The Riemann zeta function and tuning: Difference between revisions

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

Alternatively (as [[groundfault]] has found), if we instead ~~only~~ look at the record {{nowrap|{{!}}Z(''x''){{!}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs|1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}}{{ellip}} of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}}{{ellip}} and edos present in the previous list but not present here include {{EDOs|4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}}{{ellip}}. 72's removal is perhaps being the most surprising, showing ~~how strong~~ 53 is in that 72 does not improve on 53's peak. This definition may be better for measuring how accurate EDOs are without stretched or compressed octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos." Similarly, we can look at pure-tritave EDTs, etc.

Alternatively (as [[groundfault]] has found), if we do not allow octave detuning and instead look at only the record {{nowrap|{{!}}Z(''x''){{!}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs|1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}}{{ellip}} of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}}{{ellip}} and edos present in the previous list but not present here include {{EDOs|4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}}{{ellip}} with 72's removal perhaps being the most surprising, showing the strength of 53 in that 72 does not improve on 53's peak. This definition may be better for measuring how accurate EDOs are without stretched or compressed octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos."

Similarly, we can look at pure-tritave EDTs, etc.

==== Zeta integral edos ====

@@ Line 293: / Line 293: @@
 ==== Zeta peak integer edos ====
-Alternatively (as [[groundfault]] has found), if we instead only look at the record {{nowrap|{{!}}Z(''x''){{!}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs|1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}}{{ellip}} of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}}{{ellip}} and edos present in the previous list but not present here include {{EDOs|4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}}{{ellip}}. 72's removal is perhaps being the most surprising, showing how strong 53 is in that 72 does not improve on 53's peak. This definition may be better for measuring how accurate EDOs are without stretched or compressed octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos." Similarly, we can look at pure-tritave EDTs, etc.
+Alternatively (as [[groundfault]] has found), if we do not allow octave detuning and instead look at only the record {{nowrap|{{!}}Z(''x''){{!}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs|1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}}{{ellip}} of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}}{{ellip}} and edos present in the previous list but not present here include {{EDOs|4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}}{{ellip}} with 72's removal perhaps being the most surprising, showing the strength of 53 in that 72 does not improve on 53's peak. This definition may be better for measuring how accurate EDOs are without stretched or compressed octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos."
+Similarly, we can look at pure-tritave EDTs, etc.
 ==== Zeta integral edos ====