User:Overthink/Asymptotic consistency score: Difference between revisions

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== The consistency rating ==

We want our rating to prioritize lower limits over higher ones. We also want it to use the best mapping of the edo, which may or may not map each odd harmonic to its closest value. Given an EDO, we start with a trivial mapping of 〈0], which maps 1/1 to 0 steps. Note that while we will be write our mappings like vals, we are not using traditional vals with primes, but instead mappings with odds described in the introduction. From then on, we either add the closest or second-closest mapping of the next odd harmonic, whichever one leads to the least amount of inconsistencies. Note that an interval and its octave complement are counted as one, not seperately, and octave equivalence is presumed. If both mappings have the same number of inconsistencies, the closest mapping is used. Lets look at 49edo for an example:

We want our rating to prioritize lower limits over higher ones. We also want it to use the best mapping of the edo, which may or may not map each odd harmonic to its closest value. Given an EDO, we start with a trivial mapping of ⟨0], which maps 1/1 to 0 steps. Note that while we will be write our mappings like vals, we are not using traditional vals with primes, but instead mappings with odds described in the introduction. From then on, we either add the closest or second-closest mapping of the next odd harmonic, whichever one leads to the least amount of inconsistencies. Note that an interval and its octave complement are counted as one, not seperately, and octave equivalence is presumed. If both mappings have the same number of inconsistencies, the closest mapping is used. Lets look at 49edo for an example:

We add the best approximations of odds 3, 5, and 7, as 49edo is consistent in these limits. Our mapping so far is therefore 〈0 78 114 138], which reduces to 〈0 29 16 40]. When we get to odd 9, we have some inconsistencies. Using the nearest mapping of 9, we have 9/3, 9/5, and 9/7 inconsistent. Using our second nearest mapping, we only have 9/1 inconsistent, and since one inconsistency is less than three, we go with the second nearest mapping of 9 to get 〈0 29 16 40 9] reduced. Then, using the nearest mapping of 11 adds no additional inconsistencies, so we go with that to get 〈0 29 16 40 9 23]. Then, using the nearest mapping of 13 gets 13/3, 13/5, 13/7, 13/9, and 13/11 inconsistent, while the second nearest mapping only gets 13/1 inconsistent, so we go with that to get 〈0 29 16 40 9 23 35]. Finally, adding odd 15, we find the second closest mapping is best, giving us 〈0 29 16 40 9 23 35 45].

We add the best approximations of odds 3, 5, and 7, as 49edo is consistent in these limits. Our mapping so far is therefore ⟨0 78 114 138], which reduces to ⟨0 29 16 40]. When we get to odd 9, we have some inconsistencies. Using the nearest mapping of 9, we have 9/3, 9/5, and 9/7 inconsistent. Using our second nearest mapping, we only have 9/1 inconsistent, and since one inconsistency is less than three, we go with the second nearest mapping of 9 to get 〈0 29 16 40 9] reduced. Then, using the nearest mapping of 11 adds no additional inconsistencies, so we go with that to get ⟨0 29 16 40 9 23]. Then, using the nearest mapping of 13 gets 13/3, 13/5, 13/7, 13/9, and 13/11 inconsistent, while the second nearest mapping only gets 13/1 inconsistent, so we go with that to get ⟨0 29 16 40 9 23 35]. Finally, adding odd 15, we find the second closest mapping is best, giving us ⟨0 29 16 40 9 23 35 45].

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 == The consistency rating ==
-We want our rating to prioritize lower limits over higher ones. We also want it to use the best mapping of the edo, which may or may not map each odd harmonic to its closest value. Given an EDO, we start with a trivial mapping of 〈0], which maps 1/1 to 0 steps. Note that while we will be write our mappings like vals, we are not using traditional vals with primes, but instead mappings with odds described in the introduction. From then on, we either add the closest or second-closest mapping of the next odd harmonic, whichever one leads to the least amount of inconsistencies. Note that an interval and its octave complement are counted as one, not seperately, and octave equivalence is presumed. If both mappings have the same number of inconsistencies, the closest mapping is used. Lets look at 49edo for an example:
+We want our rating to prioritize lower limits over higher ones. We also want it to use the best mapping of the edo, which may or may not map each odd harmonic to its closest value. Given an EDO, we start with a trivial mapping of ⟨0], which maps 1/1 to 0 steps. Note that while we will be write our mappings like vals, we are not using traditional vals with primes, but instead mappings with odds described in the introduction. From then on, we either add the closest or second-closest mapping of the next odd harmonic, whichever one leads to the least amount of inconsistencies. Note that an interval and its octave complement are counted as one, not seperately, and octave equivalence is presumed. If both mappings have the same number of inconsistencies, the closest mapping is used. Lets look at 49edo for an example:
 {{Harmonics in equal|49}}
-We add the best approximations of odds 3, 5, and 7, as 49edo is consistent in these limits. Our mapping so far is therefore 〈0 78 114 138], which reduces to 〈0 29 16 40]. When we get to odd 9, we have some inconsistencies. Using the nearest mapping of 9, we have 9/3, 9/5, and 9/7 inconsistent. Using our second nearest mapping, we only have 9/1 inconsistent, and since one inconsistency is less than three, we go with the second nearest mapping of 9 to get 〈0 29 16 40 9] reduced. Then, using the nearest mapping of 11 adds no additional inconsistencies, so we go with that to get 〈0 29 16 40 9 23]. Then, using the nearest mapping of 13 gets 13/3, 13/5, 13/7, 13/9, and 13/11 inconsistent, while the second nearest mapping only gets 13/1 inconsistent, so we go with that to get 〈0 29 16 40 9 23 35]. Finally, adding odd 15, we find the second closest mapping is best, giving us 〈0 29 16 40 9 23 35 45].
+We add the best approximations of odds 3, 5, and 7, as 49edo is consistent in these limits. Our mapping so far is therefore ⟨0 78 114 138], which reduces to ⟨0 29 16 40]. When we get to odd 9, we have some inconsistencies. Using the nearest mapping of 9, we have 9/3, 9/5, and 9/7 inconsistent. Using our second nearest mapping, we only have 9/1 inconsistent, and since one inconsistency is less than three, we go with the second nearest mapping of 9 to get 〈0 29 16 40 9] reduced. Then, using the nearest mapping of 11 adds no additional inconsistencies, so we go with that to get ⟨0 29 16 40 9 23]. Then, using the nearest mapping of 13 gets 13/3, 13/5, 13/7, 13/9, and 13/11 inconsistent, while the second nearest mapping only gets 13/1 inconsistent, so we go with that to get ⟨0 29 16 40 9 23 35]. Finally, adding odd 15, we find the second closest mapping is best, giving us ⟨0 29 16 40 9 23 35 45].