Talk:The Riemann zeta function and tuning - Revision history

Godtone: /* Alternate list based on unmodified zeta function */ (was thinking about signing again to indicate the small edit to this comment but decided against it but accidentally left in this remnant while editing)

2025-04-26T13:30:27Z

Alternate list based on unmodified zeta function: (was thinking about signing again to indicate the small edit to this comment but decided against it but accidentally left in this remnant while editing)

← Older revision		Revision as of 13:30, 26 April 2025
Line 140:		Line 140:

	--[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 18:39, 17 April 2025 (UTC)		--[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 18:39, 17 April 2025 (UTC)

	~~Edited & correc~~

	== Top 20 (and top 10) zeta edos ==		== Top 20 (and top 10) zeta edos ==

Godtone: /* Alternate list based on unmodified zeta function */ missing bracket

2025-04-26T13:29:29Z

Alternate list based on unmodified zeta function: missing bracket

← Older revision		Revision as of 13:29, 26 April 2025
Line 135:		Line 135:

	Take a look at [[User:Godtone/zeta#Top 10]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences		Take a look at [[User:Godtone/zeta#Top 10]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences
	(please share any findings/concerns here or on {{nowrap\| [[User talk:Godtone/zeta\|the talk page for the zeta page I made]]. }}		(please share any findings/concerns here or on {{nowrap\| [[User talk:Godtone/zeta\|the talk page for the zeta page I made]]). }}

	A rather strange recurring theme is 60edo is liked by zeta a surprising amount, but looking at its low- and high-limit tuning profile it doesn't seem that remarkable to me. (A strange coincidence is some time ago I had a dream that this was a good edo. That doesn't happen often at all (dreaming about edos, let alone a specific one being good; the dream said its 11-limit was good; maybe that's true in the sense that the high errors of 5, 7 and 11 can easily cancel each-other out in ratios or composites, since zeta doesn't obey a val). Also happens to be significant as the simplest way to represent fourth-order ambiguities in my theory of functional harmony which I derived from first principles starting from [[Ringer scale]]s (especially Perfect Ringer scales), so that (other than the 12edo intervals) it represents the most xenmelodically nontrivial categories available (which correspond to areas of nontrivial harmony).)		A rather strange recurring theme is 60edo is liked by zeta a surprising amount, but looking at its low- and high-limit tuning profile it doesn't seem that remarkable to me. (A strange coincidence is some time ago I had a dream that this was a good edo. That doesn't happen often at all (dreaming about edos, let alone a specific one being good; the dream said its 11-limit was good; maybe that's true in the sense that the high errors of 5, 7 and 11 can easily cancel each-other out in ratios or composites, since zeta doesn't obey a val). Also happens to be significant as the simplest way to represent fourth-order ambiguities in my theory of functional harmony which I derived from first principles starting from [[Ringer scale]]s (especially Perfect Ringer scales), so that (other than the 12edo intervals) it represents the most xenmelodically nontrivial categories available (which correspond to areas of nontrivial harmony).)

Godtone: /* Alternate list based on unmodified zeta function */ nowrap

2025-04-26T13:28:20Z

Alternate list based on unmodified zeta function: nowrap

← Older revision		Revision as of 13:28, 26 April 2025
Line 134:		Line 134:
	Here's a list based purely on unmodified zeta, in case someone proves the alteration is wrong (though I'd still be interested in an "absolute" version of the list that includes an equal temperament if it's better than the 3rd-best-scorer so far). Because there's no accounting for the size of the equal temperament, I'm giving equal temperaments a lot of chances to appear to try to account for this bias, so that an equal temperament appears if it's better than the 10th-best-scorer so far. The other reason I give so many chances is that the resulting list is very similar and also surprisingly high-quality.		Here's a list based purely on unmodified zeta, in case someone proves the alteration is wrong (though I'd still be interested in an "absolute" version of the list that includes an equal temperament if it's better than the 3rd-best-scorer so far). Because there's no accounting for the size of the equal temperament, I'm giving equal temperaments a lot of chances to appear to try to account for this bias, so that an equal temperament appears if it's better than the 10th-best-scorer so far. The other reason I give so many chances is that the resulting list is very similar and also surprisingly high-quality.

