User talk:TallKite: Difference between revisions

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: For 11-limit, consider a chain of neutral 3rds centered on the unison: m3-hd5-m7-n2-P4-n6-P1-n3-P5-n7-M2-hA4-M6 where hd = half-dim and hA = half-aug. Consider the 6 intervals hd5-n2-n6-n3-n7-hA4. Coldly suspended seems to mean the central part of this chain. Warmly suspended seems to mean the further away parts. Except hd5 = 16/11 gets its own category. Again, this might be because 3/2 is so powerful, 16/11 sounds more like a very flat 3/2 than an interval in its own right. Note that there's a small comma 243/242 which tends to blur the difference between 11-over and 11-under.

: So your categories seem to correspond to various regions of the lattice, which makes sense to me. Not sure I understand the positive/negative classification. Personally I loosely classify imperfect intervals as supermajor-major-neutral-minor-subminor, 5ths as superperfect-perfect-halfdim-dim-(subdim) and 4ths as (superaug)-aug-halfaug-perfect-subperfect. So basically 5-limit and deviations from there, very 31-edo-like. One could add submajor, superminor, superperfect 4th, etc. to get it down to 3-limit, very 41-edo-like. If you sharpen the 5th, then in the 3-limit chain of 5ths major sounds like supermajor and minor sounds subminor. If you flatten it, you get submajor and superminor, and if you flatten a lot, neutral. Is that the logic behind positive/negative? If so, that might be a better way to describe it, rather than referring to edos. Also note that to get 11/8 and 16/11, you are presumably flattening the 5th by a quartertone. This makes the major 2nd sound minor, and the major 3rd sound diminished! --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 07:23, 2 August 2021 (UTC)

:: Thank you so much for your response! Yes, the logic is that positive polarity refers to intervals generated by sharp fifths while negative polarity refers to intervals generated by flat fifths. I appreciate all of your input and the connection between color notation and intervallic polarity makes a lot of sense. On the other hand, I still wonder if there is something related to the dissonance, harmonic entropy, or complexity of an interval that could be used to derive its intervallic polarity. (This could allow intervallic polarity to possibly be generalized to chords and/or intervals played with different timbres)

:: Thank you so much for your response! Yes, the logic is that positive polarity refers to intervals generated by sharp fifths while negative polarity refers to intervals generated by flat fifths. I appreciate all of your input and the connection between color notation and intervallic polarity makes a lot of sense. On the other hand, I still wonder if there is something related to the dissonance, harmonic entropy, or complexity of an interval that could be used to derive its intervallic polarity. (This could allow intervallic polarity to possibly be generalized to chords and/or intervals played with different timbres) --[[User:Userminusone|Userminusone]] ([[User talk:Userminusone|talk]]) 21:11, 2 August 2021 (UTC)

== having torsion vs. being enfactored ==

Hi Kite. Per your request I'm continuing discussion with you on your user page where you are more likely to see it sooner. This is a continuation of the discussion started here: [[Talk:Color notation/Temperament names]]

I'm glad you agree about torsion. I like the way you explained it, pointing to the name of RTT itself. As a nit-pick, though, I can't agree with the statement that "you can't hear periodicity blocks". That wasn't what I was trying to say. In fact, I was trying to say something like the opposite. My point was that using e.g. {{vector|-8 8 -2}} instead of {{vector|-4 4 -1}} has an audible effect on periodicity blocks but not on temperaments. For a periodicity block, it causes the size of the scale to double, but half of the notes are a redundant copy of the other half, simply offset. Because this is a real audible effect, and I understand there are maybe even some uses for it or cases where it's desirable, it has a name, "torsion". But for a temperament, though, where the comma is by definition tempered out, there is no audible effect, and thus using {{vector|-8 8 -2}} instead of {{vector|-4 4 -1}} is meaningless. It's just pathological enfactoring that is removed when the comma basis is put into canonical form.

I'm glad you agree about contorsion too. I'm not sure we do, though, because your statement about 12- and 24- ET is not how I would describe it. I would say something more like this: "Calling {{map|24 38 56}} a 'temperament' is misleading because everything it does as a temperament is already done by the simpler {{map|12 19 28}}. In other words, all of its notes are real and audible, but half of them are not used for tempering, or we could say that it is 2-enfactored. Therefore it should not be listed as a strict 'temperament'; perhaps we could call it a 'temperoid' or something like that instead." Does that check out with you?

