Defactoring algorithms: Difference between revisions

Line 29:

Being enfactored tells us that it's wasteful to use this mapping. Specifically, being 2-enfactored tells us that we have 2x as many pitches as we need. Said another way, half of the pitches in our system are bringing nothing to the table, at least not in terms of approximating intervals built out of the 5-limit primes 2, 3, and 5, which is the primary goal of a temperament.

This is the mapping for [[5-limit]] [[24-ET]]. To be clear, we're not saying there's a major problem with 24 as an [[EDO]]. The point here is only that — if you're after a 5-limit temperament — you may as well use [[12-ET]]. So we would consider 24-ET to stand for 24 Equal Temperoid.

This is the mapping for [[5-limit]] [[24-ET]]. To be clear, we're not saying there's a major problem with 24 as an [[EDO]]. The point here is only that — if you're after a 5-limit temperament — you may as well use [[12-ET]]<ref>The mathematical term for this idea is [[Wikipedia:Surjective_function|surjective]]: if we understand a mapping as a function, we expect there to be no elements in the range of this function which no domain inputs map to.</ref>. So we would consider 24-ET to stand for 24 Equal Temperoid.

Think of it this way: because every element is even, any [[JI]] interval you'd map with with the mapping must come out as an even number of steps of 24-ET, by the definition of the dot product, and every even step of 24-ET is just a step of 12-ET. Examples: {{vector|1 -2 1}}.{{map|24 38 56}} = 24 - 76 + 56 = 4, {{vector|1 1 -1}}.{{map|24 38 56}} = 24 + 38 - 56 = 6.

@@ Line 29: / Line 29: @@
 Being enfactored tells us that it's wasteful to use this mapping. Specifically, being 2-enfactored tells us that we have 2x as many pitches as we need. Said another way, half of the pitches in our system are bringing nothing to the table, at least not in terms of approximating intervals built out of the 5-limit primes 2, 3, and 5, which is the primary goal of a temperament.
-This is the mapping for [[5-limit]] [[24-ET]]. To be clear, we're not saying there's a major problem with 24 as an [[EDO]]. The point here is only that — if you're after a 5-limit temperament — you may as well use [[12-ET]]. So we would consider 24-ET to stand for 24 Equal Temperoid.
+This is the mapping for [[5-limit]] [[24-ET]]. To be clear, we're not saying there's a major problem with 24 as an [[EDO]]. The point here is only that — if you're after a 5-limit temperament — you may as well use [[12-ET]]<ref>The mathematical term for this idea is [[Wikipedia:Surjective_function|surjective]]: if we understand a mapping as a function, we expect there to be no elements in the range of this function which no domain inputs map to.</ref>. So we would consider 24-ET to stand for 24 Equal Temperoid.
 Think of it this way: because every element is even, any [[JI]] interval you'd map with with the mapping must come out as an even number of steps of 24-ET, by the definition of the dot product, and every even step of 24-ET is just a step of 12-ET. Examples: {{vector|1 -2 1}}.{{map|24 38 56}} = 24 - 76 + 56 = 4, {{vector|1 1 -1}}.{{map|24 38 56}} = 24 + 38 - 56  = 6.