	Take a look at [[User:Godtone/zeta#Top 10]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences (please share any findings/concerns here or on [[User talk:Godtone/zeta\|the talk page for the zeta page I made]]).		Take a look at [[User:Godtone/zeta#Top 10]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences
			(please share any findings/concerns here or on {{nowrap\| [[User talk:Godtone/zeta\|the talk page for the zeta page I made]]. }}

	A rather strange recurring theme is 60edo is liked by zeta a surprising amount, but looking at its low- and high-limit tuning profile it doesn't seem that remarkable to me. (A strange coincidence is some time ago I had a dream that this was a good edo. That doesn't happen often at all (dreaming about edos, let alone a specific one being good; the dream said its 11-limit was good; maybe that's true in the sense that the high errors of 5, 7 and 11 can easily cancel each-other out in ratios or composites, since zeta doesn't obey a val). Also happens to be significant as the simplest way to represent fourth-order ambiguities in my theory of functional harmony which I derived from first principles starting from [[Ringer scale]]s (especially Perfect Ringer scales), so that (other than the 12edo intervals) it represents the most xenmelodically nontrivial categories available (which correspond to areas of nontrivial harmony).)		A rather strange recurring theme is 60edo is liked by zeta a surprising amount, but looking at its low- and high-limit tuning profile it doesn't seem that remarkable to me. (A strange coincidence is some time ago I had a dream that this was a good edo. That doesn't happen often at all (dreaming about edos, let alone a specific one being good; the dream said its 11-limit was good; maybe that's true in the sense that the high errors of 5, 7 and 11 can easily cancel each-other out in ratios or composites, since zeta doesn't obey a val). Also happens to be significant as the simplest way to represent fourth-order ambiguities in my theory of functional harmony which I derived from first principles starting from [[Ringer scale]]s (especially Perfect Ringer scales), so that (other than the 12edo intervals) it represents the most xenmelodically nontrivial categories available (which correspond to areas of nontrivial harmony).)

	--[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 18:39, 17 April 2025 (UTC)		--[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 18:39, 17 April 2025 (UTC)

			Edited & correc

	== Top 20 (and top 10) zeta edos ==		== Top 20 (and top 10) zeta edos ==

Godtone: /* Alternate list based on unmodified zeta function */ add invitation to share any findings/concerns

2025-04-26T13:27:11Z

Alternate list based on unmodified zeta function: add invitation to share any findings/concerns

← Older revision		Revision as of 13:27, 26 April 2025
Line 134:		Line 134:
	Here's a list based purely on unmodified zeta, in case someone proves the alteration is wrong (though I'd still be interested in an "absolute" version of the list that includes an equal temperament if it's better than the 3rd-best-scorer so far). Because there's no accounting for the size of the equal temperament, I'm giving equal temperaments a lot of chances to appear to try to account for this bias, so that an equal temperament appears if it's better than the 10th-best-scorer so far. The other reason I give so many chances is that the resulting list is very similar and also surprisingly high-quality.		Here's a list based purely on unmodified zeta, in case someone proves the alteration is wrong (though I'd still be interested in an "absolute" version of the list that includes an equal temperament if it's better than the 3rd-best-scorer so far). Because there's no accounting for the size of the equal temperament, I'm giving equal temperaments a lot of chances to appear to try to account for this bias, so that an equal temperament appears if it's better than the 10th-best-scorer so far. The other reason I give so many chances is that the resulting list is very similar and also surprisingly high-quality.

	Take a look at [[User:Godtone/zeta#Top 10]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences.		Take a look at [[User:Godtone/zeta#Top 10]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences (please share any findings/concerns here or on [[User talk:Godtone/zeta\|the talk page for the zeta page I made]]).

	A rather strange recurring theme is 60edo is liked by zeta a surprising amount, but looking at its low- and high-limit tuning profile it doesn't seem that remarkable to me. (A strange coincidence is some time ago I had a dream that this was a good edo. That doesn't happen often at all (dreaming about edos, let alone a specific one being good; the dream said its 11-limit was good; maybe that's true in the sense that the high errors of 5, 7 and 11 can easily cancel each-other out in ratios or composites, since zeta doesn't obey a val). Also happens to be significant as the simplest way to represent fourth-order ambiguities in my theory of functional harmony which I derived from first principles starting from [[Ringer scale]]s (especially Perfect Ringer scales), so that (other than the 12edo intervals) it represents the most xenmelodically nontrivial categories available (which correspond to areas of nontrivial harmony).)		A rather strange recurring theme is 60edo is liked by zeta a surprising amount, but looking at its low- and high-limit tuning profile it doesn't seem that remarkable to me. (A strange coincidence is some time ago I had a dream that this was a good edo. That doesn't happen often at all (dreaming about edos, let alone a specific one being good; the dream said its 11-limit was good; maybe that's true in the sense that the high errors of 5, 7 and 11 can easily cancel each-other out in ratios or composites, since zeta doesn't obey a val). Also happens to be significant as the simplest way to represent fourth-order ambiguities in my theory of functional harmony which I derived from first principles starting from [[Ringer scale]]s (especially Perfect Ringer scales), so that (other than the 12edo intervals) it represents the most xenmelodically nontrivial categories available (which correspond to areas of nontrivial harmony).)

