Consistent circle - Revision history

Lucius Chiaraviglio: /* Motivation */ Grammar fix

2024-11-29T15:05:24Z

Motivation: Grammar fix

← Older revision		Revision as of 15:05, 29 November 2024
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	== Motivation ==		== Motivation ==
	The circle of fifths/fourths can be confusing to navigate in edos which are not [[telic]] in that a circle of [[3/2]]'s / [[4/3]]'s fails to "close", for example in [[31edo]] where the difference between 31 just fifths and 18 octaves is 415% of a 31edostep. Usually, in such edos, there is present other intervals, such as [[5/4]] and [[7/4]] in the case of 31edo, which are far more accurate and therefore far more reliable for navigation. In the case of 31edo, 5/4 and 7/4 are in fact so accurate that stacking either of them 31 times (and in fact, any combination of them or their [[octave complement]]s 31 times, as long as there isn't more than 31 intervals in total) will keep the result off by less than a 31edostep (meaning they form [[#Weak circle\|''weakly consistent circles'']]), even if the result isn't guaranteed to be [[consistent]] beyond floor(31/2) = 15 moves. (Alternatively, stated, this means the result is guaranteed to be consistent if you stack at most floor(31/2) = 15 of them w.r.t. a starting note.)		The circle of fifths/fourths can be confusing to navigate in edos which are not [[telic]] in that a circle of [[3/2]]'s / [[4/3]]'s fails to "close", for example in [[31edo]] where the difference between 31 just fifths and 18 octaves is 415% of a 31edostep. Usually, in such edos, there are present other intervals, such as [[5/4]] and [[7/4]] in the case of 31edo, which are far more accurate and therefore far more reliable for navigation. In the case of 31edo, 5/4 and 7/4 are in fact so accurate that stacking either of them 31 times (and in fact, any combination of them or their [[octave complement]]s 31 times, as long as there isn't more than 31 intervals in total) will keep the result off by less than a 31edostep (meaning they form [[#Weak circle\|''weakly consistent circles'']]), even if the result isn't guaranteed to be [[consistent]] beyond floor(31/2) = 15 moves. (Alternatively, stated, this means the result is guaranteed to be consistent if you stack at most floor(31/2) = 15 of them w.r.t. a starting note.)

	== Definitions ==		== Definitions ==

Godtone: /* Definitions */ slightly less confusing hopefully

2024-10-10T14:38:57Z

Definitions: slightly less confusing hopefully

← Older revision		Revision as of 14:38, 10 October 2024
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	== Definitions ==		== Definitions ==
	We define a [[consistent circle]] (abbreviatable to just ''circle''{{idiosyncratic}}) of some (usually [[JI]]) [[interval]] ''a''/''b'' as: an interval with such extremely low [[relative error]] with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimum required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual (untempered) size, which is to say it is off by less than 0.5\''N'' = 1200{{cent}} / ''N'' / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the edo if the circle occurs in a subset edo, but that the circle must have low enough error that within the full edo it is still consistent.		We define a [[consistent circle]] (abbreviatable to just ''circle''{{idiosyncratic}}) of some (usually [[JI]]) [[interval]] ''a''/''b'' as: an interval with such extremely low [[relative error]] with respect to ''N''-[[edo]] that when we stack its best approximation ''m'' > 0 times, where ''m'' is the minimum required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual (untempered) size, which is to say it is off by less than 0.5\''N'' = 1200{{cent}} / ''N'' / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the edo if the circle occurs in a subset edo, but that the circle must have low enough error that within the full edo it is still consistent.

	Note that when a/b ''does'' generate all notes of the edo (meaning ''N'' = ''m''), then that means that (''a''/''b'')<sup>''m'' = ''N''</sup> reaches ''m'' = ''N'' octaves. This will always be true in a prime edo, such as 31edo, meaning we can easily deduce that stacking 35/32 31 times gets us at 4 octaves, because 35/32's direct mapping is 4\31. This same reasoning can be applied in general if you think instead in terms of the subset edo generated.		Note that when a/b ''does'' generate all notes of the edo (meaning ''N'' = ''m''), then that means that (''a''/''b'')<sup>''m'' = ''N''</sup> reaches ''m'' = ''N'' octaves. This will always be true in a prime edo, such as 31edo, meaning we can easily deduce that stacking 35/32 31 times gets us at 4 octaves, because 35/32's direct mapping is 4\31. This same reasoning can be applied in general if you think instead in terms of the subset edo generated.

