The Regular Mapping Paradigm

I've written a web page about how some of us think about tuning theory now. Here it is:

http://x31eq.com/paradigm.html

I welcome comments. Anybody who things it applies to them, or doesn't and should. Or general suggestions for improvements.

Once it's been through a bit of peer review I'll tell the main list about it.

Graham

>I've written a web page about how some of us think about tuning theory
>now. Here it is:
>
>http://x31eq.com/paradigm.html
>
>I welcome comments. Anybody who things it applies to them, or doesn't
>and should. Or general suggestions for improvements.
>
>Once it's been through a bit of peer review I'll tell the main list
>about it.

You've been busy!

It seems kinda randomly organized and has lots of undefined
terms (rank 2, to give one example, seems to come out of nowhere),
but it is certainly better than nothing. I like the idea of an
exposition on this stuff, and I like the name
"Regular Mapping Paradigm" combined with a disclaimer about
the term "paradigm".

I also like the 'incompatible ideas' part.

And the bit about consistency, and how it's now popular to
consider the mappings (rather than the ETs) as atomic, and
to use consistent mappings. I've been meaning to say
something about that myself.

I'm not sure if it would make a good basis for a FAQ or definitive
exposition, even with lots of editing. But it could very well be
something worth sharing with the other lists.

-Carl

Been to

http://www.lumma.org/music/theory/tctmo/

lately?

-C.

At 02:34 PM 5/26/2006, you wrote:
>>I've written a web page about how some of us think about tuning theory
>>now. Here it is:
>>
>>http://x31eq.com/paradigm.html
>>
>>I welcome comments. Anybody who things it applies to them, or doesn't
>>and should. Or general suggestions for improvements.
>>
>>Once it's been through a bit of peer review I'll tell the main list
>>about it.
>
>You've been busy!
>
>It seems kinda randomly organized and has lots of undefined
>terms (rank 2, to give one example, seems to come out of nowhere),
>but it is certainly better than nothing. I like the idea of an
>exposition on this stuff, and I like the name
>"Regular Mapping Paradigm" combined with a disclaimer about
>the term "paradigm".
>
>I also like the 'incompatible ideas' part.
>
>And the bit about consistency, and how it's now popular to
>consider the mappings (rather than the ETs) as atomic, and
>to use consistent mappings. I've been meaning to say
>something about that myself.
>
>I'm not sure if it would make a good basis for a FAQ or definitive
>exposition, even with lots of editing. But it could very well be
>something worth sharing with the other lists.
>
>-Carl

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> I've written a web page about how some of us think about tuning
theory
> now. Here it is:
>
> http://x31eq.com/paradigm.html
>
> I welcome comments. Anybody who things it applies to them, or
doesn't
> and should. Or general suggestions for improvements.
>
> Once it's been through a bit of peer review I'll tell the main list
> about it.
>
>
> Graham

I like it, if anything, its a great refresher. I would be great
if you expanded the section on Group Theory. Also, now I'm really
curious about diatonic set theory. (I'm interested in enumerative
combinatorics, (Polya), but that is kind of separate from tuning
stuff, but maybe not?)

Paul Hj
>

Carl Lumma wrote:
> Been to
> > http://www.lumma.org/music/theory/tctmo/
> > lately?

No, but I'm checking it now. I might borrow some links. You've got generators, unison vectors and regular temperaments there.

Graham

Paul G Hjelmstad wrote:

> I like it, if anything, its a great refresher. I would be great
> if you expanded the section on Group Theory. Also, now I'm really
> curious about diatonic set theory. (I'm interested in enumerative
> combinatorics, (Polya), but that is kind of separate from tuning
> stuff, but maybe not?)

I don't want to frighten off mathematicians, and anyway I don't have my group theory textbook to check details. But what would you like to see? I should probably mention Lindley and Turner-Smith's book (much derided by Gene) that does bring perceptual limits into group theory.

I don't know anything about enumerative combinatorics :P

Graham

Carl Lumma wrote:

> It seems kinda randomly organized and has lots of undefined
> terms (rank 2, to give one example, seems to come out of nowhere),
> but it is certainly better than nothing. I like the idea of an
> exposition on this stuff, and I like the name
> "Regular Mapping Paradigm" combined with a disclaimer about
> the term "paradigm".

The organization is:

1) describe the existing paradigms

2) show why they proved inadequate

3) describe the new paradigm

4) discuss its implications

Certainly (2) has to come after (1) and (4) after (3). I could move (3) earlier but I like to have some suspense.

There is a problem with undefined terms. I could add a glossary but things should stll make sense as you read it. Perhaps some definitions in the introduction? I've cleaned up "rank 2" anyway, and updated it

http://x31eq.com/paradigm.html

> I also like the 'incompatible ideas' part.
> > And the bit about consistency, and how it's now popular to
> consider the mappings (rather than the ETs) as atomic, and
> to use consistent mappings. I've been meaning to say
> something about that myself.