Godtone: /* Alternate list based on unmodified zeta function */ correction as per updating of list to top 10 and as per more stuff being added to that page

2025-04-26T13:25:42Z

Alternate list based on unmodified zeta function: correction as per updating of list to top 10 and as per more stuff being added to that page

← Older revision		Revision as of 13:25, 26 April 2025
Line 132:		Line 132:

	=== Alternate list based on unmodified zeta function ===		=== Alternate list based on unmodified zeta function ===
	Here's a list based purely on unmodified zeta, in case someone proves the alteration is wrong (though I'd still be interested in an "absolute" version of the list that includes an equal temperament if it's better than the 3rd-best-scorer so far). Because there's no accounting for the size of the equal temperament, I'm giving equal temperaments a lot of chances to appear to try to account for this bias, so that an equal temperament appears if it's better than the ~~9th~~-best-scorer so far. The other reason I give so many chances is that the resulting list is very similar and also surprisingly high-quality.		Here's a list based purely on unmodified zeta, in case someone proves the alteration is wrong (though I'd still be interested in an "absolute" version of the list that includes an equal temperament if it's better than the 3rd-best-scorer so far). Because there's no accounting for the size of the equal temperament, I'm giving equal temperaments a lot of chances to appear to try to account for this bias, so that an equal temperament appears if it's better than the 10th-best-scorer so far. The other reason I give so many chances is that the resulting list is very similar and also surprisingly high-quality.

	Take a look at [[User:Godtone/zeta]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences.		Take a look at [[User:Godtone/zeta#Top 10]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences.

	A rather strange recurring theme is 60edo is liked by zeta a surprising amount, but looking at its low- and high-limit tuning profile it doesn't seem that remarkable to me. (A strange coincidence is some time ago I had a dream that this was a good edo. That doesn't happen often at all (dreaming about edos, let alone a specific one being good; the dream said its 11-limit was good; maybe that's true in the sense that the high errors of 5, 7 and 11 can easily cancel each-other out in ratios or composites, since zeta doesn't obey a val). Also happens to be significant as the simplest way to represent fourth-order ambiguities in my theory of functional harmony which I derived from first principles starting from [[Ringer scale]]s (especially Perfect Ringer scales), so that (other than the 12edo intervals) it represents the most xenmelodically nontrivial categories available (which correspond to areas of nontrivial harmony).)		A rather strange recurring theme is 60edo is liked by zeta a surprising amount, but looking at its low- and high-limit tuning profile it doesn't seem that remarkable to me. (A strange coincidence is some time ago I had a dream that this was a good edo. That doesn't happen often at all (dreaming about edos, let alone a specific one being good; the dream said its 11-limit was good; maybe that's true in the sense that the high errors of 5, 7 and 11 can easily cancel each-other out in ratios or composites, since zeta doesn't obey a val). Also happens to be significant as the simplest way to represent fourth-order ambiguities in my theory of functional harmony which I derived from first principles starting from [[Ringer scale]]s (especially Perfect Ringer scales), so that (other than the 12edo intervals) it represents the most xenmelodically nontrivial categories available (which correspond to areas of nontrivial harmony).)

Godtone: /* Top 20 (and top 10) zeta edos */ forgot to sign

2025-04-24T17:06:30Z

Top 20 (and top 10) zeta edos: forgot to sign

← Older revision		Revision as of 17:06, 24 April 2025
Line 142:		Line 142:
	== Top 20 (and top 10) zeta edos ==		== Top 20 (and top 10) zeta edos ==
	Since I got increasingly curious how high-quality the lists can be using the "top n" strategy applied to zeta peaks, I've detailed them and shared the lightly-modified code here: [[User:Godtone/zeta]] (same page I linked before).		Since I got increasingly curious how high-quality the lists can be using the "top n" strategy applied to zeta peaks, I've detailed them and shared the lightly-modified code here: [[User:Godtone/zeta]] (same page I linked before).

