Telicity: Difference between revisions

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Different EDOs have different relationships among the various primes that are used to define their [[patent val]]s, and thus, there is a need to designate these relationships. Given that EDOs specifically are defined as equal divisions of the [[octave]], the 2-prime – which defines the octave – is always listed last, with the largest prime being listed first, and other primes being listed in the middle from largest to smallest thus, for example a telic connection between the 3-prime and the 2-prime is denoted as 3-2 telicity. It should be noted that the only type of telicity available to the 3-prime is 3-2 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 in order to achieve telicity. 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 5-3 and 5-2 telicity available to it. Not only that, but in cases where multiple overlapping telic relationships exist for a given EDO without the largest tempered comma failing to satisfy the telicity equation, one can express all of these telic relationships within a single designation. For example, [[12edo]], which simultaneously demonstrates 3-2 telicity, 5-3 telicity, and 5-2 telicity, can be said to demonstrate 5-3-2 telicity.

== K-Strong Telicity ==

== k-Strong Telicity ==

While the telicity of EDOs with, say, 3-2 telicity and only a single circle of fifths, is independent, properly accounting for the same type of telicity in EDOs with multiple circles of fifths is another story, and for that, we need to work with k-Strong Telicity. k-Strong Telicity is k times as strict as normal telicity, which is to say that for any two generating intervals A and B, A^n * B^m for nonzero integers n,m should by patent val consistently be mapped to the right interval in both N EDO and kN EDO so that the error is less than 50%/k of a step in N EDO, which is to say the error is less than 1\(2kN). Note that this also requires that the mapping for intervals A and B in kN EDO should be the same as the mapping for them in N EDO, and that it requires all the other things needed for telicity by default. Using this, we can see that 12edo is a 2-strong 3-2 telic system and 53edo is a 3-strong 3-2 telic system.

== Telicity On Subgroups ==

Telicity is often most useful in the accurate modelling of subgroups of interest; it therefore makes sense to define senses of telicity for subgroups. This builds on the idea that a '''telic connection''' between two generators in a rank one temperament can be '''k-strong'''. Consider a set of generators. As we are generalising the notion, the generators need not necessarily prime, but ideally all generators are either harmonics (positive integers > 1) or at least (ideally low-complexity) intervals of significant musical interest. Then a subgroup (a set of rationals > 1 AKA a "set of generators") is '''k-strong pairwise telic''' if there is a k-strong telic connection between every pair of generators. If a subgroup is "almost" k-strong pairwise telic except for exactly '''n''' pairs of generators lacking a k-strong telic connection, then it is instead '''n-deficient k-strong pairwise-telic'''. If a subgroup is 1-strong pairwise telic it is simply "pairwise telic". This means that a subgroup can be both "pairwise telic" and "n-deficient k-strong pairwise-telic" simultaneously. This can be abbreviated as being "n-weak k-strong pairwise-telic".

There is a yet weaker - but not unuseful - notion of telicity on subgroups, where every generator can be considered as a node in a graph, and every telic connection can be considered as an edge in that graph. Then the model of the subgroup that the rank one temperament provides is said to have '''connective telicity''' if the graph is connected, meaning every generator in that subgroup can be related to every other generator directly or indirectly through a path of telic connections to other generators. Then, if every one of those telic connections is k-strong, it is said to have "k-strong connective telicity". Analogously, a model of a subgroup may have "k-strong connective telicity" except for exactly '''n''' pairs of generators that do not have a k-strong telic connection, but which demonstrate 1-strong connective telicity with respect to the subgroup nonetheless. Then the subgroup demonstrates '''n-weak k-strong connective telicity'''.

While the telicity of EDOs with, say, 3-2 telicity and only a single circle of fifths, is independent, properly accounting for the same type of telicity in EDOs with multiple circles of fifths is another story, and for that, we need to work with K-Strong Telicity. K-Strong Telicity is k times as strict as normal telicity, which is to say that for any two generating intervals A and B, A^n * B^m for nonzero integers n,m should by patent val consistently be mapped to the right interval in both N EDO and kN EDO so that the error is less than 50%/k of a step in N EDO. Note that this also requires that the mapping for intervals A and B in kN EDO should be the same as the mapping for them in N EDO, and that it requires all the other things needed for telicity by default. Using this, we can see that 12edo is a 2-strong 3-2 telic system and 53edo is a 3-strong 3-2 telic system.

