general methods for linear temperament notation

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Re: general methods for linear temperament notation

Post by cmloegcmluin »

Dave Keenan wrote: Thu Mar 18, 2021 4:32 pm I'm kind of just wishing, for pedagogical reasons, that Gene had never coined the term, in which case Joe wouldn't miss what he never had.
Ah, got it.
But you've convinced me that using the term "monzo" is bad.
I'm not sure I meant to go that far. And there's no way we'd eliminate it now if we wanted to. But I certainly wanted to make a point about pedagogy, and pine for a lost past.
Thanks for clarifying. But I can avoid propagating it further.
Dave Keenan wrote: Thu Mar 18, 2021 5:36 pm ...you'll also need the extensions ⎢ ⎥
Ah. I had assumed those were just pipes. Thanks.
In this forum, you can also use latex as follows:
Oh great! I should have tried to peek inside your earlier post to see how you did that.
Dave Keenan wrote: Thu Mar 18, 2021 6:30 pm Oh wait. You can right-align the elements if you use "array" instead of "matrix"...
Excellente!
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Re: general methods for linear temperament notation

Post by cmloegcmluin »

The last week or so I've been feeling a bit like the floor I was standing on vaporized. So I was looking on the home page of the Xen Wiki, and followed its link to Mike's Lectures On Regular Temperament Theory. At the very bottom of lecture/episode 1, he touches upon this wedgie vs. mapping dispute.

I also followed the home page to the Tour of Regular Temperaments, not that I haven't been here at some point before, but in light of the information you're sharing here, I'm seeing it in a new light. If you go to one of the pages branching off from the tour, e.g. the Apotome family, almost every temperament has both a mapping and a wedgie provided. I guess I had always assumed a wedgie was for some deeper wizardly I might never want to understand about microtonality, but now I'm thinking... there are just two camps within the xen community, with two different ways of doing the same thing!

I also ended up on your (Dave) page, and it says you're the "one of the developers of the regular mapping paradigm". That links out to the same page I commented about on Facebook yesterday, that came up in relation to the "bent tuning forks". But you're not in the list of folks thanked at the bottom, so I have to assume you're the author, or at least co-author, since the document is on Graham's page so I have to assume he's at least co-author. Anyway, I've just read it now (rather quickly), and I'm a bit confused. I read the word "paradigm" 88 times, apparently, but I don't really understand what paradigm the authors were shifting away from. To me, coming to it as a man from the future, the page feels more like a survey of popular microtonal ideas. The section of incompatible ideas at the bottom was maybe the most instructive, although I'm confused to find "resultant tones" listed there, since I know Dave has done work with those, and that's what the bent tuning forks project was all about!

Anyway, I'm just sensing all sorts of rivalry lately. What's the story here? Certainly a lot of raw information is out there on that Yahoo Groups archive, but has anyone made any serious attempt to record the modern history of tuning theory? Have I spent my entire microtonal existence amidst the fallout of a great wedgie vs. mapping war? Did either side lose or win? Was this regular mapping paradigm written in response to monzos/vals, or did the monzos/vals stuff come later and this paradigm shift was away from just primordial chaos? Was Paul's Middle Path considered a part of this paradigm? Does the regular mapping paradigm actually contain both the wedgie and mapping camps?
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Re: general methods for linear temperament notation

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Wedgies are useful for some things, but you can easily calculate them from mappings if that's what you have. At least for rank 2 temperaments the wedge product is straightforward to calculate, and for higher ranks I think there's a way to do it with determinants of matrices. The wedgie is just a sort of "reduced" wedge product (removing common factors if any). So it makes sense to start with the mappings.

I believe that "val" is short for "valuation", but I never did understand enough of the context to know why Gene used this word. The Wikipedia article, as typical for math-related topics, doesn't make much sense to a non-mathematician.

https://en.wikipedia.org/wiki/Valuation_(algebra)

I've also not been consistent about using the "row" and "column" vector notation. I have a lot of older documents that use vertical bars in place of brackets, e.g.

[<1 2 4 7|, <0 -1 -4 -10|]

And I have a tendency to use parentheses for generator counts, as in this note on Sagittal symbols for 7-limit meantone.

