Talk:The Riemann zeta function and tuning: Difference between revisions
m →Criticisms of & possible improvements to new list: make title easier to link to (& -> and) |
m →Criticisms of and possible improvements to new list: remove redundant part of list (the part where every EDO < 11 is included, therefore it makes sense to start at 12) |
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== Criticisms of and possible improvements to new list == | == Criticisms of and possible improvements to new list == | ||
If anyone doesn't think these lists are high-quality enough I first encourage naming a bunch of EDOs you don't think should be included in the list (note that 39edo isn't included in any list because it's 39et that's included, as 39edo is near a zeta valley), and then discussing how to improve it. I agree that there are a few ETs in the extended list that seem out of place, and that it seems to get slightly more dubious as we look at larger ETs. Therefore one fix is instead of looking at the absolute error (multiplying by the ET) we could still take tone-efficiency into account by multiplying by the square-root of the ET, but this is harder to motivate beyond "a balance between only caring about relative error (efficiency) and only caring about absolute error (tuning damage)", so it sort of implicates the inclusion of the latter list as a prerequisite so we have something to compare to. Another alteration is trying to improve the extended list. For example, we could include all ETs that do better than the second-best ''record-setter'' rather than the second-best ''scorer'', so that we have a bound that is more forgiving than "second-best scorer" without including too many ETs as "better than third-best scorer" might be judged to do. This is again harder to motivate, but if we take both of these alterations as a given, the list is extremely high-quality. Again, {{nowrap| ''s'' {{=}} 1/2}} and {{nowrap| ''s'' {{=}} 1}} give the same results thru 311et, with the only difference being that {{nowrap| ''s'' {{=}} 1/2}} includes 37et and 121et. The resulting sequence is: | If anyone doesn't think these lists are high-quality enough I first encourage naming a bunch of EDOs you don't think should be included in the list (note that 39edo isn't included in any list because it's 39et that's included, as 39edo is near a zeta valley), and then discussing how to improve it. I agree that there are a few ETs in the extended list that seem out of place, and that it seems to get slightly more dubious as we look at larger ETs. Therefore one fix is instead of looking at the absolute error (multiplying by the ET) we could still take tone-efficiency into account by multiplying by the square-root of the ET, but this is harder to motivate beyond "a balance between only caring about relative error (efficiency) and only caring about absolute error (tuning damage)", so it sort of implicates the inclusion of the latter list as a prerequisite so we have something to compare to. Another alteration is trying to improve the extended list. For example, we could include all ETs that do better than the second-best ''record-setter'' rather than the second-best ''scorer'', so that we have a bound that is more forgiving than "second-best scorer" without including too many ETs as "better than third-best scorer" might be judged to do. This is again harder to motivate, but if we take both of these alterations as a given, the list is extremely high-quality. Again, {{nowrap| ''s'' {{=}} 1/2}} and {{nowrap| ''s'' {{=}} 1}} give the same results thru 311et, with the only difference being that {{nowrap| ''s'' {{=}} 1/2}} includes 37et and 121et. The resulting sequence includes every ET < 11, so I'll start it at 10. This sequence is: | ||
{{EDOs| | {{EDOs| 10, 12, 14, 15, 17, 19, 22, 24, 26, 27, 29, 31, 34, 36, (37,) 38, 39, 41, 43, 46, 53, 58, 60, 63, 65, 68, 72, 77, 80, 84, 87, 94, 99, 103, 111, 118, (121,) 125, 130, 140, 152, 159, 171, 183, 217, 224, 243, 270, 282, 289, 296, 301, 311 }} | ||
I can say with confidence that every EDO >= 111 (except maybe 121) deserves to be here, though it's sad that {{EDOs| 48, 50, 56, 106, 113, 137, 149, 161, 193, 202, 229, 239, 248, 277 }} (which I mention as being present in the extended list I added) are missed (which also very much deserve to be there). | I can say with confidence that every EDO >= 111 (except maybe 121) deserves to be here, though it's sad that {{EDOs| 48, 50, 56, 106, 113, 137, 149, 161, 193, 202, 229, 239, 248, 277 }} (which I mention as being present in the extended list I added) are missed (which also very much deserve to be there). | ||