			--[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 17:06, 24 April 2025 (UTC)

Godtone: top 20 (and top 10) zeta edos

2025-04-24T17:06:16Z

top 20 (and top 10) zeta edos

← Older revision		Revision as of 17:06, 24 April 2025
Line 132:		Line 132:

	=== Alternate list based on unmodified zeta function ===		=== Alternate list based on unmodified zeta function ===
	Here's a list based purely on unmodified zeta, in case someone proves the ~~alterationis~~ wrong (though I'd still be interested in an "absolute" version of the list that includes an equal temperament if it's better than the 3rd-best-scorer so far). Because there's no accounting for the size of the equal temperament, I'm giving equal temperaments a lot of chances to appear to try to account for this bias, so that an equal temperament appears if it's better than the 9th-best-scorer so far. The other reason I give so many chances is that the resulting list is very similar and also surprisingly high-quality.		Here's a list based purely on unmodified zeta, in case someone proves the alteration is wrong (though I'd still be interested in an "absolute" version of the list that includes an equal temperament if it's better than the 3rd-best-scorer so far). Because there's no accounting for the size of the equal temperament, I'm giving equal temperaments a lot of chances to appear to try to account for this bias, so that an equal temperament appears if it's better than the 9th-best-scorer so far. The other reason I give so many chances is that the resulting list is very similar and also surprisingly high-quality.

	Take a look at [[User:Godtone/zeta]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences.		Take a look at [[User:Godtone/zeta]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences.
Line 139:		Line 139:

	--[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 18:39, 17 April 2025 (UTC)		--[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 18:39, 17 April 2025 (UTC)

			== Top 20 (and top 10) zeta edos ==
			Since I got increasingly curious how high-quality the lists can be using the "top n" strategy applied to zeta peaks, I've detailed them and shared the lightly-modified code here: [[User:Godtone/zeta]] (same page I linked before).

Fredg999: Move topic and WSArchiveLink from previous talk page (dating from before the main page was moved and the talk page didn't follow)

2025-04-19T15:23:29Z

Move topic and WSArchiveLink from previous talk page (dating from before the main page was moved and the talk page didn't follow)

← Older revision		Revision as of 15:23, 19 April 2025
Line 1:		Line 1:
	{{High priority}}		{{High priority}}
			{{WSArchiveLink}}

			== Zeta troughs? ==

			Would it be possible to use this function to calculate increasingly deep troughs as well as peaks, so as to produce a sequence of EDOs that have increasingly high levels of relative error for their size & inconsistency between various limits? That could be an interesting addition to the lists for people who want to produce intentionally perverse & dissonant music. --[[User:Yourmusic Productions\|Yourmusic Productions]] ([[User talk:Yourmusic Productions\|talk]]) 17:42, 6 April 2021 (UTC)

			: Yes, although since we are looking at the absolute value of the zeta function can be no greater than 0. So, any EDO which has a zeta value of 0 is "maximally inharmonic," in a sense. According to the Riemann hypothesis, anyway, the function can only get to 0 if we're on the critical line, which means we're weighting rationals as <math>1/(nd)^{0.5}</math>. But even on other lines there will always be local minima. Strangely, the touch tone "DTMF" frequencies were chosen, they say, to be maximally inharmonic, because they didn't want to confuse harmonics of the DTMF tones with other tones. You may think that this would mean they went for a zeta zero, but if you look you will see that all of the touch tone frequencies (which form a subset of an equal temperament) magically happen to synchronize with a local maximum of the zeta function - though a relatively small one - to within rounding error in Hz. In other words, the DTMF folks went with a relatively inharmonic region of EDO space (14-EDO ish), but then they maximized the harmonicity within that region. Very strange... [[User:Mike Battaglia\|Mike Battaglia]] ([[User talk:Mike Battaglia\|talk]]) 11:13, 9 April 2021 (UTC)

	== Review needed ==		== Review needed ==

Godtone: /* Alternate list based on unmodified zeta function */ add minor detail

2025-04-17T18:42:16Z

Alternate list based on unmodified zeta function: add minor detail

← Older revision		Revision as of 18:42, 17 April 2025
Line 128:		Line 128:
	Take a look at [[User:Godtone/zeta]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences.		Take a look at [[User:Godtone/zeta]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences.