I found this page of yours this morning which uses the word "torsion" a lot: [[Catalog of rank two temperaments]]. Per my explanation above, would you mind if I renamed uses of "torsion" to "enfactored" there? If you prefer, I could include a footnote on the first one that explains it has historically sometimes been called torsion. Alternatively... why does it matter if they are enfactored? Does anyone need to care? Just defactor them. Maybe it'd be better to just include the canonical form of these temperaments. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 16:56, 30 September 2021 (UTC)

: I stand corrected about periodicity blocks. As for that page, it's actually an attempt to find a canonical comma list. It's complicated, let's discuss it in person when you visit. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 07:42, 2 October 2021 (UTC)

:: Ah! Yes I seem to recall that someone — Paul, Mike, maybe? — mentioned to me early on that your color notation included some thinking about canonicalization. I am embarrassed to say that I haven't learned color notation yet, besides a few basics. So if you don't mind, if we have time when we meet, maybe you could show me the ropes? I expect we will need a couple sessions! --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:31, 3 October 2021 (UTC)

::: I would love to show you! --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 05:57, 5 October 2021 (UTC)

== Chessboard distance ==

I noticed this bit just now: https://en.xen.wiki/w/Commas_by_taxicab_distance#Triangularizing_proposal

FYI, "triangularized taxicab" distance like this has an established name. It's [[Wikipedia:Chebyshev_distance|Chebyshev distance]], AKA "chessboard distance," because if a 2D lattice was like a chessboard, then it's the number of moves the king piece would need to take to reach from point A to point B. I made this chart, in case it helps:

{| class="wikitable"

|+

!L-norm

!eponym

!locale

!agent

|-

|1

|Minkowsky

|Manhattan

|taxicab

|-

|2

|Euclid

|space

|crow

|-

|∞

|Chebyshev

|chessboard

|king

|}

You can see these distances are associated with different L norms. The L₁ norm and L∞ norms are each others' duals and the L₂ norm is self-dual.

These come up in tuning. When you minimize the L∞ norm on the prime error, this causes a minimization of the L1 norm on interval error. That's TIPTOP tuning. The L∞ norm of a vector is simply the max value of any of its entries; I understand it that way because your "king" can move as diagonally as necessary, and so he'll just move diagonally in every dimension until he runs out of dimensions he needs to go except for one, at which point he continues straight along that dimension. And if you minimize the L1 norm on the prime error, this causes a minimization of the L∞ norm on interval error. So if you wanted to use L∞ norm for interval error, you'd set your tuning optimizer to minimize the sum of the absolute values of errors per prime. If you have any questions, let me know -- I'm not rock solid on this stuff yet, but I think it's pretty interesting. Dave and I have attempted to improve our geometric intuition for dual norms' effects on tuning, but it's been a while since I looked at it.

Anyway, just thought you might like to revise that original paragraph to use established nomenclature, or at least reference it! You may not have been aware of it; I only just learned it myself a few months back. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:04, 19 January 2022 (UTC)

: Hmm, interesting. But actually, what I'm proposing is different from all of these. In the 2-D case I propose a shearing so that the rectangular lattice becomes triangular.

{| class="wikitable"

|+triangularized

|1

|2

|-

|1

|0

|1

|-

|2

|1

|}

: This applies to all prime subgroups, but let's assume 2.3.5 and see what the ratios are. Note that the ratios that are now two moves away are the ones with the much higher odd-limit of 15. Thus it does seem to reflect the actual musical distance better than any of the 3 ways you listed.

{| class="wikitable"

|+ratios

|5/3

|5/4

|15/8

|-

|4/3

|1/1

|3/2

|-

|16/15

|8/5

|6/5

|}

: In 3-D, 4-D etc., it's better thought of not as shearing but as higher primes cancelling lower primes that are on the opposite side of the ratio.

: I'm not following the L1 stuff. Can you give some actual examples? --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 07:47, 20 January 2022 (UTC)

:: Ah, I see. What you're talking about is completely different. I started making some corrections to my previous statements before I'd noticed you'd replied already. So I'm going to go ahead and make those rather than leave the misinformation up and correct it here, if that's okay (my incorrectness is still preserved in the edit history). I think I probably shouldn't try to say more about the Lp norms yet until I have a better handle on them, so never mind for now, especially since it's irrelevant to your purpose anyway. Sorry for the confusion. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 23:42, 22 January 2022 (UTC)

