The Riemann zeta function and tuning: Difference between revisions
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<math>\rround{x} = \abs{x - \floor{x + \frac{1}{2}}}</math> | <math>\rround{x} = \abs{x - \floor{x + \frac{1}{2}}}</math> | ||
For example, ⌊8.202⌉ would be 0.202, since it is the difference between 8.202 and the nearest integer, which is 8. Meanwhile, ⌊7.95⌉ would be 0.05, which is the difference between 7.95 and the nearest integer, which is 8. | For example, ⌊8.202⌉ would be 0.202, since it is the difference between 8.202 and the nearest integer, which is 8. Meanwhile, ⌊7.95⌉ would be 0.05, which is the difference between 7.95 and the nearest integer, which is 8. This represents the absolute relative error of the octave in equal tuning ''x'', or alternatively how much x is detuned from an edo. | ||
For any value of ''x'', we can construct a ''p''-limit [[Patent_val|generalized patent val]]. We do so by rounding {{nowrap|''x'' log<sub>2</sub>(''q'')}} to the nearest integer for each prime ''q'' up to ''p''. Now consider the function | For any value of ''x'', we can construct a ''p''-limit [[Patent_val|generalized patent val]]. We do so by rounding {{nowrap|''x'' log<sub>2</sub>(''q'')}} to the nearest integer for each prime ''q'' up to ''p''. Now consider the function | ||