(|\ [-13, 5, 1, 1> (+8, -19)
|) [6, -2, 0, -1> (-5, +12)
||\ [-7, 3, 1> (+3, -7)

But sometimes I use the column vector notation for those. And I haven't been consistent on whether or not I use commas, which makes searching a bit difficult.
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Re: general methods for linear temperament notation

Post by Dave Keenan »

cmloegcmluin wrote: Fri Mar 19, 2021 7:51 am I also ended up on your (Dave) page, and it says you're the "one of the developers of the regular mapping paradigm". That links out to the same page I commented about on Facebook yesterday, that came up in relation to the "bent tuning forks".
I certainly am one of the developers of the regular mapping paradigm, as is Herman. Mike could have listed me, along with Graham, as preferring the matrix algebra (= linear algebra) formulation. I'm not sure where Paul would sit on that, but AFAIK none of Paul's paper have used the wedge product. Mike has done a skilful job of sitting on the fence, with that lecture, but he too has not actually used wedge products in it.
But you're not in the list of folks thanked at the bottom, so I have to assume you're the author, or at least co-author, since the document is on Graham's page so I have to assume he's at least co-author.
I was not a co-author of that paper. That's all Graham's (excellent) work. He needs to put his name, and the date of first publication, at the start. It was May-2006.

That "handful of yahoos" link is very confusing. If you look at the html source, you'll see that it links to the first Footnote. Its anchor is immediately before the words: "As soon as I finished polishing this web page ...". So it has nothing to do with that list of names above the Footnotes. That's a list of people who suggested improvements on the earlier drafts. Paul Erlich is not on that list either. Maybe Paul thought the drafts were just fine as they were, or maybe he was busy. In my case, I suspect I wasn't active on the yahoo tuning groups at the time the drafts were offered for comment, and only got to read it much later. When I did, I thought it was brilliant, and told Graham so by email.

That link should really be coming from the words "only shared", since the footnote is explaining that he has since learned that it was also shared by some non-yahoos.

In the paper, Graham writes, "Dave did the first systematic search for linear temperaments". I did that in July 1999. He also writes that "the rediscovery of miracle temperament did change the way I think about the music I'd like to write.", and on another page he mentions that Miracle was "rediscovered by Paul Erlich and Dave Keenan". That was Apr-2001.
Anyway, I've just read it now (rather quickly), and I'm a bit confused. I read the word "paradigm" 88 times, apparently, but I don't really understand what paradigm the authors were shifting away from.
Er. The section after the Introduction is called "Prior paradigms". He gives 3. Basically tuning was either conventional or microtonal, and microtonal was either EDO or JI, and never the twain shall meet. Hence Paul's title "Middle Path".
To me, coming to it as a man from the future, the page feels more like a survey of popular microtonal ideas. The section of incompatible ideas at the bottom was maybe the most instructive, although I'm confused to find "resultant tones" listed there, since I know Dave has done work with those, and that's what the bent tuning forks project was all about!
Yes. And I thought I made it clear at the start of the bent tuning forks thread, that in doing so, I was moving beyond the regular mapping paradigm.
Have I spent my entire microtonal existence amidst the fallout of a great wedgie vs. mapping war?
Kind of. But ...
Did either side lose or win?
No side won, because there was no war. There was room for both. But it does mean that we now have the pedagogical problem of an unholy mixture of bits of Linear algebra terminology and notation, with bits of Exterior algebra terminology and notation, with bits of DIrac notation, and bits of Gene-speak.
Was this regular mapping paradigm written in response to monzos/vals,
Good grief no. The regular mapping paradigm was worked out on the tuning list before Gene got involved and renamed everything, and changed the math notation. But of course we didn't call it the regular mapping paradigm until Graham gave it that name, much later (May-2006). And we called them prime exponent vectors and mappings, as you can read in Middle Path.
or did the monzos/vals stuff come later and this paradigm shift was away from just primordial chaos?
The monzo/val names came later.
Was Paul's Middle Path considered a part of this paradigm?
Absolutely! I'm pretty sure it was the first paper written purely for the purpose of bringing the whole thing together in one place. It's dated Aug-2004. I was more nearly a co-author of that one. I created several of the diagrams, and Paul and I nutted out the criteria for which rank 2 temperaments should be included as being potentially musically useful.

