our uniform multiplier, we find the map 17.1 · {{map\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5}} = {{map\|17.1 27.103 39.705}} which rounds to {{map\| 17 27 40 }}. This is one of the many uniform maps for 17edo.

	Every [[~~EDO~~]] has a finite number of possible uniform maps within a given prime limit, but an infinite number of uniform maps if no prime limit is given.		Every [[edo]] has a finite number of possible uniform maps within a given prime limit, but an infinite number of uniform maps if no prime limit is given.

	To find a uniform map for some ~~<math>~~n~~</math>~~-~~EDO~~, choose any multiplier that rounds to ~~<math>~~n~~</math>~~ (anywhere from {{nowrap\|n ~~−~~ 0.5~~</math>~~}} to {{nowrap\|n + 0.5}}). For example, 16.9 rounds to 17, so we could use that to find another example for ~~17-EDO~~. Uniformly multiplying 16.9·{{map\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5}} = {{map\|16.9 26.786 39.241}}, which rounds to {{map\|17 27 39}}.		To find a uniform map for some ''n''-edo, choose any multiplier that rounds to ''n'' (anywhere from {{nowrap\|''n'' − 0.5}} to {{nowrap\|''n'' + 0.5}}). For example, 16.9 rounds to 17, so we could use that to find another example for 17edo. Uniformly multiplying 16.9 · {{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 }} = {{map\| 16.9 26.786 39.241 }}, which rounds to {{map\| 17 27 39 }}.

	Subgroup uniform maps are also possible. For example, a uniform map on the 2.5.9/7 subgroup would be found by uniformly multiplying {{map\|log<sub>2</sub>2 log<sub>2</sub>5 log<sub>2</sub>(9/7)}}.		Subgroup uniform maps are also possible. For example, a uniform map on the 2.5.9/7 subgroup would be found by uniformly multiplying {{map\|log<sub>2</sub>2 log<sub>2</sub>5 log<sub>2</sub>(9/7)}}.

	== Integer uniform map ==		== Integer uniform map, or simple map ==
	A uniform map whose multiplier is an integer is called an '''integer uniform map'''. For example, we could use the integer 17 itself directly. So 17 · {{map\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5}} = {{map\|17 26.944 39.473}}, which also rounds to {{map\|17 27 39}}. Every ~~EDO~~ has one integer uniform map.		{{Main\| patent val }}

			A uniform map whose multiplier is an integer is called an '''integer uniform map'''. For example, we could use the integer 17 itself directly. So 17 · {{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 }} = {{map\| 17 26.944 39.473 }}, which also rounds to {{map\| 17 27 39 }}. Every edo has one integer uniform map.

	~~== Simple map ==~~
	Another name for an integer uniform map is a '''simple map'''. The two different terms provide two different ways of presenting the same object, which can be helpful in different contexts:		Another name for an integer uniform map is a '''simple 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 well. 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 well.

	To define "simple map" independently of "integer uniform map", it 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\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 ... log<sub>2</sub><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\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 log<sub>2</sub>7}} = {{map\|19 30.115 44.117 53.340}} which rounds to {{map\|19 30 44 53}}.

	~~So 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\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5}} ~~= {{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~~}}.		To define ''simple map'' independently of ''integer uniform map'', it is a specific type of map used for [[edo]]s. Every edo has one simple map. The simple map for ''n''-edo is found by taking the [[just intonation point]] (JIP) for some prime limit ''p'', {{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, multiplying it by ''n'', then individually rounding entries to the nearest integer. For example, the simple map for 7-limit 19edo is 19·{{map\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 log<sub>2</sub>7 }} = {{map\| 19 30.115 44.117 53.340 }} which rounds to {{map\| 19 30 44 53 }}.

	=~~= Vs~~. ~~related terminology ==~~		So 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\| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 }} = {{map\| 17 26.944 39.473 }} which rounds to {{map\| 17 27 39 }}. The approximation of prime 5 is really bad here; it is 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 }}.
	~~A uniform~~ map is the same ~~thing as a [[generalized patent val]]~~, ~~or GPV~~.

	An integer uniform map or simple map is the same thing as a [[patent val]].		== Terminology ==
			A uniform map is the same thing as a [[generalized patent val]], or GPV. An integer uniform map or simple map is the same thing as a [[patent val]].

	As for the difference between "map" and "val", there is none, at least in most [[RTT]] cases; this is discussed here: [[Map]].		As for the difference between ''map'' and ''val'', there is none, at least in most [[RTT]] cases; this is discussed here: [[Map]].

