Talk:The Riemann zeta function and tuning: Difference between revisions
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::: Yeah, that's the plan. To be clear, my reasoning for posting the link was asking for feedback on the writing that is there now. It's obviously a work-in-progress / incomplete page as it stands. | ::: Yeah, that's the plan. To be clear, my reasoning for posting the link was asking for feedback on the writing that is there now. It's obviously a work-in-progress / incomplete page as it stands. | ||
::: – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 23:04, 14 April 2025 (UTC) | ::: – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 23:04, 14 April 2025 (UTC) | ||
== Criticisms of & 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: | |||
{{EDOs| 1, 2, 3, 4, 5, 6, 7, 8, 9, 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). | |||
The way I am judging is looking how many occurrences there are of that EDO in the <code>optimal_edo_sequence</code>s for odd-limits 23 thru 123. For EDOs 72 or smaller it's possible to evaluate manually fairly easily, so that this is mainly for larger EDOs, because EG it's not obvious that 106et would be performant. Some EDOs like 190 appear very rarely by this metric, but as 190 has about the same score as 193 (so that IIRC zeta slightly prefers 190 to 193) it seems worth including. Examples of large EDOs present in none of these zeta lists discusssed so far but appearing abundantly in the <code>optimal_edo_sequence</code>s for odd-limits 23 thru 123 are [[181edo]] and [[258edo]]], where the former is notably only two off from [[183edo]]. 181 and 183 appear the same number of times (25), where ~half of the time only one of the two appear and the other ~half both appear. An example of a medium EDO that appears in a variety of altered zeta EDO lists but which hasn't appeared any of these is [[62edo]], which appears in the <code>optimal_edo_sequence</code> for every odd-limit 19 thru 77 except 27, as well as in the 93-, 95- and 97-odd-limit. | |||
To my eye, the only EDOs that seem out of place in the extended list that I documented are {{EDOs| 38, 39, 45, 60, 96, 176, 212 }}, which I suspected intuitively, but confirmed via checking number of occurrences in the <code>optimal_edo_sequence</code>s for odd-limits 19 thru 123: 38 has one appearance (39-odd-limit), 39 has zero (though that one's unfair cuz it's not octave-tempered but zeta tells us it really should be), 45 has zero, 60 has two (31- and 33-odd-limit), 96 has zero, 176 has zero and 212 has zero. But for some reason, zeta prefers 60et over 63et generally speaking. | |||
--[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 18:10, 16 April 2025 (UTC) | |||