Comments on the regular mapping paradigm

I think its true a new paradigm has emerged, but I think its a little
murky what the newness really is. But it's a great idea to try to put
it all together.

One thing which might be worth mentioning is that the regular mapping
paradigm reflects a paradigm shift a large hunk of pure math underwent
50 years ago, with the emergence of homological algebra, category
theory, schemes and the like. The paradigm was a mapping
paradigm--that it is the mappings between the abstract objects you are
looking at, more than the objects themselves, which is really crucial.

So far as shifts in points of view goes, I arrived on the tuning list
with it firmly fixed in my mind that homomorphic mapping was the key
concept; hence my insistence that it be given a name. My big paradigm
shift was that I thought mappings to equal temperaments sufficed;
instead of thinking about meantone or pajara as linear temperaments, I
thought just looking at them as intersections of kernels of mappings
was good enough. So, pajara would be just whatever 12 and 22 have in
common, namely tempering out 50/49 and 64/63. The biggest deal for me
was learning to look at linear temperaments.

The point is, I had a paradigm shift but I'm not sure it's exactly the
same as you are saying, but certainly seeing that something like
miracle was worth thinking about (instead of just 72-et as a mapping,
with eg 225/224 in the kernel, which *was* a part of my paradigm) was
a big shift for me. I can't speak for other people's paradigms, but I
guess what I'm saying is that a higher-rank paradigm, not just a
mapping paradigm, seems to me to be a big part of it.

Whether or not anyone else had this paradigm, you could say my
paradigm was the "JI as a group and equal temperaments as homomorphic
mappings" paradigm. It might help to look more closely at the various
paradigms. As you point out, there was a JI paradigm, and an equal
temperaments (but without the emphasis on mappings) paradigm, and a
diatonic paradigm, but I don't think that covers it. What were other
theorists thinking? Fokker blocks is another thing; was there a
"fundamental region" paradigm at work?

>I think its true a new paradigm has emerged, but I think its a little
>murky what the newness really is. But it's a great idea to try to put
>it all together.

To me it's crystal-clear: The first rigorous search and evaluation
of rank 2 temperaments.

>So far as shifts in points of view goes, I arrived on the tuning list
>with it firmly fixed in my mind that homomorphic mapping was the key
>concept; hence my insistence that it be given a name. My big paradigm
>shift was that I thought mappings to equal temperaments sufficed;
>instead of thinking about meantone or pajara as linear temperaments, I
>thought just looking at them as intersections of kernels of mappings
>was good enough. So, pajara would be just whatever 12 and 22 have in
>common, namely tempering out 50/49 and 64/63. The biggest deal for me
>was learning to look at linear temperaments.

Ha! You're still saying "linear" when you mean "rank 2". And Paul
was slitting my throat for this a year ago.

>Fokker blocks is another thing;

Part of what's happened here is the integration of approaches
like MOS, Fokker blocks, ETs and non-octave ETs, etc.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

Graham says:

"In the regular mapping paradigm, miracle temperament is a rank 2
temperament with the mapping

<< 1, 1, 3, 3, 2],
< 0, 6, -7, -2, 15]]"

Well, not precisely in my paradigm, where miracle *can* have this
mapping, not that it must have it. Trivially, that includes using
2/secor for a generator, but also eg 7/4 as a generator with 5 as a
period. Take the 4x4 matrix from [5, 7/4, 225/224, 1029/1024] and it
has determininat -1, and so is unimdular and invertible. I'd call that
miracle, and consider 43 tones to the 5 a miracle scale.

Gene Ward Smith wrote:
> I think its true a new paradigm has emerged, but I think its a little
> murky what the newness really is. But it's a great idea to try to put
> it all together.

The interesting question is how many of us think the same way, and to what extent.

> One thing which might be worth mentioning is that the regular mapping
> paradigm reflects a paradigm shift a large hunk of pure math underwent
> 50 years ago, with the emergence of homological algebra, category
> theory, schemes and the like. The paradigm was a mapping
> paradigm--that it is the mappings between the abstract objects you are
> looking at, more than the objects themselves, which is really crucial. The trouble is even I don't understand that, and I certainly wouldn't expect my readers to. Perhaps you could write a separate document explaining the implications of all this to mathematicians.

> So far as shifts in points of view goes, I arrived on the tuning list
> with it firmly fixed in my mind that homomorphic mapping was the key
> concept; hence my insistence that it be given a name. My big paradigm
> shift was that I thought mappings to equal temperaments sufficed;
> instead of thinking about meantone or pajara as linear temperaments, I
> thought just looking at them as intersections of kernels of mappings
> was good enough. So, pajara would be just whatever 12 and 22 have in
> common, namely tempering out 50/49 and 64/63. The biggest deal for me
> was learning to look at linear temperaments. Interesting! I was thinking in terms of linear temperaments (as we called them then) as being the key idea, and I'd done lots of work with matrices. But I hadn't expected there to be such a pluralism of regular temperaments and my thinking is different now in subtle ways.

> The point is, I had a paradigm shift but I'm not sure it's exactly the
> same as you are saying, but certainly seeing that something like
> miracle was worth thinking about (instead of just 72-et as a mapping,
> with eg 225/224 in the kernel, which *was* a part of my paradigm) was
> a big shift for me. I can't speak for other people's paradigms, but I
> guess what I'm saying is that a higher-rank paradigm, not just a
> mapping paradigm, seems to me to be a big part of it.

The idea of a paradigm is that it's shared by a group of people. So wherever we came from we've arrived at the same place now.

> Whether or not anyone else had this paradigm, you could say my
> paradigm was the "JI as a group and equal temperaments as homomorphic
> mappings" paradigm. It might help to look more closely at the various
> paradigms. As you point out, there was a JI paradigm, and an equal
> temperaments (but without the emphasis on mappings) paradigm, and a
> diatonic paradigm, but I don't think that covers it. What were other
> theorists thinking? Fokker blocks is another thing; was there a
> "fundamental region" paradigm at work?

There's also an "anything goes" paradigm that lies outside everything else. And gamelan tunings don't really fit into this.

I think Fokker's work here was outside the paradigm of his time, and mostly ignored as a result. It's one of the things that prompted our paradigm shift. Not an immediate cause, which is why it's in a later section.

Graham

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@...> wrote:
> > Graham says:
> > "In the regular mapping paradigm, miracle temperament is a rank 2
> temperament with the mapping
> > << 1, 1, 3, 3, 2],
> < 0, 6, -7, -2, 15]]"
> > Well, not precisely in my paradigm, where miracle *can* have this
> mapping, not that it must have it. Trivially, that includes using
> 2/secor for a generator, but also eg 7/4 as a generator with 5 as a
> period. Take the 4x4 matrix from [5, 7/4, 225/224, 1029/1024] and it
> has determininat -1, and so is unimdular and invertible. I'd call that
> miracle, and consider 43 tones to the 5 a miracle scale.

I don't think you're talking about a different mapping, in the sense of a way of approximating JI intervals as tempered intervals. Rather you're talking about different generators used for the scales, which result in different ways of writing the mapping.

Perhaps I should explain that somewhere. Ideally I'd explain the mapping before the generators but that makes it harder to give examples.

Graham