@@ Line 398: / Line 398: @@
 : For 11-limit, consider a chain of neutral 3rds centered on the unison: m3-hd5-m7-n2-P4-n6-P1-n3-P5-n7-M2-hA4-M6 where hd = half-dim and hA = half-aug. Consider the 6 intervals hd5-n2-n6-n3-n7-hA4. Coldly suspended seems to mean the central part of this chain. Warmly suspended seems to mean the further away parts. Except hd5 = 16/11 gets its own category. Again, this might be because 3/2 is so powerful, 16/11 sounds more like a very flat 3/2 than an interval in its own right. Note that there's a small comma 243/242 which tends to blur the difference between 11-over and 11-under.
 : So your categories seem to correspond to various regions of the lattice, which makes sense to me. Not sure I understand the positive/negative classification. Personally I loosely classify imperfect intervals as supermajor-major-neutral-minor-subminor, 5ths as superperfect-perfect-halfdim-dim-(subdim) and 4ths as (superaug)-aug-halfaug-perfect-subperfect. So basically 5-limit and deviations from there, very 31-edo-like. One could add submajor, superminor, superperfect 4th, etc. to get it down to 3-limit, very 41-edo-like. If you sharpen the 5th, then in the 3-limit chain of 5ths major sounds like supermajor and minor sounds subminor. If you flatten it, you get submajor and superminor, and if you flatten a lot, neutral. Is that the logic behind positive/negative? If so, that might be a better way to describe it, rather than referring to edos. Also note that to get 11/8 and 16/11, you are presumably flattening the 5th by a quartertone. This makes the major 2nd sound minor, and the major 3rd sound diminished! --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 07:23, 2 August 2021 (UTC)
-:: Thank you so much for your response! Yes, the logic is that positive polarity refers to intervals generated by sharp fifths while negative polarity refers to intervals generated by flat fifths. I appreciate all of your input and the connection between color notation and intervallic polarity makes a lot of sense. On the other hand, I still wonder if there is something related to the dissonance, harmonic entropy, or complexity of an interval that could be used to derive its intervallic polarity. (This could allow intervallic polarity to possibly be generalized to chords and/or intervals played with different timbres)
+:: Thank you so much for your response! Yes, the logic is that positive polarity refers to intervals generated by sharp fifths while negative polarity refers to intervals generated by flat fifths. I appreciate all of your input and the connection between color notation and intervallic polarity makes a lot of sense. On the other hand, I still wonder if there is something related to the dissonance, harmonic entropy, or complexity of an interval that could be used to derive its intervallic polarity. (This could allow intervallic polarity to possibly be generalized to chords and/or intervals played with different timbres) --[[User:Userminusone|Userminusone]] ([[User talk:Userminusone|talk]]) 21:11, 2 August 2021 (UTC)
+== having torsion vs. being enfactored ==
+Hi Kite. Per your request I'm continuing discussion with you on your user page where you are more likely to see it sooner. This is a continuation of the discussion started here: [[Talk:Color notation/Temperament names]]
+I'm glad you agree about torsion. I like the way you explained it, pointing to the name of RTT itself. As a nit-pick, though, I can't agree with the statement that "you can't hear periodicity blocks". That wasn't what I was trying to say. In fact, I was trying to say something like the opposite. My point was that using e.g. {{vector|-8 8 -2}} instead of {{vector|-4 4 -1}} has an audible effect on periodicity blocks but not on temperaments. For a periodicity block, it causes the size of the scale to double, but half of the notes are a redundant copy of the other half, simply offset. Because this is a real audible effect, and I understand there are maybe even some uses for it or cases where it's desirable, it has a name, "torsion". But for a temperament, though, where the comma is by definition tempered out, there is no audible effect, and thus using {{vector|-8 8 -2}} instead of {{vector|-4 4 -1}} is meaningless. It's just pathological enfactoring that is removed when the comma basis is put into canonical form.
+I'm glad you agree about contorsion too. I'm not sure we do, though, because your statement about 12- and 24- ET is not how I would describe it. I would say something more like this: "Calling {{map|24 38 56}} a 'temperament' is misleading because everything it does as a temperament is already done by the simpler {{map|12 19 28}}. In other words, all of its notes are real and audible, but half of them are not used for tempering, or we could say that it is 2-enfactored. Therefore it should not be listed as a strict 'temperament'; perhaps we could call it a 'temperoid' or something like that instead." Does that check out with you?
