Telicity: Difference between revisions

Line 27:

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.

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. ~~This is why~~ telicity is ~~useful-~~ 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 the '''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)).

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 the '''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)).

== Integer and Rational Telicity ==

@@ Line 27: / Line 27: @@
 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.
-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.  This is why telicity is useful- 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 the '''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)).
+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 the '''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)).
 == Integer and Rational Telicity ==