Consistent circle: Difference between revisions

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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.

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 ===

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 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.
+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 ===