The Riemann zeta function and tuning: Difference between revisions

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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.

Now, one nitpick to notice above is that this expression technically involves all 'unreduced' rationals, e.g. there will be a cosine error term not just for 3/2, but also for 6/4, 9/6, etc. However, we can easily show that the same expression also measures the cosine relative error for ~~unreduced~~ rationals:

Now, one nitpick to notice above is that this expression technically involves all 'unreduced' rationals, e.g. there will be a cosine error term not just for 3/2, but also for 6/4, 9/6, etc. However, we can easily show that the same expression also measures the cosine relative error for reduced rationals:

==From Unreduced Rationals to Reduced Rationals==

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 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.
-Now, one nitpick to notice above is that this expression technically involves all 'unreduced' rationals, e.g. there will be a cosine error term not just for 3/2, but also for 6/4, 9/6, etc. However, we can easily show that the same expression also measures the cosine relative error for unreduced rationals:
+Now, one nitpick to notice above is that this expression technically involves all 'unreduced' rationals, e.g. there will be a cosine error term not just for 3/2, but also for 6/4, 9/6, etc. However, we can easily show that the same expression also measures the cosine relative error for reduced rationals:
 ==From Unreduced Rationals to Reduced Rationals==