Telicity: Difference between revisions

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(''~~Under construction~~'')

'''Telicity''' is a property of both [[EDO]]s and [[commas]] and how they relate to each other. Specifically, for EDOs, it is the quality or state of being able, by means of tempering one or more commas smaller than half a step, to successfully stack a number of instances of a given prime's [[patent interval]] to connect with an interval belonging to a chain created by a lower prime's patent interval – designated as the '''telos''' – 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. For commas, telicity is the quality of being able to join two distinct prime interval chains in the aforementioned manner by being tempered.

Stated more mathematically, where "N" is the number of steps in a given EDO, "r" is the ratio of an interval in one of the two prime chains in question, and "M" is the monzo of "r", the equation {N, round(log2(3)*N), round(log2(5)*N), round(log2(7)*N), round(log2(11)*N), ...}.{M} = round(log2(r)*N) must hold true along both prime chains up to and including the point of connection.

Commas and EDOs that demonstrate this property are known as '''telic'''. When a given EDO is telic in a given multiprime relationship by more than one means, it can be said to be '''multitelic'''.

'''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 a 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, or '''telos''', 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.

== Further Details ==

~~Stated more mathematically, where "N" is the number of steps in a given EDO, "r" is the ratio of an interval in one of the two prime chains, and "M" is the monzo of "r", the equation {N, round~~(~~log2(3)*N), round(log2(5)*N), round(log2(7)*N), round(log2(11)*N), ...}.{M} = round(log2(r)*N)~~ ''~~must~~'' ~~hold true along ''both'' prime chains up to and including the point of connection.~~

(''Under construction'')

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. The size of a comma's telicity range is evaluated by comparing the size of a given EDO's step with the size of the comma in question – if the comma is more than half the size of the EDO's step, said EDO is outside the comma's telicity range. For example, [[81/80]] has a telicity range of 27, as 27 is the largest EDO with a step size that is more that twice the size of the comma, while [[Mercator's comma]] has a telicity range of 165, as 165 is the largest EDO with a step size that is more that twice the size of that comma. In addition, the concept of telicity can be used to evaluate the usefulness of EDOs relative to the [http://musictheory.zentral.zone/huntsystem2.html#2 JND] of human pitch perception, and, since an interval smaller than 3.5 cents is unlikely to be noticed by even the most trained listeners, it can be said that the JND has a telicity range of 171, meaning that EDOs that are higher than [[171edo|171]] are not all that suitable for use as musical systems outside of pitch bends.

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-(''Under construction'')
+'''Telicity''' is a property of both [[EDO]]s and [[commas]] and how they relate to each other.  Specifically, for EDOs, it is the quality or state of being able, by means of tempering one or more commas smaller than half a step, to successfully stack a number of instances of a given prime's [[patent interval]] to connect with an interval belonging to a chain created by a lower prime's patent interval – designated as the '''telos''' – 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.  For commas, telicity is the quality of being able to join two distinct prime interval chains in the aforementioned manner by being tempered.
+Stated more mathematically, where "N" is the number of steps in a given EDO, "r" is the ratio of an interval in one of the two prime chains in question, and "M" is the monzo of "r", the equation {N, round(log2(3)*N), round(log2(5)*N), round(log2(7)*N), round(log2(11)*N), ...}.{M} = round(log2(r)*N) must hold true along both prime chains up to and including the point of connection.
+Commas and EDOs that demonstrate this property are known as '''telic'''.  When a given EDO is telic in a given multiprime relationship by more than one means, it can be said to be '''multitelic'''.
-'''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 a 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, or '''telos''', 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.
+== Further Details ==
-Stated more mathematically, where "N" is the number of steps in a given EDO, "r" is the ratio of an interval in one of the two prime chains, and "M" is the monzo of "r", the equation {N, round(log2(3)*N), round(log2(5)*N), round(log2(7)*N), round(log2(11)*N), ...}.{M} = round(log2(r)*N) ''must'' hold true along ''both'' prime chains up to and including the point of connection.
+(''Under construction'')
 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.  The size of a comma's telicity range is evaluated by comparing the size of a given EDO's step with the size of the comma in question – if the comma is more than half the size of the EDO's step, said EDO is outside the comma's telicity range.  For example, [[81/80]] has a telicity range of 27, as 27 is the largest EDO with a step size that is more that twice the size of the comma, while [[Mercator's comma]] has a telicity range of 165, as 165 is the largest EDO with a step size that is more that twice the size of that comma.  In addition, the concept of telicity can be used to evaluate the usefulness of EDOs relative to the [http://musictheory.zentral.zone/huntsystem2.html#2 JND] of human pitch perception, and, since an interval smaller than 3.5 cents is unlikely to be noticed by even the most trained listeners, it can be said that the JND has a telicity range of 171, meaning that EDOs that are higher than [[171edo|171]] are not all that suitable for use as musical systems outside of pitch bends.