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?