Telicity: Difference between revisions

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* F((81/80)^3) = 1 (!)

Because F is not a linear function, it does not satisfy any of the conditions above. So even though rounding sometimes gives us better approximations, it doesn't conserve ~~interval~~ interval arithmetic, and conserving interval arithmetic is why we care about RTT in the first place.

Because F is not a linear function, it does not satisfy any of the conditions above. So even though rounding sometimes gives us better approximations, it doesn't conserve interval arithmetic, and conserving interval arithmetic is why we care about RTT in the first place.

However, people who care about ''both'' good approximations ''and'' conserving interval arithmetic are seemingly presented with a dilemma- either go with a linear map and risk bad approximations, or go with a non-linear map and risk inconsistent interval arithmetic. What makes telicity so useful is that when one is able to work with the section of the EDO's harmonic lattice in which both mapping methods lead to the same result and limit the usable portion of the harmonic lattice to this section through means of '''harmonic lattice resets''' – which, as per the name, are places in which the harmonic lattice of an EDO is considered to "reset" either to the [[unison]] or to something that can easily access the unison – one has the best of both worlds. In order to do this mathematically, however, one has to use an equation that sets the results of both mapping methods against each other, hence why telicity is defined by equation val(N)⋅monzo(r) = round(N⋅log2(r)).

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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 Range and Usable EDO Sizes ==

Each telic comma has a numerical range in which it can possibly be telic, that is, a '''telicity range'''. The size of a comma's telicity range is inversely 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 equal temperament's step with the size of the comma in question – if the comma is more than half the size of the equal temperament's step, said equal temperament is outside the comma's telicity range. For example, in EDO systems, [[81/80]] has a telicity range of 27, as 27 is the largest EDO with a step size that is more than 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.

Each telic comma has a numerical range in which it can possibly be telic, that is, a '''telicity range'''. The size of a comma's telicity range is inversely 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 equal temperament's step with the size of the comma in question – if the comma is more than half the size of the equal temperament's step, said equal temperament is outside the comma's telicity range. For example, in EDO systems, [[81/80]] has a telicity range of 27, as 27 is the largest EDO with a step size that is more than 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, if you take JND-sized commas as commas to be tempered. (If you instead want to have the melodic JND mapped to a single edostep, ~[[311edo|311]] might be a reasonable upper bound, due to being barely above the JND and due to its theoretical significance.) Note that the melodic JND varies widely from person to person and depending on context.

It should also be noted that while the tempering of a given comma that is larger than half a step in a given equal temperament can sometimes be accomplished to join primes without either prime exceeding the 50% relative error threshold in said equal temperament, the fact remains that the equal temperament in question lies outside the comma's telicity range. Thus, such a phenomenon in relation to telicity is analyzed as being a result of the equal temperament in question tempering out two or more commas that actually do 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 satisfy the equation for telicity on account of being less than half an EDO-step in size.

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Given that different EDOs can temper out different commas to achieve the same type of telicity – for example, [[12edo]] tempers out the [[Pythagorean comma]] to achieve 3-2 telicity, while [[53edo]] tempers out Mercator's comma to achieve 3-2 telicity – it can thus be argued that sequences of different EDOs demonstrating one or more types of telicity can be compiled. For instance, the first nine EDOs to demonstrate 3-2 telicity specifically form the sequence of {{EDOs| 1, 2, 5, 12, 24, 53, 106, 159, 306}}. In addition, one can compare multiple such telicity sequences, and see how frequently the various prime chains connect to one another across various EDOs, revealing which portions of the harmonic lattice are best utilized by any given EDO. Furthermore, this also enables one to examine the properties of the various prime chains themselves and provides cause to look for unexpectedly useful commas that, as of yet, are still unknown. As if all this weren't enough, telicity also useful in notation systems for establishing good positions for the "resets" in JI harmonic lattice representation that inevitably come about due to EDOs being closed systems in terms of their own harmonic lattices. All this makes telicity a viable endgame for the application of [[Consistent #Consistency to distance d|consistency to distance ''d'']], with which the concept of telicity is closely related.