== Applications ==

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 Different EDOs have different relationships among the various primes that are used to define their [[patent val]]s, and thus, there is a need to designate these relationships. Given that EDOs specifically are defined as equal divisions of the [[octave]], the 2-prime – which defines the octave – is always listed last, with the largest prime being listed first, and other primes being listed in the middle from largest to smallest thus, for example a telic connection between the 3-prime and the 2-prime is denoted as 3-2 telicity.  It should be noted that the only type of telicity available to the 3-prime is 3-2 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 in order to achieve telicity.  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 5-3 and 5-2 telicity available to it.  Not only that, but in cases where multiple overlapping telic relationships exist for a given EDO without the largest tempered comma failing to satisfy the telicity equation, one can express all of these telic relationships within a single designation.  For example, [[12edo]], which simultaneously demonstrates 3-2 telicity, 5-3 telicity, and 5-2 telicity, can be said to demonstrate 5-3-2 telicity.
-== K-Strong Telicity ==
+== k-Strong Telicity ==
+While the telicity of EDOs with, say, 3-2 telicity and only a single circle of fifths, is independent, properly accounting for the same type of telicity in EDOs with multiple circles of fifths is another story, and for that, we need to work with k-Strong Telicity.  k-Strong Telicity is k times as strict as normal telicity, which is to say that for any two generating intervals A and B, A^n * B^m for nonzero integers n,m should by patent val consistently be mapped to the right interval in both N EDO and kN EDO so that the error is less than 50%/k of a step in N EDO, which is to say the error is less than 1\(2kN).  Note that this also requires that the mapping for intervals A and B in kN EDO should be the same as the mapping for them in N EDO, and that it requires all the other things needed for telicity by default.  Using this, we can see that 12edo is a 2-strong 3-2 telic system and 53edo is a 3-strong 3-2 telic system.
+== Telicity On Subgroups ==
+Telicity is often most useful in the accurate modelling of subgroups of interest; it therefore makes sense to define senses of telicity for subgroups. This builds on the idea that a '''telic connection''' between two generators in a rank one temperament can be '''k-strong'''. Consider a set of generators. As we are generalising the notion, the generators need not necessarily prime, but ideally all generators are either harmonics (positive integers > 1) or at least (ideally low-complexity) intervals of significant musical interest. Then a subgroup (a set of rationals > 1 AKA a "set of generators") is '''k-strong pairwise telic''' if there is a k-strong telic connection between every pair of generators. If a subgroup is "almost" k-strong pairwise telic except for exactly '''n''' pairs of generators lacking a k-strong telic connection, then it is instead '''n-deficient k-strong pairwise-telic'''. If a subgroup is 1-strong pairwise telic it is simply "pairwise telic". This means that a subgroup can be both "pairwise telic" and "n-deficient k-strong pairwise-telic" simultaneously. This can be abbreviated as being "n-weak k-strong pairwise-telic".
+There is a yet weaker - but not unuseful - notion of telicity on subgroups, where every generator can be considered as a node in a graph, and every telic connection can be considered as an edge in that graph. Then the model of the subgroup that the rank one temperament provides is said to have '''connective telicity''' if the graph is connected, meaning every generator in that subgroup can be related to every other generator directly or indirectly through a path of telic connections to other generators. Then, if every one of those telic connections is k-strong, it is said to have "k-strong connective telicity". Analogously, a model of a subgroup may have "k-strong connective telicity" except for exactly '''n''' pairs of generators that do not have a k-strong telic connection, but which demonstrate 1-strong connective telicity with respect to the subgroup nonetheless. Then the subgroup demonstrates '''n-weak k-strong connective telicity'''.
-While the telicity of EDOs with, say, 3-2 telicity and only a single circle of fifths, is independent, properly accounting for the same type of telicity in EDOs with multiple circles of fifths is another story, and for that, we need to work with K-Strong Telicity.  K-Strong Telicity is k times as strict as normal telicity, which is to say that for any two generating intervals A and B, A^n * B^m for nonzero integers n,m should by patent val consistently be mapped to the right interval in both N EDO and kN EDO so that the error is less than 50%/k of a step in N EDO.  Note that this also requires that the mapping for intervals A and B in kN EDO should be the same as the mapping for them in N EDO, and that it requires all the other things needed for telicity by default.  Using this, we can see that 12edo is a 2-strong 3-2 telic system and 53edo is a 3-strong 3-2 telic system.
 == Applications ==