Godtone

80 EDO is a great no-limit system for conceptualising and internalising harmonic series interval categories/structures through '''RINGER 80''' which contains the entirety of the no-127's no-135's no-141's 145-odd-limit and in which ~84.44% of all intervals present are mapped consistently. This '''RINGER 80''' uses the best-performing val for 125-odd-limit consistency by various metrics (squared error, sum of error, number of inconsistencies, number of inconsistencies if we require <25% error, etc.). To achieve the CS ([[constant structure]]) property, the primes '''31, 47, 53, 61, 67, 73, 79, 107, 109''' are sharpened by 1 step compared to their flat patent val mapping (AKA are mapped to their second-best mapping); all other primes are patent val. This scale has a few remarkable properties. Firstly, all the intervals that are not inconsistent are mapped - at worst - to their second-best mapping, meaning you will never have a categorical/interval mapping exceeding 15 cents (which seems like a very reasonable result for a RINGER scale). Secondly, all of the "filler harmonics" beyond the 125-odd-limit fit in an obvious way; note how there are no warts beyond the 113-prime-limit (which the 125-odd-limit corresponds to due to a '''record prime gap''' from 113 to 127) meaning all composite harmonics were either already part of the 113-prime-limit or if prime the primes were patent val and sharp-tending. "In an obvious way" also means that every superparticular '''(n+1)/n''' in the 125-odd-limit that was mapped to 2 steps is split into '''(2n+2)/(2n+1)''' and '''(2n+1)/(2n)''', retaining the lowest possible complexity. Finally, note that while composite odd harmonics start going missing after the 125th harmonic, that prime harmonics are very much not lacking. This scale exists inside the no-127's no-151's no-163's 179-prime-limit, meaning that '''all primes up to and including 179''' are present excepting those three, making it full of prime flavour (on top of its high compositeness due to the 125-odd-limit corresponding to a record-prime-gap). My only personal qualm with this scale is that prime 73 is not patent val when I'd like it to be, but keeping it warted allows the introduction of the 145th harmonic which adds a lot of low-complexity and consistent intervals, and not warting it means no longer using the best-performing mapping and using the "147.2th" harmonic instead of the much-preferable "145th". Specifically, all intervals made in ratio with the 145th harmonic that simplify are mapped consistently. Finally, its worth noting that I used [[#My_Python_3_code]] to find and hone this scale.