The Riemann zeta function and tuning: Difference between revisions

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This, however, requires that an edo's zeta peak be record-holding with both pure and detuned octaves. It is actually debatable whether or not this constraint is a good idea, as it does not account for tuning tendencies that are skewed sharpward or flatward. If detuned octaves are allowed, the two smallest additional edos that are strict zeta edos would be [[72edo]] and [[954edo]].

~~==== Absolute zeta peak edos ====~~

If we consider that zeta is a measure of relative error (that is, error measured relative to the step size), we realize that plenty of equal temperaments* are excluded simply because, though practically speaking they have great tuning properties, they are not as "efficient" with their number of tones as the last record peak. Arguably what we're interested in is a sequence of edos that generally do increasingly better at tuning JI in terms of lowering the average cent error. Therefore, it suffices to multiply the score by the size of the equal temperament. Surprisingly, the list for ''s'' = 1/2 — which is supposedly where high-limit information is maximized — is ''almost identical'' to the one for ''s'' = 1 — which is the smallest value of ''s'' that we can assume to be meaningful without assuming that the analytic continuation preserves the tuning properties we are interested in — so that we have reassurance from the ''s'' = 1 list that the ''s'' = 1/2 list is meaningful wherever they agree. This is important because surprisingly, the two lists of equal temperament are ''identical up to [[311edo|311et]]'', with only one edo, [[8edo]], omitted from the list for ''s'' = 1. This list is {{EDOs| 1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 17, 19, 22, 24, 26, 27, 31, 34, 41, 46, 53, 58, 65, 68, 72, 84, 87, 94, 99, 111, 118, 130, 140, 152, 171, 183, 198, 212, 217, 224, 243, 270, 311, ... }}.

<nowiki>*</nowiki> Note importantly that we speak of "equal temperaments" rather than "edos" because generally a record peak ''does not'' correspond to an edo, which can have tangible consequences (a significant example is discussed in the next section).

~~===== Extended list of absolute zeta peak edos =====~~

If you look at the graph of zeta (for any zeta graph of interest), another issue quickly becomes evident: many equal temperaments of interest fail to have peaks of record height by only small amounts, so that we intuitively want to include them in a more comprehensive list. However, trying to "fix" this issue quickly leads into another issue: how many "nearly record" edos should we include, and why? The smallest alteration we can make is to allow an equal temperament that does better than the second-best-scoring equal temperament so far. But sometimes we have two very strong equal temperaments appear in quick succession, and given the motivation is to find a more comprehensive list anyways, here we'll include any equal temperament that does better than the third-best-scoring equal temperament so far. The motivation for this cutoff is that you intuitively might expect that the three best equal temperaments found so far represent roughly how good we can do in a given range of step sizes, so that they define what is "normal" for that range, that is, it's the heuristic of the "rule of three". Again, the list for ''s'' = 1/2 is almost identical to ''s'' = 1 for equal temperaments up to 311et, though this time the differences are less trivial: [[176edo|176et]] and [[202edo|202et]] only appear for ''s'' = 1/2, so are put in brackets. The list is {{EDOs| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 34, 36, 38, 39*, 41, 43, 45, 46, 48, 50, 53, 56, 58, 60, 63, 65, 68, 72, 77, 80, 84, 87, 89, 94, 96, 99, 103, 106, 111, 113, 118, 121, 125, 130, 137, 140, 145, 149, 152, 159, 161, 166, 171, (176,) 183, 190, 193, 198, (202,) 212, 217, 224, 229, 239, 243, 248, 255, 270, 277, 282, 289, 301, 311, ... }}.