At that stage, I had already described regular mapping in my Nov 2003 paper https://www.dkeenan.com/Music/MicroGuitar.pdf, but it was specific to the micro-guitar application. In that paper I use the term "prime exponent list" rather than "prime exponent vector" to make it sound less mathy. And I'm pretty sure Graham Breed described it using matrix math on his web site around the same time, maybe earlier, but he doesn't date anything.

It's hard for me to imagine how it could be unclear to you whether Middle Path is part of the paradigm. Can you explain? Is it because it doesn't contain the words monzo or val?
Does the regular mapping paradigm actually contain both the wedgie and mapping camps?
Sure. Although I think the wedgie camp mostly consisted of Gene. That is, people who actually computed things using wedge products, as opposed to people who talked about it. I don't think anyone else had access to the software he was using, or knew how to use it for Exterior algebra if they did. I think it was in Maple.

In Dec 2001, Graham wrote his own Exterior algebra (wedge product) library in Python. I think he later decided it was easier to revert to doing things with mappings and matrix math.

I even wrote a tutorial on how to calculate a wedge product by hand at one stage. I couldn't tell you now. It's almost impossible to figure out which terms to add and which to subtract. It has that in common with calculating the determinant of a matrix. I wish I could find it. But I can only find this mention of it: https://yahootuninggroupsultimatebackup ... 55058.html

Have you actually computed things using wedge products, Herman? @herman.miller What are they good for exactly?
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Re: general methods for linear temperament notation

Post by cmloegcmluin »

Dave Keenan wrote: Sat Mar 20, 2021 2:21 am At that stage, I had already described regular mapping in my Nov 2003 paper https://www.dkeenan.com/Music/MicroGuitar.pdf, but it was specific to the micro-guitar application. In that paper I use the term "prime exponent list" rather than "prime exponent vector" to make it sound less mathy. And I'm pretty sure Graham Breed described it using matrix math on his web site around the same time, maybe earlier, but he doesn't date anything.
Ah, I see. That's the paper I recently discovered in my XH18 and complimented you on!

Perhaps if I started over going through my 1/1 journals (beginning in 1985) but this time keeping my eyes peeled for prime exponent vectors, I might not find any. Their use seems to be a key historical development. Although you did say that they're an obvious idea to use them for coordinates in a JI lattice. So perhaps it's rather the introduction of either mappings or wedgies which is associated with this new paradigm of regular temperaments, as opposed to just EDs or JI.
It's hard for me to imagine how it could be unclear to you whether Middle Path is part of the paradigm. Can you explain? Is it because it doesn't contain the words monzo or val?
Yes I can easily explain. It was because there were no dates, and I personally wasn't able to effectively figure out from the "prior paradigms" section of that history paper whether this was somehow a reaction to a world in which the Middle Path paper already existed or not. I'm sure it's all quite clear to you, having lived through it, but it was a total fog of possibilities for me. Thank you for summarizing and clarifying all of this! The story is clear enough to me now.
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Re: general methods for linear temperament notation

Post by Dave Keenan »

cmloegcmluin wrote: Sat Mar 20, 2021 2:53 am Perhaps if I started over going through my 1/1 journals (beginning in 1985) but this time keeping my eyes peeled for prime exponent vectors, I might not find any.
You might, if there's an article by Joe Monzo. But he may have used a different term.
Their use seems to be a key historical development. Although you did say that they're an obvious idea to use them for coordinates in a JI lattice. So perhaps it's rather the introduction of either mappings or wedgies which is associated with this new paradigm of regular temperaments, as opposed to just EDs or JI.
Yes. It's using the mapping to transform the PEV that's important. And yes, PEV already existed so it is introduction of mapping that was important. Hence the regular mapping paradigm.
Yes I can easily explain. It was because there were no dates, and I personally wasn't able to effectively figure out from the "prior paradigms" section of that history paper whether this was somehow a reaction to a world in which the Middle Path paper already existed or not. I'm sure it's all quite clear to you, having lived through it, but it was a total fog of possibilities for me. Thank you for summarizing and clarifying all of this! The story is clear enough to me now.
Ah yes. Of course. A pretty useless "history" with no date of publication and no dates of events listed in it. It doesn't actually claim to be a history, but it could so easily be one, with a few dates.

I researched those dates from the tuning list archive and my emails.
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Re: general methods for linear temperament notation

Post by cmloegcmluin »

Dave Keenan wrote: Sat Mar 20, 2021 3:16 am regular mapping paradigm.
Right! I'll get it straight. Thanks.
I researched those dates from the tuning list archive and my emails.
Thanks for doing all that. Perhaps I should pester Graham to tack some more dates and names onto that thing, benefitting from all your research just now.