	Otherwise, the difference in terminology between uniform maps, integer uniform maps, patent vals, and GPVs, reflects an inversion in conceptual framing. For patent vals and GPVs, patent vals are considered the base case, and GPVs a generalization thereof, whereas for uniform maps and integer uniform maps, uniform maps are considered the base case and integer uniform maps a specialization thereof. There is an argument that uniform maps are the more fundamental and important concept to regular temperament theory and therefore that this framing is the superior of the two.		Otherwise, the difference in terminology between uniform maps, integer uniform maps, patent vals, and GPVs, reflects an inversion in conceptual framing. For patent vals and GPVs, patent vals are considered the base case, and GPVs a generalization thereof, whereas for uniform maps and integer uniform maps, uniform maps are considered the base case and integer uniform maps a specialization thereof. There is an argument that uniform maps are the more fundamental and important concept to regular temperament theory and therefore that this framing is the superior of the two.

ArrowHead294 at 16:52, 7 October 2024

2024-10-07T16:52:33Z

Cmloegcmluin: /* Simple map */

2024-08-13T14:48:29Z

Simple map

← Older revision		Revision as of 14:48, 13 August 2024
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	== Simple map ==		== Simple map ==

	Another name for an [[integer uniform map]] is a '''simple map'''. The two different terms provide two different ways of presenting the same object, which can be helpful in different contexts:		Another name for an integer uniform map is a '''simple 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 well. 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 well.

Cmloegcmluin: "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

2023-05-07T17:51:58Z

"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

← Older revision		Revision as of 17:51, 7 May 2023
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	[[File:Uniform maps.png\|thumb\|right\|500px\|This visualizes EDO maps. We could call it the uniform map continuum. Generator size decreases to the right, so the EDO number goes up. Each colored cell is the closest approximation of steps to a prime for that generator size. We can see uniform maps as any set of step counts for primes found as perfectly vertical lines drawn through such a diagram. And integer uniform maps would be any such vertical line that is also drawn straight through a number which appears in the row for n-ED2.]]		[[File:Uniform maps.png\|thumb\|right\|500px\|This visualizes EDO maps. We could call it the uniform map continuum. Generator size decreases to the right, so the EDO number goes up. Each colored cell is the closest approximation of steps to a prime for that generator size. We can see uniform maps as any set of step counts for primes found as perfectly vertical lines drawn through such a diagram. And integer uniform maps would be any such vertical line that is also drawn straight through a number which appears in the row for n-ED2.]]

	What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found [[Patent_val#Generalized_patent_val\|here]]). The map with the overall best tuning accuracy for an EDO will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ~~GPV~~ sequence]]s given for many [[regular temperament]]s across the wiki.		What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found [[Patent_val#Generalized_patent_val\|here]]). The map with the overall best tuning accuracy for an EDO will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ET sequence]]s given for many [[regular temperament]]s across the wiki.

	For an example, choosing 17.1 as our uniform multiplier, we find the map 17.1·{{map\|log₂2 log₂3 log₂5}} = {{map\|17.1 27.103 39.705}} which rounds to {{map\|17 27 40}}. This is one of the many uniform maps for 17-EDO.		For an example, choosing 17.1 as our uniform multiplier, we find the map 17.1·{{map\|log₂2 log₂3 log₂5}} = {{map\|17.1 27.103 39.705}} which rounds to {{map\|17 27 40}}. This is one of the many uniform maps for 17-EDO.

Cmloegcmluin: /* Simple map */ include a definition independent of integer uniform map, for the benefit of those who may be being redirected here

2021-12-29T16:23:13Z

Simple map: include a definition independent of integer uniform map, for the benefit of those who may be being redirected here

← Older revision		Revision as of 16:23, 29 December 2021
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	* 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 well. 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 well.

			To define "simple map" independently of "integer uniform map", it 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}}.

	So 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}}.		So 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}}.

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

2021-12-29T16:19:37Z

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

← Older revision		Revision as of 16:19, 29 December 2021
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	== Integer uniform map ==		== Integer uniform map ==

	A uniform map whose multiplier is an integer is called an '''integer uniform map'''~~, or [[simple map]]~~. For example, we could use the integer 17 itself directly. So 17·{{map\|log₂2 log₂3 log₂5}} = {{map\|17 26.944 39.473}}, which also rounds to {{map\|17 27 39}}.		A uniform map whose multiplier is an integer is called an '''integer uniform map'''. For example, we could use the integer 17 itself directly. So 17·{{map\|log₂2 log₂3 log₂5}} = {{map\|17 26.944 39.473}}, which also rounds to {{map\|17 27 39}}. Every EDO has one integer uniform map.

			== Simple map ==

			Another name for an [[integer uniform map]] is a '''simple 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.