The equal temperaments added relative to the non-extended list of only things that are records proper are: {{EDOs| 6, 8, 11, 13, 16, 21, 29, 36, 38, 39*, 43, 45, 48, 50, 56, 60, 63, 77, 80, 89, 96, 103, 106, 113, 121, 125, 137, 145, 149, 159, 161, 166, (176,) 190, 193, (202,) 229, 239, 248, 255, 277, 282, 289, 301 }}. * 39et is a notable example because [[39edo]] corresponds to a zeta valley, so it's surprising that it would be included here; the reason that it is included is because this is ''not'' 39edo, but 39 ''equal temperament'', corresponding to a 3.8{{cent}} flat-tempered octave so that it is actually ~39.124edo, that is, it corresponds to the 173rd zeta peak, known by the shorthand 173zpi (where i stands for index). Therefore, this may prove a good testcase for investigating the effects of zeta-based octave-tempering, though given the size of the stretch, the difference is likely to be subtle, but the fact that it "changes zeta's mind" this much is itself interesting. You can also interpret this result differently, which is as evidence that you should not include equal temperaments worse than the third-best-scoring equal temperament so far, given the somewhat dubious inclusion of 39et, however it should be noted that this is more to do with that at the very beginning of the list there aren't many equal temperaments to "beat" so that beating the third-best-scoring equal temperament so far is easy, though arguably this isn't a flaw because people are often more likely to try a smaller equal temperament. It's also perhaps worth noting that 37et almost makes this extended list, but the omission of 37et is much better addressed by no-3's zeta.

==== List of record zeta edos ====

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Notice that there is a very large jump from [[79edo]] to [[5941edo]]. We know that record {{nowrap|{{abs|Z(x)}}}} scores, both with tempered octaves and pure octaves, grow logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval {{nowrap|[0, ''c'' log(''x'')]}}, the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos. So, it would be more productive to find edos which zeta score is simply less than a given threshold.

=== Absolute zeta peak edos ===

If we consider that zeta is a measure of relative error (that is, error measured relative to the step size), we realize that plenty of equal temperaments* are excluded simply because, though practically speaking they have great tuning properties, they are not as "efficient" with their number of tones as the last record peak. Arguably what we're interested in is a sequence of edos that generally do increasingly better at tuning JI in terms of lowering the average cent error. Therefore, it suffices to multiply the score by the size of the equal temperament. Surprisingly, the list for ''s'' = 1/2 — which is supposedly where high-limit information is maximized — is ''almost identical'' to the one for ''s'' = 1 — which is the smallest value of ''s'' that we can assume to be meaningful without assuming that the analytic continuation preserves the tuning properties we are interested in — so that we have reassurance from the ''s'' = 1 list that the ''s'' = 1/2 list is meaningful wherever they agree. This is important because surprisingly, the two lists of equal temperament are ''identical up to [[311edo|311et]]'', with only one edo, [[8edo]], omitted from the list for ''s'' = 1. This list is {{EDOs| 1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 17, 19, 22, 24, 26, 27, 31, 34, 41, 46, 53, 58, 65, 68, 72, 84, 87, 94, 99, 111, 118, 130, 140, 152, 171, 183, 198, 212, 217, 224, 243, 270, 311, ... }}.

<nowiki>*</nowiki> Note importantly that we speak of "equal temperaments" rather than "edos" because generally a record peak ''does not'' correspond to an edo, which can have tangible consequences (a significant example is discussed in the next section).

==== Extended list of absolute zeta peak edos ====

If you look at the graph of zeta (for any zeta graph of interest), another issue quickly becomes evident: many equal temperaments of interest fail to have peaks of record height by only small amounts, so that we intuitively want to include them in a more comprehensive list. However, trying to "fix" this issue quickly leads into another issue: how many "nearly record" edos should we include, and why? The smallest alteration we can make is to allow an equal temperament that does better than the second-best-scoring equal temperament so far. But sometimes we have two very strong equal temperaments appear in quick succession, and given the motivation is to find a more comprehensive list anyways, here we'll include any equal temperament that does better than the third-best-scoring equal temperament so far. The motivation for this cutoff is that you intuitively might expect that the three best equal temperaments found so far represent roughly how good we can do in a given range of step sizes, so that they define what is "normal" for that range, that is, it's the heuristic of the "rule of three". Again, the list for ''s'' = 1/2 is almost identical to ''s'' = 1 for equal temperaments up to 311et, though this time the differences are less trivial: [[176edo|176et]] and [[202edo|202et]] only appear for ''s'' = 1/2, so are put in brackets. The list is {{EDOs| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 34, 36, 38, 39*, 41, 43, 45, 46, 48, 50, 53, 56, 58, 60, 63, 65, 68, 72, 77, 80, 84, 87, 89, 94, 96, 99, 103, 106, 111, 113, 118, 121, 125, 130, 137, 140, 145, 149, 152, 159, 161, 166, 171, (176,) 183, 190, 193, 198, (202,) 212, 217, 224, 229, 239, 243, 248, 255, 270, 277, 282, 289, 301, 311, … }}.