+I found this page of yours this morning which uses the word "torsion" a lot: [[Catalog of rank two temperaments]]. Per my explanation above, would you mind if I renamed uses of "torsion" to "enfactored" there? If you prefer, I could include a footnote on the first one that explains it has historically sometimes been called torsion. Alternatively... why does it matter if they are enfactored? Does anyone need to care? Just defactor them. Maybe it'd be better to just include the canonical form of these temperaments. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 16:56, 30 September 2021 (UTC)
+: I stand corrected about periodicity blocks. As for that page, it's actually an attempt to find a canonical comma list. It's complicated, let's discuss it in person when you visit. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 07:42, 2 October 2021 (UTC)
+:: Ah! Yes I seem to recall that someone — Paul, Mike, maybe? — mentioned to me early on that your color notation included some thinking about canonicalization. I am embarrassed to say that I haven't learned color notation yet, besides a few basics. So if you don't mind, if we have time when we meet, maybe you could show me the ropes? I expect we will need a couple sessions! --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:31, 3 October 2021 (UTC)
+::: I would love to show you! --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 05:57, 5 October 2021 (UTC)
+== Chessboard distance ==
+I noticed this bit just now: https://en.xen.wiki/w/Commas_by_taxicab_distance#Triangularizing_proposal
+FYI, "triangularized taxicab" distance like this has an established name. It's [[Wikipedia:Chebyshev_distance|Chebyshev distance]], AKA "chessboard distance," because if a 2D lattice was like a chessboard, then it's the number of moves the king piece would need to take to reach from point A to point B. I made this chart, in case it helps:
+{| class="wikitable"
+|+
+!L-norm
+!eponym
+!locale
+!agent
+|-
+|1
+|Minkowsky
+|Manhattan
+|taxicab
+|-
+|2
+|Euclid
+|space
+|crow
+|-
+|∞
+|Chebyshev
+|chessboard
+|king
+|}
+You can see these distances are associated with different L norms. The L₁ norm and L∞ norms are each others' duals and the L₂ norm is self-dual.
+These come up in tuning. When you minimize the L∞ norm on the prime error, this causes a minimization of the L1 norm on interval error. That's TIPTOP tuning. The L∞ norm of a vector is simply the max value of any of its entries; I understand it that way because your "king" can move as diagonally as necessary, and so he'll just move diagonally in every dimension until he runs out of dimensions he needs to go except for one, at which point he continues straight along that dimension. And if you minimize the L1 norm on the prime error, this causes a minimization of the L∞ norm on interval error. So if you wanted to use L∞ norm for interval error, you'd set your tuning optimizer to minimize the sum of the absolute values of errors per prime. If you have any questions, let me know -- I'm not rock solid on this stuff yet, but I think it's pretty interesting. Dave and I have attempted to improve our geometric intuition for dual norms' effects on tuning, but it's been a while since I looked at it.
+Anyway, just thought you might like to revise that original paragraph to use established nomenclature, or at least reference it! You may not have been aware of it; I only just learned it myself a few months back. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:04, 19 January 2022 (UTC)
+: Hmm, interesting. But actually, what I'm proposing is different from all of these. In the 2-D case I propose a shearing so that the rectangular lattice becomes triangular.
+{| class="wikitable"
+|+triangularized
+|1
+|1
+|2
+|-
+|1
+|0
+|1
+|-
+|2
+|1
+|1
+|}
+: This applies to all prime subgroups, but let's assume 2.3.5 and see what the ratios are. Note that the ratios that are now two moves away are the ones with the much higher odd-limit of 15. Thus it does seem to reflect the actual musical distance better than any of the 3 ways you listed.
+{| class="wikitable"
+|+ratios
+|5/3
+|5/4
+|15/8
+|-
+|4/3
+|1/1
+|3/2
+|-
+|16/15
+|8/5
+|6/5
+|}
+: In 3-D, 4-D etc., it's better thought of not as shearing but as higher primes cancelling lower primes that are on the opposite side of the ratio.
+: I'm not following the L1 stuff. Can you give some actual examples? --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 07:47, 20 January 2022 (UTC)
+:: Ah, I see. What you're talking about is completely different. I started making some corrections to my previous statements before I'd noticed you'd replied already. So I'm going to go ahead and make those rather than leave the misinformation up and correct it here, if that's okay (my incorrectness is still preserved in the edit history). I think I probably shouldn't try to say more about the Lp norms yet until I have a better handle on them, so never mind for now, especially since it's irrelevant to your purpose anyway. Sorry for the confusion. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 23:42, 22 January 2022 (UTC)