Consistent circle: Difference between revisions

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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 ~~(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.

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.

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.

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.

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.

=== Vs. subgroup telicity ===

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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 (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.
+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.
-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.
+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.
+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.
 === Vs. subgroup telicity ===