The equal temperaments added relative to the non-extended list of only things that are records proper are: {{EDOs| 6, 8, 11, 13, 16, 21, 29, 36, 38, 39*, 43, 45, 48, 50, 56, 60, 63, 77, 80, 89, 96, 103, 106, 113, 121, 125, 137, 145, 149, 159, 161, 166, (176,) 190, 193, (202,) 229, 239, 248, 255, 277, 282, 289, 301 }}. * 39et is a notable example because [[39edo]] corresponds to a zeta valley, so it's surprising that it would be included here; the reason that it is included is because this is ''not'' 39edo, but 39 ''equal temperament'', corresponding to a 3.8{{cent}} flat-tempered octave so that it is actually ~39.124edo, that is, it corresponds to the 173rd zeta peak, known by the shorthand 173zpi (where i stands for index). Therefore, this may prove a good testcase for investigating the effects of zeta-based octave-tempering, though given the size of the stretch, the difference is likely to be subtle, but the fact that it "changes zeta's mind" this much is itself interesting. You can also interpret this result differently, which is as evidence that you should not include equal temperaments worse than the third-best-scoring equal temperament so far, given the somewhat dubious inclusion of 39et, however it should be noted that this is more to do with that at the very beginning of the list there aren't many equal temperaments to "beat" so that beating the third-best-scoring equal temperament so far is easy, though arguably this isn't a flaw because people are often more likely to try a smaller equal temperament. It's also perhaps worth noting that 37et almost makes this extended list, but the omission of 37et is much better addressed by no-3's zeta.

