Consistent circle: Difference between revisions

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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 ==

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 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 ==

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'')''m'' = ''N'' 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.

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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.

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 ===

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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.

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 ===

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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.

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 ===

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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.)

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 ==

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== Comparing with telicity ==

At first glance, it would appear that the concept of telicity and having a circle are identical, however they are not upon closer inspection of their definitions: a circle concerns any rational interval with respect to closing at some ''equave'', while telicity (usually) concerns primes. (The case where telicity does not refer to primes is dealt with in [[#Vs. subgroup telicity]].) This means that "closure" is usually concerning being closed w.r.t. a psychoacoustic equave — by default the [[octave]] — while telicity allows closing w.r.t. any prime ''up to octave-reduction'', so is ~~conceptualised~~ differently, because the target at which the circle is closed is no longer a specific equave. In other words, consistent circles concern closure of some rational w.r.t. the equave while telicity concerns reliability of connection between generators.

At first glance, it would appear that the concept of telicity and having a circle are identical, however they are not upon closer inspection of their definitions: a circle concerns any rational interval with respect to closing at some ''equave'', while telicity (usually) concerns primes. (The case where telicity does not refer to primes is dealt with in [[#Vs. subgroup telicity]].) This means that "closure" is usually concerning being closed w.r.t. a psychoacoustic equave — by default the [[octave]] — while telicity allows closing w.r.t. any prime ''up to octave-reduction'', so is conceptualized differently, because the target at which the circle is closed is no longer a specific equave. In other words, consistent circles concern closure of some rational w.r.t. the equave while telicity concerns reliability of connection between generators.

Another key difference is that ''being telic'' is more strict a requirement than ''having a consistent circle of some kind''; an edo can [[#have a sub-weak circle]] without qualifying for even 0.5-strong 2-a/b telicity (which would usually not be considered as being telic), or it can [[#have a weak circle]] without qualifying for 1-strong 2-a/b telicity (again not usually considered as telic), because these do not require reliability of the full circle, but rather a weaker sense of reliability that ~~nonetheless~~ is sufficient for many of its practical applications.

Another key difference is that ''being telic'' is often a more strict requirement than ''having a consistent circle of some kind''; an edo can [[#having a sub-weak circle|have a sub-weak circle]] without qualifying for even 0.5-strong 2-a/b telicity (which would usually not be considered as being telic anyways), or it can [[#have a weak circle|having a weak circle]] without qualifying for 1-strong 2-a/b telicity (again not usually considered as being telic), because these do not require reliability of the full circle, but rather a weaker sense of reliability that is sufficient for many of its practical/musical applications.

~~Further~~, [[#having a super-strong circle]] is arbitrarily stricter than 2-a/b telicity, which means that in general, it corresponds to ''s''-strong 2-a/b telicity, with ''s'' = GCD(''N'', round(''N'' log2(a/b))) / 2.

Furthermore, [[#having a super-strong circle]] is arbitrarily stricter than 2-a/b telicity, which means that in general, it corresponds to ''s''-strong 2-a/b telicity, with ''s'' = GCD(''N'', round(''N'' log2(a/b))) / 2.

=== Vs. subgroup telicity ===

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* [[Telicity]]

* [[Circle of fifths]]

* [[Fractional-octave temperaments]]

[[Category: pages with idiosyncratic terms]]

@@ Line 1: / Line 1: @@
-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 ==
-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 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 ==
-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.
@@ Line 28: / Line 28: @@
 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 ===
@@ Line 37: / Line 37: @@
 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 ===
@@ Line 50: / Line 50: @@
 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 ===
@@ Line 57: / Line 59: @@
 (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 ==
@@ Line 64: / Line 68: @@
 == Comparing with telicity ==
-At first glance, it would appear that the concept of telicity and having a circle are identical, however they are not upon closer inspection of their definitions: a circle concerns any rational interval with respect to closing at some ''equave'', while telicity (usually) concerns primes. (The case where telicity does not refer to primes is dealt with in [[#Vs. subgroup telicity]].) This means that "closure" is usually concerning being closed w.r.t. a psychoacoustic equave — by default the [[octave]] — while telicity allows closing w.r.t. any prime ''up to octave-reduction'', so is conceptualised differently, because the target at which the circle is closed is no longer a specific equave. In other words, consistent circles concern closure of some rational w.r.t. the equave while telicity concerns reliability of connection between generators.
+At first glance, it would appear that the concept of telicity and having a circle are identical, however they are not upon closer inspection of their definitions: a circle concerns any rational interval with respect to closing at some ''equave'', while telicity (usually) concerns primes. (The case where telicity does not refer to primes is dealt with in [[#Vs. subgroup telicity]].) This means that "closure" is usually concerning being closed w.r.t. a psychoacoustic equave — by default the [[octave]] — while telicity allows closing w.r.t. any prime ''up to octave-reduction'', so is conceptualized differently, because the target at which the circle is closed is no longer a specific equave. In other words, consistent circles concern closure of some rational w.r.t. the equave while telicity concerns reliability of connection between generators.
-Another key difference is that ''being telic'' is more strict a requirement than ''having a consistent circle of some kind''; an edo can [[#have a sub-weak circle]] without qualifying for even 0.5-strong 2-a/b telicity (which would usually not be considered as being telic), or it can [[#have a weak circle]] without qualifying for 1-strong 2-a/b telicity (again not usually considered as telic), because these do not require reliability of the full circle, but rather a weaker sense of reliability that nonetheless is sufficient for many of its practical applications.
+Another key difference is that ''being telic'' is often a more strict requirement than ''having a consistent circle of some kind''; an edo can [[#having a sub-weak circle|have a sub-weak circle]] without qualifying for even 0.5-strong 2-a/b telicity (which would usually not be considered as being telic anyways), or it can [[#have a weak circle|having a weak circle]] without qualifying for 1-strong 2-a/b telicity (again not usually considered as being telic), because these do not require reliability of the full circle, but rather a weaker sense of reliability that is sufficient for many of its practical/musical applications.
-Further, [[#having a super-strong circle]] is arbitrarily stricter than 2-a/b telicity, which means that in general, it corresponds to ''s''-strong 2-a/b telicity, with ''s'' = GCD(''N'', round(''N'' log<sub>2</sub>(a/b))) / 2.
+Furthermore, [[#having a super-strong circle]] is arbitrarily stricter than 2-a/b telicity, which means that in general, it corresponds to ''s''-strong 2-a/b telicity, with ''s'' = GCD(''N'', round(''N'' log<sub>2</sub>(a/b))) / 2.
 === Vs. subgroup telicity ===
@@ Line 78: / Line 82: @@
 * [[Telicity]]
 * [[Circle of fifths]]
+* [[Fractional-octave temperaments]]
 [[Category: pages with idiosyncratic terms]]