User:Overthink/Asymptotic consistency score: Difference between revisions

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As 35edo is consistent in the 7-odd-limit, we use the best mappings of odds 3 through 7, with a mapping of ⟨35| 0 20 11 28]. When we get to odd 9, the closest mapping is best, giving us a mapping of ⟨35| 0 20 11 28 6]. The ratio between odds 9 and 3 in this mapping is not mapped to the same interval as the ratio between odds 3 and 1, even though the fraction 9/3 simplifies to 3/1. Here, we map 9/6 (9/3 reduced) to 21 steps, while we map 3/2 to 20 steps. This effectively means 3/2 has two different mappings, one at 20 steps and the other at 21, so 35edo is a dual-fifth system. Note that 49edo should technically also be a dual-fifth system, since harmonic 3 is (just barely) more than 1/3 of an edostep sharp, but the sharpness of harmonics 5, 7, and 11 means it is best to use the harmonic 9 mapped to 3*3, rather than take the best approximation of 9 and have dual 3's.

While the aim of this page is to create a metric for overall consistency of an EDO, this mapping based on odds rather than primes may itself be an interesting topic to explore.

While the aim of this page is to create a metric for overall consistency of an EDO, this mapping based on odds rather than primes may itself be an interesting topic to explore, even though it was really created mostly to simplify calculations.

== The consistency metric ==

We want to give each edo a single score for how well it does in terms of consistency. Given n-edo, we start with the trivial mapping ⟨n| 0]. We then add each odd one by one, and look at how many additional intervals are consistent and inconsistent. When we add odd q, each consistent interval increases the score by 1/q<sup>2</sup>, and each inconsistent interval decreases the score by 3/q<sup>2</sup>. However, it is impossible to calculate this score precisely, as there would be infinitely many terms. Using a program I made on scratch, considering EDOs up to 311 and odds up to 255, here is a sequence of EDOs that have better consistency:

{{edos|(1, 3, 4,) 5, 10, 15, 22, 29, 31, 41, 46, 53, 87, 130, 140, 183, 270, 311.}}

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 As 35edo is consistent in the 7-odd-limit, we use the best mappings of odds 3 through 7, with a mapping of ⟨35| 0 20 11 28]. When we get to odd 9, the closest mapping is best, giving us a mapping of ⟨35| 0 20 11 28 6]. The ratio between odds 9 and 3 in this mapping is not mapped to the same interval as the ratio between odds 3 and 1, even though the fraction 9/3 simplifies to 3/1. Here, we map 9/6 (9/3 reduced) to 21 steps, while we map 3/2 to 20 steps. This effectively means 3/2 has two different mappings, one at 20 steps and the other at 21, so 35edo is a dual-fifth system. Note that 49edo should technically also be a dual-fifth system, since harmonic 3 is (just barely) more than 1/3 of an edostep sharp, but the sharpness of harmonics 5, 7, and 11 means it is best to use the harmonic 9 mapped to 3*3, rather than take the best approximation of 9 and have dual 3's.
-While the aim of this page is to create a metric for overall consistency of an EDO, this mapping based on odds rather than primes may itself be an interesting topic to explore.
+While the aim of this page is to create a metric for overall consistency of an EDO, this mapping based on odds rather than primes may itself be an interesting topic to explore, even though it was really created mostly to simplify calculations.
 == The consistency metric ==
+We want to give each edo a single score for how well it does in terms of consistency. Given n-edo, we start with the trivial mapping ⟨n| 0]. We then add each odd one by one, and look at how many additional intervals are consistent and inconsistent. When we add odd q, each consistent interval increases the score by 1/q<sup>2</sup>, and each inconsistent interval decreases the score by 3/q<sup>2</sup>. However, it is impossible to calculate this score precisely, as there would be infinitely many terms. Using a program I made on scratch, considering EDOs up to 311 and odds up to 255, here is a sequence of EDOs that have better consistency:
+{{edos|(1, 3, 4,) 5, 10, 15, 22, 29, 31, 41, 46, 53, 87, 130, 140, 183, 270, 311.}}