Talk:The Riemann Zeta Function and Tuning

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Zeta troughs?

Would it be possible to use this function to calculate increasingly deep troughs as well as peaks, so as to produce a sequence of EDOs that have increasingly high levels of relative error for their size & inconsistency between various limits? That could be an interesting addition to the lists for people who want to produce intentionally perverse & dissonant music. --Yourmusic Productions (talk) 17:42, 6 April 2021 (UTC)

Yes, although since we are looking at the absolute value of the zeta function can be no greater than 0. So, any EDO which has a zeta value of 0 is "maximally inharmonic," in a sense. According to the Riemann hypothesis, anyway, the function can only get to 0 if we're on the critical line, which means we're weighting rationals as [math]1/(nd)^{0.5}[/math]. But even on other lines there will always be local minima. Strangely, the touch tone "DTMF" frequencies were chosen, they say, to be maximally inharmonic, because they didn't want to confuse harmonics of the DTMF tones with other tones. You may think that this would mean they went for a zeta zero, but if you look you will see that all of the touch tone frequencies (which form a subset of an equal temperament) magically happen to synchronize with a local *maximum* of the zeta function - though a relatively small one - to within rounding error in Hz. In other words, the DTMF folks went with a relatively inharmonic region of EDO space (14-EDO ish), but then they maximized the harmonicity within that region. Very strange... Mike Battaglia (talk) 11:13, 9 April 2021 (UTC)