Isn't it past your bedtime? :)
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Re: general methods for linear temperament notation

Post by herman.miller »

There's basically two things that I find wedgies / wedge products useful for. One is to calculate the TOP tuning of a temperament, taking three elements at a time to find commas, and calculating the TOP tuning for each of those commas. In most cases, one of those will be the TOP tuning for the temperament. If you don't have linear programming software, that's a relatively easy way to calculate it. The second use is to calculate a measure of complexity (adding the absolute values of each element weighted by the prime weights of the primes that relate to that element).
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Re: general methods for linear temperament notation

Post by herman.miller »

I think the easiest way to explain how to calculate wedge products by hand is to explicitly write out the basis vectors (you might call them e1, e2, e3, e4 for a 7-limit temperament ... or maybe e2, e3, e5, e7 would be preferable). Take 7-limit meantone as an example.

[<1 2 4 7] <0 -1 -4 -10]>

We want to write this as a wedge product of (1 * e1 + 2 * e2 + 4 * e3 + 7 * e4) and (0 * e1 - 1 * e2 - 4 * e3 - 10 * e4). So what you do is to take all the wedge products of each of the terms of the first expression with each term of the second one. Any of the basis vectors wedged with itself is 0, so we can skip those.

(1 * e1 ^ -1 * e2) + (1 * e1 ^ -4 * e3) + (1 * e1 ^ -10 * e4)
+ (2 * e2 ^ 0 * e1) + (2 * e2 ^ -4 * e3) + (2 * e2 ^ -10 * e4)
+ (4 * e3 ^ 0 * e1) + (4 * e3 ^ -1 * e2) + (4 * e3 ^ -10 * e4)
+ (7 * e4 ^ 0 * e1) + (7 * e4 ^ -1 * e2) + (7 * e4 ^ -4 * e3)

Now we can combine the e1 ^ e2 terms with the e2 ^ e1 because e1 ^ e2 = -(e2 ^ e1), and so on. When you do the algebra you end up with this for a result:

(-1 * e1 ^ e2) + (-4 * e1 ^ e3) + (-10 * e1 ^ e4) + (-4 * e2 ^ e3) + (-13 * e2 ^ e4) + (-12 * e3 ^ e4)

And we write this as <<-1 -4 -10 -4 -13 -12]]. In cases like this we like to reverse the signs if the first one ends up negative (that's just a matter of which order you do the wedge product). So we get <<1 4 10 4 13 12]].
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Re: general methods for linear temperament notation

Post by cmloegcmluin »

Dave Keenan wrote: Thu Mar 18, 2021 6:40 am I notice that he not only uses the same notation |a b c ...> for prime exponents and generator counts, he doesn't use plus signs and he even calls the latter "tempered monzos" — yikes — helping making my case that Gene's monzo/val terminology is unhelpful and just raises a barrier to entry.
As you know I've been pushing myself to level up my understanding of RTT recently. I had an insight that helps explain to me Mike's decision here. If you forgive the problem of needlessly renaming vector to monzo, then "tempered monzo" is reasonable (as is using the same notation), because actually prime exponent vectors and generator count vectors are the same thing when you consider that JI can be represented as a mapping matrix:

\( \left[ \begin{array} {rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right] \)

herman.miller wrote: Sat Mar 20, 2021 9:33 am [One use I find for wedgies / wedge products] is to calculate a measure of complexity (adding the absolute values of each element weighted by the prime weights of the primes that relate to that element).
What are the prime weights in this context?
(you might call them e1, e2, e3, e4 for a 7-limit temperament ... or maybe e2, e3, e5, e7 would be preferable).
I started out using e2, e3, e5, e7, but because the method for working out the dual of the wedgie involves the indices of these primes (to know whether to flip the signs or not), and because I wanted to leave things generalized to temperaments of stuff other than simple primes, I settled on e1, e2, e3, e4 as preferable.
Take 7-limit meantone as an example....
Just wanted to say Herman that I always look forward to reading through your explanations of things. For me, they're just what I'm looking for: methodical, unambiguous, and give just the right amount of "why" as they go along. So, thanks!
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