			So 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. related terminology ==		== Vs. related terminology ==
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	A uniform map is the same thing as a [[generalized patent val]], or GPV.		A uniform map is the same thing as a [[generalized patent val]], or GPV.

	An integer uniform map is the same thing as a [[patent val]].		An integer uniform map or simple map is the same thing as a [[patent val]].

	As for the difference between "map" and "val", there is none, at least in most [[RTT]] cases; this is discussed here: [[Map]].		As for the difference between "map" and "val", there is none, at least in most [[RTT]] cases; this is discussed here: [[Map]].

Cmloegcmluin at 13:38, 25 December 2021

2021-12-25T13:38:43Z

← Older revision		Revision as of 13:38, 25 December 2021
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	Otherwise, the difference in terminology between uniform maps, integer uniform maps, patent vals, and GPVs, reflects an inversion in conceptual framing. For patent vals and GPVs, patent vals are considered the base case, and GPVs a generalization thereof, whereas for uniform maps and integer uniform maps, uniform maps are considered the base case and integer uniform maps a specialization thereof. There is an argument that uniform maps are the more fundamental and important concept to regular temperament theory and therefore that this framing is the superior of the two.		Otherwise, the difference in terminology between uniform maps, integer uniform maps, patent vals, and GPVs, reflects an inversion in conceptual framing. For patent vals and GPVs, patent vals are considered the base case, and GPVs a generalization thereof, whereas for uniform maps and integer uniform maps, uniform maps are considered the base case and integer uniform maps a specialization thereof. There is an argument that uniform maps are the more fundamental and important concept to regular temperament theory and therefore that this framing is the superior of the two.

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

← Older revision		Revision as of 17:51, 7 May 2023
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	[[File:Uniform maps.png\|thumb\|right\|500px\|This visualizes EDO maps. We could call it the uniform map continuum. Generator size decreases to the right, so the EDO number goes up. Each colored cell is the closest approximation of steps to a prime for that generator size. We can see uniform maps as any set of step counts for primes found as perfectly vertical lines drawn through such a diagram. And integer uniform maps would be any such vertical line that is also drawn straight through a number which appears in the row for n-ED2.]]		[[File:Uniform maps.png\|thumb\|right\|500px\|This visualizes EDO maps. We could call it the uniform map continuum. Generator size decreases to the right, so the EDO number goes up. Each colored cell is the closest approximation of steps to a prime for that generator size. We can see uniform maps as any set of step counts for primes found as perfectly vertical lines drawn through such a diagram. And integer uniform maps would be any such vertical line that is also drawn straight through a number which appears in the row for n-ED2.]]

	What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found [[Patent_val#Generalized_patent_val\|here]]). The map with the overall best tuning accuracy for an EDO will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ~~GPV~~ sequence]]s given for many [[regular temperament]]s across the wiki.		What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found [[Patent_val#Generalized_patent_val\|here]]). The map with the overall best tuning accuracy for an EDO will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ET sequence]]s given for many [[regular temperament]]s across the wiki.

	For an example, choosing 17.1 as our uniform multiplier, we find the map 17.1·{{map\|log₂2 log₂3 log₂5}} = {{map\|17.1 27.103 39.705}} which rounds to {{map\|17 27 40}}. This is one of the many uniform maps for 17-EDO.		For an example, choosing 17.1 as our uniform multiplier, we find the map 17.1·{{map\|log₂2 log₂3 log₂5}} = {{map\|17.1 27.103 39.705}} which rounds to {{map\|17 27 40}}. This is one of the many uniform maps for 17-EDO.

← Older revision		Revision as of 16:52, 7 October 2024
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	A '''uniform map''' is any [[map]] found by ''uniformly'' multiplying every entry of the [[just intonation point]] (JIP) for some prime limit <math>p</math>, {{map\|~~log₂2 log₂3 log₂5~~ ... ~~log₂~~<math>p</math>}}, by the same positive real number before rounding them to integers.		A '''uniform map''' is any [[map]] found by ''uniformly'' multiplying every entry of the [[just intonation point]] (JIP) for some prime limit <math>p</math>, {{map\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 ... log<sub>2</sub><math>p</math>}}, by the same positive real number before rounding them to integers.

