Talk:The Riemann zeta function and tuning: Difference between revisions
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: Re: "When we talk about how well an equal temperament (ET) approximates just intonation, we're essentially asking: "How accurately can this system represent the harmonic series?"" | : Re: "When we talk about how well an equal temperament (ET) approximates just intonation, we're essentially asking: "How accurately can this system represent the harmonic series?"" | ||
: I believe this is wrong/misleading. The reason the list of EDOs given by zeta records is so sparse is because it measures only pure relative error and doesn't care whether you are looking at 2edo or 2000edo. It's fundamentally not fair to characterise it as "how accurately it can represent"; it's "how tone-efficiently it can represent for its size, with no other considerations". I've shown you get a lot more interesting EDOs/ETs if you multiply the resulting score by the size of the EDO/ET, and yet more if you consider things that, while not record peaks, are still better than one of the best | : I believe this is wrong/misleading. The reason the list of EDOs given by zeta records is so sparse is because it measures only pure relative error and doesn't care whether you are looking at 2edo or 2000edo. It's fundamentally not fair to characterise it as "how accurately it can represent"; it's "how tone-efficiently it can represent for its size, with no other considerations". I've shown you get a lot more interesting EDOs/ETs if you multiply the resulting score by the size of the EDO/ET, and yet more if you consider things that, while not record peaks, are still better than one of the best ''n'' peaks found so far (where I suggest ''n'' = 3 as recovering almost all interesting tuning info). --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 18:56, 14 April 2025 (UTC) | ||
: Also, I'm not convinced that zeta integral is a useless metric to the point of not including it because it represents how well an equal temperament does when detuned so that it is a "peak" in a more general sense. In other words, because of this property, I firmly believe the zeta integral list makes more sense to think of as a "zeta EDO list" than the zeta peak list. You could say taking the values at the EDOs makes the most sense but after consideration of how zeta works/behaves, allowing octave tempering can be seen as a method of accounting for the "tendency" of an equal temperament (whether it generally tunes primes sharp or flat), hence resulting in only favouring systems that tend close to just. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 19:01, 14 April 2025 (UTC) | |||