The Riemann zeta function and tuning: Difference between revisions

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<span style="line-height: 1.5;">It is extremely noteworthy to mention how "composite" rationals are treated differently than with TE error. In addition to our usual error metric on the primes, we also go to each rational, look for the best "direct" or "patent" mapping of that rational within the EDO, and add 'that' to the EDO's score. In particular, we do this even when the best mapping for some rational doesn't match up with the mapping you'd get from it just looking at the primes.

So, for instance, in 16-EDO, the best mapping for 3/2 is 9 steps out of 16, and using that mapping, we get that 9/8 is 2 steps, since {{nowrap|9 * 2 − 16 {{=}} 2}}. However, there is a better mapping for 9/8 at 3 steps—one which ignores the fact that it is no longer equal to two 3/2's. This can be particularly useful for playing chords: 16-EDO's "direct mapping" for 9 is useful when playing the chord 4:5:7:9, and the "indirect" or "prime-based" mapping for 9 is useful when playing the "major 9" chord 8:10:12:15:18. We can think of the zeta function as rewarding equal temperaments not just for having a good approximation of the primes, but also for having good "extra" approximations of rationals which can be used in this way. And although 16-EDO is pretty high error, similar phenomena can be found for any EDO which becomes [[~~consistency|~~inconsistent]] for some chord of interest.

So, for instance, in 16-EDO, the best mapping for 3/2 is 9 steps out of 16, and using that mapping, we get that 9/8 is 2 steps, since {{nowrap|9 * 2 − 16 {{=}} 2}}. However, there is a better mapping for 9/8 at 3 steps—one which ignores the fact that it is no longer equal to two 3/2's. This can be particularly useful for playing chords: 16-EDO's "direct mapping" for 9 is useful when playing the chord 4:5:7:9, and the "indirect" or "prime-based" mapping for 9 is useful when playing the "major 9" chord 8:10:12:15:18. We can think of the zeta function as rewarding equal temperaments not just for having a good approximation of the primes, but also for having good "extra" approximations of rationals which can be used in this way. And although 16-EDO is pretty high error, similar phenomena can be found for any EDO which becomes [[inconsistent]] for some chord of interest.

One way to frame this in the usual group-theoretic paradigm is to consider the group in which each strictly positive rational number is given its own linearly independent basis element. In other words, look at the [https://en.wikipedia.org/wiki/Free_group free group] over the strictly positive rationals, which we'll call ''"meta-JI."'' The zeta function can then be thought of as yielding an error for all meta-JI [[Patent_val|generalized patent vals]]. Whether this can be extended to all meta-JI vals, or modified to yield something nice like a "norm" on the group of meta-JI vals, is an open question. Regardless, this may be a useful conceptual bridge to understand how to relate the zeta function to "ordinary" regular temperament theory.

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 <span style="line-height: 1.5;">It is extremely noteworthy to mention how "composite" rationals are treated differently than with TE error. In addition to our usual error metric on the primes, we also go to each rational, look for the best "direct" or "patent" mapping of that rational within the EDO, and add 'that' to the EDO's score. In particular, we do this even when the best mapping for some rational doesn't match up with the mapping you'd get from it just looking at the primes.
-So, for instance, in 16-EDO, the best mapping for 3/2 is 9 steps out of 16, and using that mapping, we get that 9/8 is 2 steps, since {{nowrap|9 * 2 − 16 {{=}} 2}}. However, there is a better mapping for 9/8 at 3 steps—one which ignores the fact that it is no longer equal to two 3/2's. This can be particularly useful for playing chords: 16-EDO's "direct mapping" for 9 is useful when playing the chord 4:5:7:9, and the "indirect" or "prime-based" mapping for 9 is useful when playing the "major 9" chord 8:10:12:15:18. We can think of the zeta function as rewarding equal temperaments not just for having a good approximation of the primes, but also for having good "extra" approximations of rationals which can be used in this way. And although 16-EDO is pretty high error, similar phenomena can be found for any EDO which becomes [[consistency|inconsistent]] for some chord of interest.
+So, for instance, in 16-EDO, the best mapping for 3/2 is 9 steps out of 16, and using that mapping, we get that 9/8 is 2 steps, since {{nowrap|9 * 2 − 16 {{=}} 2}}. However, there is a better mapping for 9/8 at 3 steps—one which ignores the fact that it is no longer equal to two 3/2's. This can be particularly useful for playing chords: 16-EDO's "direct mapping" for 9 is useful when playing the chord 4:5:7:9, and the "indirect" or "prime-based" mapping for 9 is useful when playing the "major 9" chord 8:10:12:15:18. We can think of the zeta function as rewarding equal temperaments not just for having a good approximation of the primes, but also for having good "extra" approximations of rationals which can be used in this way. And although 16-EDO is pretty high error, similar phenomena can be found for any EDO which becomes [[inconsistent]] for some chord of interest.
 One way to frame this in the usual group-theoretic paradigm is to consider the group in which each strictly positive rational number is given its own linearly independent basis element. In other words, look at the [https://en.wikipedia.org/wiki/Free_group free group] over the strictly positive rationals, which we'll call ''"meta-JI."'' The zeta function can then be thought of as yielding an error for all meta-JI [[Patent_val|generalized patent vals]]. Whether this can be extended to all meta-JI vals, or modified to yield something nice like a "norm" on the group of meta-JI vals, is an open question. Regardless, this may be a useful conceptual bridge to understand how to relate the zeta function to "ordinary" regular temperament theory.