Telicity: Difference between revisions

Line 19:

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.

== ~~Simple~~ and ~~Compound~~ Telicity ==

== Integer and Rational Telicity ==

~~While harmonics~~, and especially primes, are fairly simple as both [[equave]]s and [[generator]]s when it comes to telicity, ~~hence the term~~ '''~~simple~~ telicity''' ~~for this type of telicity~~, when the equave and or the generator are combinations of primes, things are more complicated, leading to the term '''~~compound~~ telicity''' as a descriptor for this second type of telicity.

Harmonics, and especially primes, are fairly simple as both [[equave]]s and [[generator]]s when it comes to telicity, and since all of these are integers, this type of telicity can be referred to specifically as '''integer telicity'''. However, when the equave and or the generator are combinations of primes, things are more complicated, leading to the broader term '''rational telicity''' as a descriptor for this second type of telicity.

== Telicity On Subgroups ==

@@ Line 19: / Line 19: @@
 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.
-== Simple and Compound Telicity ==
+== Integer and Rational Telicity ==
-While harmonics, and especially primes, are fairly simple as both [[equave]]s and [[generator]]s when it comes to telicity, hence the term '''simple telicity''' for this type of telicity, when the equave and or the generator are combinations of primes, things are more complicated, leading to the term '''compound telicity''' as a descriptor for this second type of telicity.
+Harmonics, and especially primes, are fairly simple as both [[equave]]s and [[generator]]s when it comes to telicity, and since all of these are integers, this type of telicity can be referred to specifically as '''integer telicity'''.  However, when the equave and or the generator are combinations of primes, things are more complicated, leading to the broader term '''rational telicity''' as a descriptor for this second type of telicity.
 == Telicity On Subgroups ==