	[[File:Uniform maps.png\|thumb\|right\|500px\|This visualizes EDO maps. We could call it the uniform map continuum. Generator size decreases to the right, so the EDO number goes up. Each colored cell is the closest approximation of steps to a prime for that generator size. We can see uniform maps as any set of step counts for primes found as perfectly vertical lines drawn through such a diagram. And integer uniform maps would be any such vertical line that is also drawn straight through a number which appears in the row for n-ED2.]]		[[File:Uniform maps.png\|thumb\|right\|500px\|This visualizes EDO maps. We could call it the uniform map continuum. Generator size decreases to the right, so the EDO number goes up. Each colored cell is the closest approximation of steps to a prime for that generator size. We can see uniform maps as any set of step counts for primes found as perfectly vertical lines drawn through such a diagram. And integer uniform maps would be any such vertical line that is also drawn straight through a number which appears in the row for n-ED2.]]
Line 5:		Line 5:
	What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found [[Patent_val#Generalized_patent_val\|here]]). The map with the overall best tuning accuracy for an EDO will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ET sequence]]s given for many [[regular temperament]]s across the wiki.		What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found [[Patent_val#Generalized_patent_val\|here]]). The map with the overall best tuning accuracy for an EDO will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ET sequence]]s given for many [[regular temperament]]s across the wiki.

	For an example, choosing 17.1 as our uniform multiplier, we find the map 17.1·{{map\|~~log₂2 log₂3 log₂5~~}} = {{map\|17.1 27.103 39.705}} which rounds to {{map\|17 27 40}}. This is one of the many uniform maps for 17-EDO.		For an example, choosing 17.1 as our uniform multiplier, we find the map {{nowrap\|17.1 · }}{{map\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5}} = {{map\|17.1 27.103 39.705}} which rounds to {{map\|17 27 40}}. This is one of the many uniform maps for 17-EDO.

	Every [[EDO]] has a finite number of possible uniform maps within a given prime limit, but an infinite number of uniform maps if no prime limit is given.		Every [[EDO]] has a finite number of possible uniform maps within a given prime limit, but an infinite number of uniform maps if no prime limit is given.

	To find a uniform map for some <math>n</math>-EDO, choose any multiplier that rounds to <math>n</math> (anywhere from ~~<math>~~n - 0.5</math> to ~~<math>~~n + 0.5~~</math>~~). For example, 16.9 rounds to 17, so we could use that to find another example for 17-EDO. Uniformly multiplying 16.9·{{map\|~~log₂2 log₂3 log₂5~~}} = {{map\|16.9 26.786 39.241}}, which rounds to {{map\|17 27 39}}.		To find a uniform map for some <math>n</math>-EDO, choose any multiplier that rounds to <math>n</math> (anywhere from {{nowrap\|n − 0.5</math>}} to {{nowrap\|n + 0.5}}). For example, 16.9 rounds to 17, so we could use that to find another example for 17-EDO. Uniformly multiplying 16.9·{{map\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5}} = {{map\|16.9 26.786 39.241}}, which rounds to {{map\|17 27 39}}.

	Subgroup uniform maps are also possible. For example, a uniform map on the 2.5.9/7 subgroup would be found by uniformly multiplying {{map\|~~log₂2 log₂5 log₂~~(9/7)}}.		Subgroup uniform maps are also possible. For example, a uniform map on the 2.5.9/7 subgroup would be found by uniformly multiplying {{map\|log<sub>2</sub>2 log<sub>2</sub>5 log<sub>2</sub>(9/7)}}.

	== Integer uniform map ==		== Integer uniform map ==
			A uniform map whose multiplier is an integer is called an '''integer uniform map'''. For example, we could use the integer 17 itself directly. So 17 · {{map\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5}} = {{map\|17 26.944 39.473}}, which also rounds to {{map\|17 27 39}}. Every EDO has one integer uniform map.
	A uniform map whose multiplier is an integer is called an '''integer uniform map'''. For example, we could use the integer 17 itself directly. So ~~17·~~{{map\|~~log₂2 log₂3 log₂5~~}} = {{map\|17 26.944 39.473}}, which also rounds to {{map\|17 27 39}}. Every EDO has one integer uniform map.

	== Simple map ==		== Simple map ==

	Another name for an integer uniform map is a '''simple map'''. The two different terms provide two different ways of presenting the same object, which can be helpful in different contexts:		Another name for an integer uniform map is a '''simple 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 well. 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 well.

	To define "simple map" independently of "integer uniform map", it 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}}.		To define "simple map" independently of "integer uniform map", it 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\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 ... log<sub>2</sub><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\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 log<sub>2</sub>7}} = {{map\|19 30.115 44.117 53.340}} which rounds to {{map\|19 30 44 53}}.

	So 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}}.		So 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\|log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5}} = {{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. related terminology ==		== Vs. related terminology ==

	A uniform map is the same thing as a [[generalized patent val]], or GPV.		A uniform map is the same thing as a [[generalized patent val]], or GPV.