Godtone: /* Motivation */

2024-10-10T14:36:39Z

Motivation

← Older revision		Revision as of 14:36, 10 October 2024
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	== Motivation ==		== Motivation ==
	The circle of fifths/fourths can be confusing to navigate in edos which are not [[telic]] in that a circle of [[3/2]]'s / [[4/3]]'s fails to "close", for example in [[31edo]] where the difference between 31 just fifths and 18 octaves is 415% of a 31edostep. Usually, in such edos, there is present other intervals, such as [[5/4]] and [[7/4]] in the case of 31edo, which are far more accurate and therefore far more reliable for navigation. In the case of 31edo, 5/4 and 7/4 are in fact so accurate that stacking either of them 31 times (and in fact, any combination of them or their [[octave complement]]s 31 times, as long as there isn't more than 31 intervals in total) will keep the result off by less than a 31edostep (meaning they form [[#Weak circle\|''weakly consistent circles'']]), even if the result isn't guaranteed to be [[consistent]] beyond floor(31/2) moves. (Alternatively, stated, this means the result is guaranteed to be consistent if you stack at most floor(31/2) = 15 of them w.r.t. a starting note.)		The circle of fifths/fourths can be confusing to navigate in edos which are not [[telic]] in that a circle of [[3/2]]'s / [[4/3]]'s fails to "close", for example in [[31edo]] where the difference between 31 just fifths and 18 octaves is 415% of a 31edostep. Usually, in such edos, there is present other intervals, such as [[5/4]] and [[7/4]] in the case of 31edo, which are far more accurate and therefore far more reliable for navigation. In the case of 31edo, 5/4 and 7/4 are in fact so accurate that stacking either of them 31 times (and in fact, any combination of them or their [[octave complement]]s 31 times, as long as there isn't more than 31 intervals in total) will keep the result off by less than a 31edostep (meaning they form [[#Weak circle\|''weakly consistent circles'']]), even if the result isn't guaranteed to be [[consistent]] beyond floor(31/2) = 15 moves. (Alternatively, stated, this means the result is guaranteed to be consistent if you stack at most floor(31/2) = 15 of them w.r.t. a starting note.)

	== Definitions ==		== Definitions ==

Eliora: /* See also */ the topic of fractional-8ve temps is directly relevant to such circles

2024-09-08T19:16:53Z

See also: the topic of fractional-8ve temps is directly relevant to such circles

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	* [[Telicity]]		* [[Telicity]]
	* [[Circle of fifths]]		* [[Circle of fifths]]
			* [[Fractional-octave temperaments]]

	[[Category: pages with idiosyncratic terms]]		[[Category: pages with idiosyncratic terms]]

Godtone: update instances of the term to bold correctly as per page move

2024-07-10T20:25:31Z

update instances of the term to bold correctly as per page move

← Older revision		Revision as of 20:25, 10 July 2024
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	Intuitively, a ~~'''consistent'''~~ [[circle]] of ~a/b's in an [[edo]] describes a case where ''a''/''b'' is so accurately approximated that you can rely on navigating with it being [[consistent]] with respect to the "circle of notes" defined. It is closely related to the concept of [[telicity]], except that a circle can be of ''any'' [[JI]] interval, as long as it is [[consistency#mathematical definition\|consistently mapped]] to the unison when octave-reduced. (It also admits generalization to any [[equave]].) Note that the precise term ''consistent circle of ~a/b's'' may be shortened to ''circle of a/b's''{{idiosyncratic}} in writing for the sake of brevity wherever it's unambiguous (or in informal writing).		Intuitively, a [[consistent circle]] of ~a/b's in an [[edo]] describes a case where ''a''/''b'' is so accurately approximated that you can rely on navigating with it being [[consistent]] with respect to the "circle of notes" defined. It is closely related to the concept of [[telicity]], except that a circle can be of ''any'' [[JI]] interval, as long as it is [[consistency#mathematical definition\|consistently mapped]] to the unison when octave-reduced. (It also admits generalization to any [[equave]].) Note that the precise term ''consistent circle of ~a/b's'' may be shortened to ''circle of a/b's''{{idiosyncratic}} in writing for the sake of brevity wherever it's unambiguous (or in informal writing).

	== Motivation ==		== Motivation ==
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	== Definitions ==		== Definitions ==
	We define a ~~'''consistent'''~~ [[circle]] (abbreviatable to just ''circle''{{idiosyncratic}}) of some (usually [[JI]]) [[interval]] ''a''/''b'' as: an interval with such extremely low [[relative error]] with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimum required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual (untempered) size, which is to say it is off by less than 0.5\''N'' = 1200{{cent}} / ''N'' / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the edo if the circle occurs in a subset edo, but that the circle must have low enough error that within the full edo it is still consistent.		We define a [[consistent circle]] (abbreviatable to just ''circle''{{idiosyncratic}}) of some (usually [[JI]]) [[interval]] ''a''/''b'' as: an interval with such extremely low [[relative error]] with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimum required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual (untempered) size, which is to say it is off by less than 0.5\''N'' = 1200{{cent}} / ''N'' / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the edo if the circle occurs in a subset edo, but that the circle must have low enough error that within the full edo it is still consistent.