=== ''k''-ary-peak edos ===

@@ Line 363: / Line 363: @@
 This, however, requires that an edo's zeta peak be record-holding with both pure and detuned octaves. It is actually debatable whether or not this constraint is a good idea, as it does not account for tuning tendencies that are skewed sharpward or flatward. If detuned octaves are allowed, the two smallest additional edos that are strict zeta edos would be [[72edo]] and [[954edo]].
-==== Absolute zeta peak edos ====
-If we consider that zeta is a measure of relative error (that is, error measured relative to the step size), we realize that plenty of equal temperaments* are excluded simply because, though practically speaking they have great tuning properties, they are not as "efficient" with their number of tones as the last record peak. Arguably what we're interested in is a sequence of edos that generally do increasingly better at tuning JI in terms of lowering the average cent error. Therefore, it suffices to multiply the score by the size of the equal temperament. Surprisingly, the list for ''s'' = 1/2 — which is supposedly where high-limit information is maximized — is ''almost identical'' to the one for ''s'' = 1 — which is the smallest value of ''s'' that we can assume to be meaningful without assuming that the analytic continuation preserves the tuning properties we are interested in — so that we have reassurance from the ''s'' = 1 list that the ''s'' = 1/2 list is meaningful wherever they agree. This is important because surprisingly, the two lists of equal temperament are ''identical up to [[311edo|311et]]'', with only one edo, [[8edo]], omitted from the list for ''s'' = 1. This list is {{EDOs| 1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 17, 19, 22, 24, 26, 27, 31, 34, 41, 46, 53, 58, 65, 68, 72, 84, 87, 94, 99, 111, 118, 130, 140, 152, 171, 183, 198, 212, 217, 224, 243, 270, 311, ... }}.
-<nowiki>*</nowiki> Note importantly that we speak of "equal temperaments" rather than "edos" because generally a record peak ''does not'' correspond to an edo, which can have tangible consequences (a significant example is discussed in the next section).
-===== Extended list of absolute zeta peak edos =====
-If you look at the graph of zeta (for any zeta graph of interest), another issue quickly becomes evident: many equal temperaments of interest fail to have peaks of record height by only small amounts, so that we intuitively want to include them in a more comprehensive list. However, trying to "fix" this issue quickly leads into another issue: how many "nearly record" edos should we include, and why? The smallest alteration we can make is to allow an equal temperament that does better than the second-best-scoring equal temperament so far. But sometimes we have two very strong equal temperaments appear in quick succession, and given the motivation is to find a more comprehensive list anyways, here we'll include any equal temperament that does better than the third-best-scoring equal temperament so far. The motivation for this cutoff is that you intuitively might expect that the three best equal temperaments found so far represent roughly how good we can do in a given range of step sizes, so that they define what is "normal" for that range, that is, it's the heuristic of the "rule of three". Again, the list for ''s'' = 1/2 is almost identical to ''s'' = 1 for equal temperaments up to 311et, though this time the differences are less trivial: [[176edo|176et]] and [[202edo|202et]] only appear for ''s'' = 1/2, so are put in brackets. The list is {{EDOs| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 34, 36, 38, 39*, 41, 43, 45, 46, 48, 50, 53, 56, 58, 60, 63, 65, 68, 72, 77, 80, 84, 87, 89, 94, 96, 99, 103, 106, 111, 113, 118, 121, 125, 130, 137, 140, 145, 149, 152, 159, 161, 166, 171, (176,) 183, 190, 193, 198, (202,) 212, 217, 224, 229, 239, 243, 248, 255, 270, 277, 282, 289, 301, 311, ... }}.
-The equal temperaments added relative to the non-extended list of only things that are records proper are: {{EDOs| 6, 8, 11, 13, 16, 21, 29, 36, 38, 39*, 43, 45, 48, 50, 56, 60, 63, 77, 80, 89, 96, 103, 106, 113, 121, 125, 137, 145, 149, 159, 161, 166, (176,) 190, 193, (202,) 229, 239, 248, 255, 277, 282, 289, 301 }}. * 39et is a notable example because [[39edo]] corresponds to a zeta valley, so it's surprising that it would be included here; the reason that it is included is because this is ''not'' 39edo, but 39 ''equal temperament'', corresponding to a 3.8{{cent}} flat-tempered octave so that it is actually ~39.124edo, that is, it corresponds to the 173rd zeta peak, known by the shorthand 173zpi (where i stands for index). Therefore, this may prove a good testcase for investigating the effects of zeta-based octave-tempering, though given the size of the stretch, the difference is likely to be subtle, but the fact that it "changes zeta's mind" this much is itself interesting. You can also interpret this result differently, which is as evidence that you should not include equal temperaments worse than the third-best-scoring equal temperament so far, given the somewhat dubious inclusion of 39et, however it should be noted that this is more to do with that at the very beginning of the list there aren't many equal temperaments to "beat" so that beating the third-best-scoring equal temperament so far is easy, though arguably this isn't a flaw because people are often more likely to try a smaller equal temperament. It's also perhaps worth noting that 37et almost makes this extended list, but the omission of 37et is much better addressed by no-3's zeta.
 ==== List of record zeta edos ====
