Telicity: Difference between revisions

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== 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'''.

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 be 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'''. (Having '''connective telicity''' is defined as having "1-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'''.

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 == 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'''.
+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 be 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'''. (Having '''connective telicity''' is defined as having "1-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'''.