Telicity: Difference between revisions

Line 17:

== 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. 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.

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 ==

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 ~~seven~~ EDOs to demonstrate 3-2 telicity specifically ~~are~~ {{EDOs| 2, 5, 12, 24, 53, 106, 159 }}. 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.

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.

[[Category:EDO theory pages]]

[[Category:Temperament]]

[[Category:Terms]]

@@ Line 17: / Line 17: @@
 == 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.  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.
+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 ==
-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 seven EDOs to demonstrate 3-2 telicity specifically are {{EDOs| 2, 5, 12, 24, 53, 106, 159 }}.  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.
+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.
 [[Category:EDO theory pages]]
 [[Category:Temperament]]
 [[Category:Terms]]