User talk:Xenwolf: Difference between revisions

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'''Telicity''' is a property of [[EDO]]s, which involves the given EDO being able to stack a number of instances of a given prime's [[patent interval]] to connect with an interval belonging to a chain created by lower prime's [[patent interval]] without either accumulating 50% relative error or more at any point in the process on the part of either prime's interval chain, or, creating as mismatch in results between the [[direct mapping]] and the more complicated traditional mapping for any interval along the chain. Furthermore, since the 2-prime simply results in manifestations of the [[unison]] at different registers, thus leading to things like [[octave reduction]], the 2-prime is not a factor that affects what qualifies as telicity ''unless'' an interval in 2-prime chain itself serves as the target of a higher prime's interval chain.

Given this definition, two things are immediately evident. Firstly, any sort of telicity involving the 2-prime cannot afford to temper out commas greater than half an EDO-step in size due to the unison being such a foundational interval to both EDOs and JI, and, the resultant inability to temper out commas greater than half a step in size without exceeding the 50% relative error threshold. Secondly, the only type of telicity available to the 3-prime is 3-to-2 telicity, as 2 is the only positive prime lower than 3, and since octave ~~equvalency~~ renders the unison as the only available target, that means that the 3-prime requires a complete [[circle of fifths]] without accumulating 50% relative error or more. However, higher primes have more options for achieving a form of telicity as there are multiple lower primes to chose from to potentially connect with, For instance, the 5-prime has both 5-to-3 and 5-to-2 telicity available to it.

Given this definition, two things are immediately evident. Firstly, any sort of telicity involving the 2-prime cannot afford to temper out commas greater than half an EDO-step in size due to the unison being such a foundational interval to both EDOs and JI, and, the resultant inability to temper out commas greater than half a step in size without exceeding the 50% relative error threshold. Secondly, the only type of telicity available to the 3-prime is 3-to-2 telicity, as 2 is the only positive prime lower than 3, and since octave equivalency renders the unison as the only available target, that means that the 3-prime requires a complete [[circle of fifths]] without accumulating 50% relative error or more. However, higher primes have more options for achieving a form of telicity as there are multiple lower primes to chose from to potentially connect with, For instance, the 5-prime has both 5-to-3 and 5-to-2 telicity available to it.

Combinations of primes are more complicated, but it's safe to say that there are more types of telicity available in such cases- namely "full telicity" and "partial telicity". Full telicity for combinations involving multiple primes occurs when the EDO in question is able to stack a number of instances of a given combination's patent interval to connect with an interval belonging to a chain created by the patent interval for a prime that is lower than the lowest prime in the initial combination after octave reduction is taken into account. In contrast, partial telicity for combinations involving multiple primes occurs when the EDO in question is able to stack a number of instances of a given combination's patent interval to connect with an interval belonging to a chain created by the patent interval for a prime that is lower than the highest prime in the initial combination after octave reduction is taken into account.

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 '''Telicity''' is a property of [[EDO]]s, which involves the given EDO being able to stack a number of instances of a given prime's [[patent interval]] to connect with an interval belonging to a chain created by lower prime's [[patent interval]] without either accumulating 50% relative error or more at any point in the process on the part of either prime's interval chain, or, creating as mismatch in results between the [[direct mapping]] and the more complicated traditional mapping for any interval along the chain.  Furthermore, since the 2-prime simply results in manifestations of the [[unison]] at different registers, thus leading to things like [[octave reduction]], the 2-prime is not a factor that affects what qualifies as telicity ''unless'' an interval in 2-prime chain itself serves as the target of a higher prime's interval chain.
-Given this definition, two things are immediately evident.  Firstly, any sort of telicity involving the 2-prime cannot afford to temper out commas greater than half an EDO-step in size due to the unison being such a foundational interval to both EDOs and JI, and, the resultant inability to temper out commas greater than half a step in size without exceeding the 50% relative error threshold.  Secondly, the only type of telicity available to the 3-prime is 3-to-2 telicity, as 2 is the only positive prime lower than 3, and since octave equvalency renders the unison as the only available target, that means that the 3-prime requires a complete [[circle of fifths]] without accumulating 50% relative error or more.  However, higher primes have more options for achieving a form of telicity as there are multiple lower primes to chose from to potentially connect with, For instance, the 5-prime has both 5-to-3 and 5-to-2 telicity available to it.
+Given this definition, two things are immediately evident.  Firstly, any sort of telicity involving the 2-prime cannot afford to temper out commas greater than half an EDO-step in size due to the unison being such a foundational interval to both EDOs and JI, and, the resultant inability to temper out commas greater than half a step in size without exceeding the 50% relative error threshold.  Secondly, the only type of telicity available to the 3-prime is 3-to-2 telicity, as 2 is the only positive prime lower than 3, and since octave equivalency renders the unison as the only available target, that means that the 3-prime requires a complete [[circle of fifths]] without accumulating 50% relative error or more.  However, higher primes have more options for achieving a form of telicity as there are multiple lower primes to chose from to potentially connect with, For instance, the 5-prime has both 5-to-3 and 5-to-2 telicity available to it.
 Combinations of primes are more complicated, but it's safe to say that there are more types of telicity available in such cases- namely "full telicity" and "partial telicity".  Full telicity for combinations involving multiple primes occurs when the EDO in question is able to stack a number of instances of a given combination's patent interval to connect with an interval belonging to a chain created by the patent interval for a prime that is lower than the lowest prime in the initial combination after octave reduction is taken into account.  In contrast, partial telicity for combinations involving multiple primes occurs when the EDO in question is able to stack a number of instances of a given combination's patent interval to connect with an interval belonging to a chain created by the patent interval for a prime that is lower than the highest prime in the initial combination after octave reduction is taken into account.