The Riemann zeta function and tuning: Difference between revisions

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==== Zeta peak edos ====

If we examine the increasingly larger peak values of {{nowrap|{{!}}Z(''x''){{!}}}}, we find they occur with values of ''x'' such that {{nowrap|Z'(''x'') {{=}} 0}} near to integers, so that there is a sequence of [[edo]]s {{EDOs| 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664, 14348, 16808, 28742, 34691, 36269, 57578, 58973, 95524, 102557, 112985, 148418, 212147, 241200,}} … of '''zeta peak edos'''. This is listed in the On-Line Encyclopedia of Integer Sequences as {{OEIS|A117536}}. Note that these peaks occur close to integer values, but are never exactly located at an integer; this can be interpreted as the zeta function suggesting [[stretched and compressed tuning|stretched or compressed ~~octaves~~]] for the edo in question, similar to the [[TOP tuning]] (although the two tunings are in general not the same). As a result, this list can also be thought of as "tempered-octave zeta peak edos."

If we examine the increasingly larger peak values of {{nowrap|{{!}}Z(''x''){{!}}}}, we find they occur with values of ''x'' such that {{nowrap|Z'(''x'') {{=}} 0}} near to integers, so that there is a sequence of [[edo]]s {{EDOs| 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664, 14348, 16808, 28742, 34691, 36269, 57578, 58973, 95524, 102557, 112985, 148418, 212147, 241200,}} … of '''zeta peak edos'''. This is listed in the On-Line Encyclopedia of Integer Sequences as {{OEIS|A117536}}. Note that these peaks occur close to integer values, but are never exactly located at an integer; this can be interpreted as the zeta function suggesting detuned ([[stretched and compressed tuning|stretched or compressed]]) octaves for the edo in question, similar to the [[TOP tuning]] (although the two tunings are in general not the same). As a result, this list can also be thought of as "tempered-octave zeta peak edos."

==== Zeta peak integer edos ====

@@ Line 293: / Line 293: @@
 ==== Zeta peak edos ====
-If we examine the increasingly larger peak values of {{nowrap|{{!}}Z(''x''){{!}}}}, we find they occur with values of ''x'' such that {{nowrap|Z'(''x'') {{=}} 0}} near to integers, so that there is a sequence of [[edo]]s {{EDOs| 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664, 14348, 16808, 28742, 34691, 36269, 57578, 58973, 95524, 102557, 112985, 148418, 212147, 241200,}} … of '''zeta peak edos'''. This is listed in the On-Line Encyclopedia of Integer Sequences as {{OEIS|A117536}}. Note that these peaks occur close to integer values, but are never exactly located at an integer; this can be interpreted as the zeta function suggesting [[stretched and compressed tuning|stretched or compressed octaves]] for the edo in question, similar to the [[TOP tuning]] (although the two tunings are in general not the same). As a result, this list can also be thought of as "tempered-octave zeta peak edos."
+If we examine the increasingly larger peak values of {{nowrap|{{!}}Z(''x''){{!}}}}, we find they occur with values of ''x'' such that {{nowrap|Z'(''x'') {{=}} 0}} near to integers, so that there is a sequence of [[edo]]s {{EDOs| 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664, 14348, 16808, 28742, 34691, 36269, 57578, 58973, 95524, 102557, 112985, 148418, 212147, 241200,}} … of '''zeta peak edos'''. This is listed in the On-Line Encyclopedia of Integer Sequences as {{OEIS|A117536}}. Note that these peaks occur close to integer values, but are never exactly located at an integer; this can be interpreted as the zeta function suggesting detuned ([[stretched and compressed tuning|stretched or compressed]]) octaves for the edo in question, similar to the [[TOP tuning]] (although the two tunings are in general not the same). As a result, this list can also be thought of as "tempered-octave zeta peak edos."
 ==== Zeta peak integer edos ====