Consistent circle: Difference between revisions

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Another example from before is that 31edo is a weak circle of ~5/4's and ~7/4's, but note that 31edo ''is'' a circle of ~[[35/32]]'s (meaning that 31edo is a (strong) circle of [[septimal neutral second]]s), where 35/32 = (5/4)/(8/7) = 5/4 * 7/4 / 2.

== 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 (hence any [[equave]]) ''up to octave-reduction'', so is conceptualised differently. 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 ''k''-strong telicity is often more strict than a consistent circle; 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 qualifying), or it can [[have a weak circle]] without qualifying for 1-strong 2-a/b telicity (which again would usually not qualify), 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.

=== Vs. subgroup telicity ===

There is an application of telicity that concerns rationals in general — [[Telicity#Telicity on subgroups|subgroup telicity]] — which can be seen as roughly equivalent to using one of the generators as an equave and the other as the interval a/b, in which case a ''consistent circle'' refers to 1-strong pairwise telicity iff the circle generates all notes w.r.t. [[Octave reduction#Generalizations|equave-reduction]], but notice this "iff". Similarly, iff the circle generates all notes as mentioned prior, then qualifying for 2-strong pairwise telicity is equivalent to having a strongly consistent circle of a/b's w.r.t. the equave, and qualifying for 0.5-strong pairwise telicity to having a weakly consistent circle of a/b's w.r.t. the equave. But notice how wordy and technical it sounds in the case where the concepts overlap - hence using the concept of "circle" might be preferred as more easily understood in the case where it's applicable, and needs to be used in inapplicable cases where the definition of telicity is too strict or otherwise too niche.

Consider, for example, that "the only type of telicity available to the 3-prime is 3-2 telicity" - this is in a sense true for consistent circles too, iff you assume your [[equave]] is the [[octave]], but otherwise not. In other words, consistent circles refer to less strict types of a/b-2 telicity in the case where the equave is the octave, and to less strict types of a/b-E telicity if E is some alternative equave.

== See also ==

@@ Line 62: / Line 62: @@
 Another example from before is that 31edo is a weak circle of ~5/4's and ~7/4's, but note that 31edo ''is'' a circle of ~[[35/32]]'s (meaning that 31edo is a (strong) circle of [[septimal neutral second]]s), where 35/32 = (5/4)/(8/7) = 5/4 * 7/4 / 2.
+== 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 (hence any [[equave]]) ''up to octave-reduction'', so is conceptualised differently. 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 ''k''-strong telicity is often more strict than a consistent circle; 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 qualifying), or it can [[have a weak circle]] without qualifying for 1-strong 2-a/b telicity (which again would usually not qualify), 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.
+=== Vs. subgroup telicity ===
+There is an application of telicity that concerns rationals in general — [[Telicity#Telicity on subgroups|subgroup telicity]] — which can be seen as roughly equivalent to using one of the generators as an equave and the other as the interval a/b, in which case a ''consistent circle'' refers to 1-strong pairwise telicity iff the circle generates all notes w.r.t. [[Octave reduction#Generalizations|equave-reduction]], but notice this "iff". Similarly, iff the circle generates all notes as mentioned prior, then qualifying for 2-strong pairwise telicity is equivalent to having a strongly consistent circle of a/b's w.r.t. the equave, and qualifying for 0.5-strong pairwise telicity to having a weakly consistent circle of a/b's w.r.t. the equave. But notice how wordy and technical it sounds in the case where the concepts overlap - hence using the concept of "circle" might be preferred as more easily understood in the case where it's applicable, and needs to be used in inapplicable cases where the definition of telicity is too strict or otherwise too niche.
+Consider, for example, that "the only type of telicity available to the 3-prime is 3-2 telicity" - this is in a sense true for consistent circles too, iff you assume your [[equave]] is the [[octave]], but otherwise not. In other words, consistent circles refer to less strict types of a/b-2 telicity in the case where the equave is the octave, and to less strict types of a/b-E telicity if E is some alternative equave.
 == See also ==