	Note that when a/b ''does'' generate all notes of the edo (meaning ''N'' = ''m''), then that means that (''a''/''b'')<sup>''m'' = ''N''</sup> reaches ''m'' = ''N'' octaves. This will always be true in a prime edo, such as 31edo, meaning we can easily deduce that stacking 35/32 31 times gets us at 4 octaves, because 35/32's direct mapping is 4\31. This same reasoning can be applied in general if you think instead in terms of the subset edo generated.		Note that when a/b ''does'' generate all notes of the edo (meaning ''N'' = ''m''), then that means that (''a''/''b'')<sup>''m'' = ''N''</sup> reaches ''m'' = ''N'' octaves. This will always be true in a prime edo, such as 31edo, meaning we can easily deduce that stacking 35/32 31 times gets us at 4 octaves, because 35/32's direct mapping is 4\31. This same reasoning can be applied in general if you think instead in terms of the subset edo generated.

FloraC: FloraC moved page Circle to Consistent circle over redirect: Follow the lemma in the intro

2024-07-10T15:47:45Z

FloraC moved page Circle to Consistent circle over redirect: Follow the lemma in the intro

← Older revision	Revision as of 15:47, 10 July 2024
(No difference)

Godtone: /* Having a super-strong circle */ terminological clarification

2024-06-29T20:53:39Z

Having a super-strong circle: terminological clarification

← Older revision		Revision as of 20:53, 29 June 2024
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	(We use this weaker/more generous bound rather than the default bound for closing error because such a circle is already going "above-and-beyond" in terms of what's necessary to produce a consistent circle.)		(We use this weaker/more generous bound rather than the default bound for closing error because such a circle is already going "above-and-beyond" in terms of what's necessary to produce a consistent circle.)

			When a circle satisfies this strongest sense of consistency (which is only possible if the circle does not generate the full edo), we say its consistency is ''super-strong''.

	== Examples ==		== Examples ==

Godtone: /* Weakly consistent circle */ fix link

2024-06-29T20:52:43Z

Weakly consistent circle: fix link

← Older revision		Revision as of 20:52, 29 June 2024
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	This is a much more common type of circle that is still generally reliable for most purposes, hence useful to distinguish.		This is a much more common type of circle that is still generally reliable for most purposes, hence useful to distinguish.

	When a circle satisfies this consistency but no stronger, we say its consistency is ''weak''. If it fails to satisfy this bound but generates a subset, it might qualify for a [[having a sub-weak circle]].		When a circle satisfies this consistency but no stronger, we say its consistency is ''weak''. If it fails to satisfy this bound but generates a subset, it might qualify for [[#having a sub-weak circle]].

	=== Strongly consistent circle ===		=== Strongly consistent circle ===

Godtone: /* Having a sub-weak circle */ terminological clarification

2024-06-29T20:52:09Z

Having a sub-weak circle: terminological clarification

← Older revision		Revision as of 20:52, 29 June 2024
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	This means that if one is satisfied with the circle in the subset edo, one may find it to be sufficiently accurate for navigation with in the larger edo, because of familiarity with it the subset edo.		This means that if one is satisfied with the circle in the subset edo, one may find it to be sufficiently accurate for navigation with in the larger edo, because of familiarity with it the subset edo.

			When a circle satisfies this consistency but no stronger (which is only possible if the circle does not generate the full edo), we say its consistency is ''sub-weak''.

	=== Having a super-strong circle ===		=== Having a super-strong circle ===

Godtone: /* Strongly consistent circle */ increase (improve) size of clickable text

2024-06-29T20:51:11Z

Strongly consistent circle: increase (improve) size of clickable text

← Older revision		Revision as of 20:51, 29 June 2024
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	For example, [[12edo]], [[53edo]] and [[665edo]] are the first three strong circles of ~3/2, given that the remnant is less than a quarter of an edostep in each case.		For example, [[12edo]], [[53edo]] and [[665edo]] are the first three strong circles of ~3/2, given that the remnant is less than a quarter of an edostep in each case.

	When a circle satisfies this consistency but no stronger (which is ~~guaranteed~~ [[#Having a super-strong circle\|if]] it generates the whole edo), we say its consistency is ''strong''.		When a circle satisfies this consistency but no stronger (which is [[#Having a super-strong circle\|guaranteed if]] it generates the whole edo), we say its consistency is ''strong''.

	=== Is vs. has ===		=== Is vs. has ===