	A rather strange recurring theme is 60edo is liked by zeta a surprising amount, but looking at its low- and high-limit tuning profile it doesn't seem that remarkable to me. (A strange coincidence is some time ago I had a dream that this was a good edo. That doesn't happen often at all (dreaming about edos, let alone a specific one being good). Also happens to be significant as the simplest way to represent fourth-order ambiguities in my theory of functional harmony which I derived from first principles starting from [[Ringer scale]]s (especially Perfect Ringer scales), so that (other than the 12edo intervals) it represents the most xenmelodically nontrivial categories available (which correspond to areas of nontrivial harmony).)		A rather strange recurring theme is 60edo is liked by zeta a surprising amount, but looking at its low- and high-limit tuning profile it doesn't seem that remarkable to me. (A strange coincidence is some time ago I had a dream that this was a good edo. That doesn't happen often at all (dreaming about edos, let alone a specific one being good; the dream said its 11-limit was good; maybe that's true in the sense that the high errors of 5, 7 and 11 can easily cancel each-other out in ratios or composites, since zeta doesn't obey a val). Also happens to be significant as the simplest way to represent fourth-order ambiguities in my theory of functional harmony which I derived from first principles starting from [[Ringer scale]]s (especially Perfect Ringer scales), so that (other than the 12edo intervals) it represents the most xenmelodically nontrivial categories available (which correspond to areas of nontrivial harmony).)

	--[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 18:39, 17 April 2025 (UTC)		--[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 18:39, 17 April 2025 (UTC)

Godtone: /* Criticisms of and possible improvements to new list */ offer alternate list and optimal edo sequences

2025-04-17T18:39:48Z

Criticisms of and possible improvements to new list: offer alternate list and optimal edo sequences

← Older revision		Revision as of 18:39, 17 April 2025
Line 122:		Line 122:

	::: I've been trying to figure out how to make it more clear why I believe that the alteration is correct. So it seems like it must be correct to think of cosine as being a function that gives a reward or punishment in the -1 to 1 range so that confirms it is in some sense directly related to the step size. The question is rather whether multiplying by ''x'' is for sure the right adjustment to make it absolute. The reason I believe it is correct is because of how we expect the relative and absolute scoring function to behave w.r.t. contorted systems. Consider 53edo which is very strong in the 5-limit, so that its 5-limit mapping is preserved for quite a few multiples. The cosine will consider 53''x''-edo as having underlying relative errors as ''x'' times as off, and in the absolute sense this corresponds to the maximum offness being 1/''x'' the size. Therefore, if we want to adjust for the fact that we are judging with a -1 to 1 range for harmonics that can only be up to 1/''x'' as off, isn't the correct adjustment necessarily to multiply by ''x''? --[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 15:42, 17 April 2025 (UTC)		::: I've been trying to figure out how to make it more clear why I believe that the alteration is correct. So it seems like it must be correct to think of cosine as being a function that gives a reward or punishment in the -1 to 1 range so that confirms it is in some sense directly related to the step size. The question is rather whether multiplying by ''x'' is for sure the right adjustment to make it absolute. The reason I believe it is correct is because of how we expect the relative and absolute scoring function to behave w.r.t. contorted systems. Consider 53edo which is very strong in the 5-limit, so that its 5-limit mapping is preserved for quite a few multiples. The cosine will consider 53''x''-edo as having underlying relative errors as ''x'' times as off, and in the absolute sense this corresponds to the maximum offness being 1/''x'' the size. Therefore, if we want to adjust for the fact that we are judging with a -1 to 1 range for harmonics that can only be up to 1/''x'' as off, isn't the correct adjustment necessarily to multiply by ''x''? --[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 15:42, 17 April 2025 (UTC)

			=== Alternate list based on unmodified zeta function ===
			Here's a list based purely on unmodified zeta, in case someone proves the alterationis wrong (though I'd still be interested in an "absolute" version of the list that includes an equal temperament if it's better than the 3rd-best-scorer so far). Because there's no accounting for the size of the equal temperament, I'm giving equal temperaments a lot of chances to appear to try to account for this bias, so that an equal temperament appears if it's better than the 9th-best-scorer so far. The other reason I give so many chances is that the resulting list is very similar and also surprisingly high-quality.

			Take a look at [[User:Godtone/zeta]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences.

			A rather strange recurring theme is 60edo is liked by zeta a surprising amount, but looking at its low- and high-limit tuning profile it doesn't seem that remarkable to me. (A strange coincidence is some time ago I had a dream that this was a good edo. That doesn't happen often at all (dreaming about edos, let alone a specific one being good). Also happens to be significant as the simplest way to represent fourth-order ambiguities in my theory of functional harmony which I derived from first principles starting from [[Ringer scale]]s (especially Perfect Ringer scales), so that (other than the 12edo intervals) it represents the most xenmelodically nontrivial categories available (which correspond to areas of nontrivial harmony).)

			--[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 18:39, 17 April 2025 (UTC)