== See also ==

* [[Consistent circle]]

[[Category:EDO theory pages]]

[[Category:Terms]]

@@ Line 25: / Line 25: @@
 * F((81/80)^3) = 1 (!)
-Because F is not a linear function, it does not satisfy any of the conditions above. So even though rounding sometimes gives us better approximations, it doesn't conserve interval interval arithmetic, and conserving interval arithmetic is why we care about RTT in the first place.
+Because F is not a linear function, it does not satisfy any of the conditions above. So even though rounding sometimes gives us better approximations, it doesn't conserve interval arithmetic, and conserving interval arithmetic is why we care about RTT in the first place.
 However, people who care about ''both'' good approximations ''and'' conserving interval arithmetic are seemingly presented with a dilemma- either go with a linear map and risk bad approximations, or go with a non-linear map and risk inconsistent interval arithmetic.  What makes telicity so useful is that when one is able to work with the section of the EDO's harmonic lattice in which both mapping methods lead to the same result and limit the usable portion of the harmonic lattice to this section through means of '''harmonic lattice resets''' – which, as per the name, are places in which the harmonic lattice of an EDO is considered to "reset" either to the [[unison]] or to something that can easily access the unison – one has the best of both worlds.  In order to do this mathematically, however, one has to use an equation that sets the results of both mapping methods against each other, hence why telicity is defined by equation val(N)⋅monzo(r) = round(N⋅log2(r)).
@@ Line 42: / Line 42: @@
 == 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'''.
@@ Line 48: / Line 48: @@
 == Telicity Range and Usable EDO Sizes ==
-Each telic comma has a numerical range in which it can possibly be telic, that is, a '''telicity range'''.  The size of a comma's telicity range is inversely 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 equal temperament's step with the size of the comma in question – if the comma is more than half the size of the equal temperament's step, said equal temperament is outside the comma's telicity range.  For example, in EDO systems, [[81/80]] has a telicity range of 27, as 27 is the largest EDO with a step size that is more than 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.
+Each telic comma has a numerical range in which it can possibly be telic, that is, a '''telicity range'''.  The size of a comma's telicity range is inversely 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 equal temperament's step with the size of the comma in question – if the comma is more than half the size of the equal temperament's step, said equal temperament is outside the comma's telicity range.  For example, in EDO systems, [[81/80]] has a telicity range of 27, as 27 is the largest EDO with a step size that is more than 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, if you take JND-sized commas as commas to be tempered. (If you instead want to have the melodic JND mapped to a single edostep, ~[[311edo|311]] might be a reasonable upper bound, due to being barely above the JND and due to its theoretical significance.) Note that the melodic JND varies widely from person to person and depending on context.
 It should also be noted that while the tempering of a given comma that is larger than half a step in a given equal temperament can sometimes be accomplished to join primes without either prime exceeding the 50% relative error threshold in said equal temperament, the fact remains that the equal temperament in question lies outside the comma's telicity range.  Thus, such a phenomenon in relation to telicity is analyzed as being a result of the equal temperament in question tempering out two or more commas that actually do 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 satisfy the equation for telicity on account of being less than half an EDO-step in size.
@@ Line 56: / Line 56: @@
 Given that different EDOs can temper out different commas to achieve the same type of telicity – for example, [[12edo]] tempers out the [[Pythagorean comma]] to achieve 3-2 telicity, while [[53edo]] tempers out Mercator's comma to achieve 3-2 telicity – it can thus be argued that sequences of different EDOs demonstrating one or more types of telicity can be compiled.  For instance, the first nine EDOs to demonstrate 3-2 telicity specifically form the sequence of {{EDOs| 1, 2, 5, 12, 24, 53, 106, 159, 306}}.  In addition, one can compare multiple such telicity sequences, and see how frequently the various prime chains connect to one another across various EDOs, revealing which portions of the harmonic lattice are best utilized by any given EDO.  Furthermore, this also enables one to examine the properties of the various prime chains themselves and provides cause to look for unexpectedly useful commas that, as of yet, are still unknown.  As if all this weren't enough, telicity also useful in notation systems for establishing good positions for the "resets" in JI harmonic lattice representation that inevitably come about due to EDOs being closed systems in terms of their own harmonic lattices.  All this makes telicity a viable endgame for the application of [[Consistent #Consistency to distance d|consistency to distance ''d'']], with which the concept of telicity is closely related.
+== See also ==
+* [[Consistent circle]]
 [[Category:EDO theory pages]]
 [[Category:Terms]]