@@ Line 569: / Line 559: @@
 Notice that there is a very large jump from [[79edo]] to [[5941edo]]. We know that record {{nowrap|{{abs|Z(x)}}}} scores, both with tempered octaves and pure octaves, grow logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval {{nowrap|[0, ''c'' log(''x'')]}}, the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos. So, it would be more productive to find edos which zeta score is simply less than a given threshold.
+=== Absolute zeta peak edos ===
+If we consider that zeta is a measure of relative error (that is, error measured relative to the step size), we realize that plenty of equal temperaments* are excluded simply because, though practically speaking they have great tuning properties, they are not as "efficient" with their number of tones as the last record peak. Arguably what we're interested in is a sequence of edos that generally do increasingly better at tuning JI in terms of lowering the average cent error. Therefore, it suffices to multiply the score by the size of the equal temperament. Surprisingly, the list for ''s'' = 1/2 — which is supposedly where high-limit information is maximized — is ''almost identical'' to the one for ''s'' = 1 — which is the smallest value of ''s'' that we can assume to be meaningful without assuming that the analytic continuation preserves the tuning properties we are interested in — so that we have reassurance from the ''s'' = 1 list that the ''s'' = 1/2 list is meaningful wherever they agree. This is important because surprisingly, the two lists of equal temperament are ''identical up to [[311edo|311et]]'', with only one edo, [[8edo]], omitted from the list for ''s'' = 1. This list is {{EDOs| 1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 17, 19, 22, 24, 26, 27, 31, 34, 41, 46, 53, 58, 65, 68, 72, 84, 87, 94, 99, 111, 118, 130, 140, 152, 171, 183, 198, 212, 217, 224, 243, 270, 311, ... }}.
+<nowiki>*</nowiki> Note importantly that we speak of "equal temperaments" rather than "edos" because generally a record peak ''does not'' correspond to an edo, which can have tangible consequences (a significant example is discussed in the next section).
+==== Extended list of absolute zeta peak edos ====
+If you look at the graph of zeta (for any zeta graph of interest), another issue quickly becomes evident: many equal temperaments of interest fail to have peaks of record height by only small amounts, so that we intuitively want to include them in a more comprehensive list. However, trying to "fix" this issue quickly leads into another issue: how many "nearly record" edos should we include, and why? The smallest alteration we can make is to allow an equal temperament that does better than the second-best-scoring equal temperament so far. But sometimes we have two very strong equal temperaments appear in quick succession, and given the motivation is to find a more comprehensive list anyways, here we'll include any equal temperament that does better than the third-best-scoring equal temperament so far. The motivation for this cutoff is that you intuitively might expect that the three best equal temperaments found so far represent roughly how good we can do in a given range of step sizes, so that they define what is "normal" for that range, that is, it's the heuristic of the "rule of three". Again, the list for ''s'' = 1/2 is almost identical to ''s'' = 1 for equal temperaments up to 311et, though this time the differences are less trivial: [[176edo|176et]] and [[202edo|202et]] only appear for ''s'' = 1/2, so are put in brackets. The list is {{EDOs| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 34, 36, 38, 39*, 41, 43, 45, 46, 48, 50, 53, 56, 58, 60, 63, 65, 68, 72, 77, 80, 84, 87, 89, 94, 96, 99, 103, 106, 111, 113, 118, 121, 125, 130, 137, 140, 145, 149, 152, 159, 161, 166, 171, (176,) 183, 190, 193, 198, (202,) 212, 217, 224, 229, 239, 243, 248, 255, 270, 277, 282, 289, 301, 311, … }}.
+The equal temperaments added relative to the non-extended list of only things that are records proper are: {{EDOs| 6, 8, 11, 13, 16, 21, 29, 36, 38, 39*, 43, 45, 48, 50, 56, 60, 63, 77, 80, 89, 96, 103, 106, 113, 121, 125, 137, 145, 149, 159, 161, 166, (176,) 190, 193, (202,) 229, 239, 248, 255, 277, 282, 289, 301 }}. * 39et is a notable example because [[39edo]] corresponds to a zeta valley, so it's surprising that it would be included here; the reason that it is included is because this is ''not'' 39edo, but 39 ''equal temperament'', corresponding to a 3.8{{cent}} flat-tempered octave so that it is actually ~39.124edo, that is, it corresponds to the 173rd zeta peak, known by the shorthand 173zpi (where i stands for index). Therefore, this may prove a good testcase for investigating the effects of zeta-based octave-tempering, though given the size of the stretch, the difference is likely to be subtle, but the fact that it "changes zeta's mind" this much is itself interesting. You can also interpret this result differently, which is as evidence that you should not include equal temperaments worse than the third-best-scoring equal temperament so far, given the somewhat dubious inclusion of 39et, however it should be noted that this is more to do with that at the very beginning of the list there aren't many equal temperaments to "beat" so that beating the third-best-scoring equal temperament so far is easy, though arguably this isn't a flaw because people are often more likely to try a smaller equal temperament. It's also perhaps worth noting that 37et almost makes this extended list, but the omission of 37et is much better addressed by no-3's zeta.
 === ''k''-ary-peak edos ===