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 ''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 ===

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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'' 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 ===