Telicity: 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 patent 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 patent interval chain. Given this, 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. Therefore, for the sake of consistency in definition, the commas that can be tempered out to achieve telicity ''must'' be less than half an EDO-step in size. Commas that join prime chains together when tempered out are called ''telic'' commas, and each telic comma has an effective range called the ''telicity range'', and the size of a comma's telicity range is negatively ~~related~~ to the size of the comma itself – that is, the smaller the comma in question, the larger the size of that comma's telicity range.

'''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 patent 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 patent interval chain. Given this, 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. Therefore, for the sake of consistency in definition, the commas that can be tempered out to achieve telicity ''must'' be less than half an EDO-step in size. Commas that join prime chains together when tempered out are called ''telic'' commas, and each telic comma has an effective range called the ''telicity range'', and the size of a comma's telicity range is negatively correlated to the size of the comma itself – that is, the smaller the comma in question, the larger the size of that comma's telicity range.

It should be noted that 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. It should also be noted that while the tempering of commas larger than half an EDO step can sometimes be accomplished to join primes without either prime exceeding the 50% relative error threshold, such a phenomenon in relation to telicity is analyzed as being a result of the EDO in question tempering out two or more commas that actually meet the strict criteria requirements, and thus, having two or more forms of the same type of telicity. As an example, 31edo tempers out [[81/80]], which is larger than half of this EDO's step size, but this can be attributed to 31edo tempering out both the [[Würschmidt comma]] (393216/390625) and the [[Semicomma]] (2109375/2097152), with the former equating an octave-reduced chain of eight [[5/4]] intervals with [[3/2]], and the latter equating an octave-reduced chain of seven 5/4 intervals with [[32/27]], while both commas additionally meet the criteria for telicity on account of being less than half an EDO step in size.

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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 patent 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 patent interval chain. Given this, 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. Therefore, for the sake of consistency in definition, the commas that can be tempered out to achieve telicity ''must'' be less than half an EDO-step in size. Commas that join prime chains together when tempered out are called ''telic'' commas, and each telic comma has an effective range called the ''telicity range'', and the size of a comma's telicity range is negatively ~~related~~ to the size of the comma itself – that is, the smaller the comma in question, the larger the size of that comma's telicity range.		'''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 patent 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 patent interval chain. Given this, 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. Therefore, for the sake of consistency in definition, the commas that can be tempered out to achieve telicity ''must'' be less than half an EDO-step in size. Commas that join prime chains together when tempered out are called ''telic'' commas, and each telic comma has an effective range called the ''telicity range'', and the size of a comma's telicity range is negatively correlated to the size of the comma itself – that is, the smaller the comma in question, the larger the size of that comma's telicity range.

	It should be noted that 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. It should also be noted that while the tempering of commas larger than half an EDO step can sometimes be accomplished to join primes without either prime exceeding the 50% relative error threshold, such a phenomenon in relation to telicity is analyzed as being a result of the EDO in question tempering out two or more commas that actually meet the strict criteria requirements, and thus, having two or more forms of the same type of telicity. As an example, 31edo tempers out [[81/80]], which is larger than half of this EDO's step size, but this can be attributed to 31edo tempering out both the [[Würschmidt comma]] (393216/390625) and the [[Semicomma]] (2109375/2097152), with the former equating an octave-reduced chain of eight [[5/4]] intervals with [[3/2]], and the latter equating an octave-reduced chain of seven 5/4 intervals with [[32/27]], while both commas additionally meet the criteria for telicity on account of being less than half an EDO step in size.		It should be noted that 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. It should also be noted that while the tempering of commas larger than half an EDO step can sometimes be accomplished to join primes without either prime exceeding the 50% relative error threshold, such a phenomenon in relation to telicity is analyzed as being a result of the EDO in question tempering out two or more commas that actually meet the strict criteria requirements, and thus, having two or more forms of the same type of telicity. As an example, 31edo tempers out [[81/80]], which is larger than half of this EDO's step size, but this can be attributed to 31edo tempering out both the [[Würschmidt comma]] (393216/390625) and the [[Semicomma]] (2109375/2097152), with the former equating an octave-reduced chain of eight [[5/4]] intervals with [[3/2]], and the latter equating an octave-reduced chain of seven 5/4 intervals with [[32/27]], while both commas additionally meet the criteria for telicity on account of being less than half an EDO step in size.