User talk:TallKite: Difference between revisions

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== 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~~]]

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 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?

Line 411:

: 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 402: / Line 402: @@
 == 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]]
+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 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?
@@ Line 411: / Line 411: @@
 : 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)