Aha!

> I'm not sure if it would make a good basis for a FAQ or definitive
> exposition, even with lots of editing. But it could very well be
> something worth sharing with the other lists.

I don't think of it as a FAQ. It is an exposition but the definitive one would have to come from outside the community, or at least have some benefit of hindsight.

Graham

Graham Breed wrote:
> I've written a web page about how some of us think about tuning theory > now. Here it is:
> > http://x31eq.com/paradigm.html
> > I welcome comments. Anybody who things it applies to them, or doesn't > and should. Or general suggestions for improvements.
> > Once it's been through a bit of peer review I'll tell the main list > about it.

One minor thing; Harry Partch's chromelodeon was tuned to the 43-note scale, not anything related to the 7-and-12 paradigm. He just retuned all the reeds and left the keyboard as it was (retuning the reeds alone would have been a pretty big task). Maybe you knew that, but someone could get the wrong impression from reading the page.

You might want to mention what a generalized keyboard is before the comment about scales with the same mapping: "They'll work with the same generalized keyboard mappings". Someone reading this for the first time might wonder what a "keyboard mapping" is and what it would mean for one of those to be "generalized" (until they get down to the section on generalized keyboards later on).

You should also mention what the "scale tree" is (or link to Wilson's article describing it).

>> It seems kinda randomly organized and has lots of undefined
>> terms (rank 2, to give one example, seems to come out of nowhere),
>> but it is certainly better than nothing. I like the idea of an
>> exposition on this stuff, and I like the name
>> "Regular Mapping Paradigm" combined with a disclaimer about
>> the term "paradigm".
>
>The organization is:
>
>1) describe the existing paradigms
>
>2) show why they proved inadequate
>
>3) describe the new paradigm
>
>4) discuss its implications

About half way through I saw a control statement alluding to
this. It wasn't completely clear. Maybe you should put the
above 4 headings into the document.

-Carl

Carl Lumma wrote:
>>>It seems kinda randomly organized and has lots of undefined
>>>terms (rank 2, to give one example, seems to come out of nowhere),
>>>but it is certainly better than nothing. I like the idea of an
>>>exposition on this stuff, and I like the name
>>>"Regular Mapping Paradigm" combined with a disclaimer about
>>>the term "paradigm".
>>
>>The organization is:
>>
>>1) describe the existing paradigms
>>
>>2) show why they proved inadequate
>>
>>3) describe the new paradigm
>>
>>4) discuss its implications
> > > About half way through I saw a control statement alluding to
> this. It wasn't completely clear. Maybe you should put the
> above 4 headings into the document.

The current second-level headings are:

Introduction
Related Paradigms
Paradigm Breaking Developments
Key Ideas
Miracle Temperament
Compatible Ideas
Incompatible Ideas

I could group the last three together.

Graham

Herman Miller wrote:

> One minor thing; Harry Partch's chromelodeon was tuned to the 43-note > scale, not anything related to the 7-and-12 paradigm. He just retuned > all the reeds and left the keyboard as it was (retuning the reeds alone > would have been a pretty big task). Maybe you knew that, but someone > could get the wrong impression from reading the page.

Yes, but it's fiddly to explain. Perhaps I should say that none of the instruments has anything to do with the 7-and-12 paradigm, and in a footnote say that the chromelodeon still looks like it does. What about the second chromelodeon?

> You might want to mention what a generalized keyboard is before the > comment about scales with the same mapping: "They'll work with the same > generalized keyboard mappings". Someone reading this for the first time > might wonder what a "keyboard mapping" is and what it would mean for one > of those to be "generalized" (until they get down to the section on > generalized keyboards later on).

Yes, give them some idea and provide a link. I can't reach the Anaphorian Embassy site to check my links. I have my own page talking about them though.

> You should also mention what the "scale tree" is (or link to Wilson's > article describing it).

Yes! Do you have a good link?

Graham

At 12:36 AM 5/27/2006, you wrote:
>Carl Lumma wrote:
>>>>It seems kinda randomly organized and has lots of undefined
>>>>terms (rank 2, to give one example, seems to come out of nowhere),
>>>>but it is certainly better than nothing. I like the idea of an
>>>>exposition on this stuff, and I like the name
>>>>"Regular Mapping Paradigm" combined with a disclaimer about
>>>>the term "paradigm".
>>>
>>>The organization is:
>>>
>>>1) describe the existing paradigms
>>>
>>>2) show why they proved inadequate
>>>
>>>3) describe the new paradigm
>>>
>>>4) discuss its implications
>>
>>
>> About half way through I saw a control statement alluding to
>> this. It wasn't completely clear. Maybe you should put the
>> above 4 headings into the document.
>
>The current second-level headings are:
>
>Introduction
>Related Paradigms
>Paradigm Breaking Developments
>Key Ideas
>Miracle Temperament
>Compatible Ideas
>Incompatible Ideas
>
>I could group the last three together.

Whaddabout:

Introduction
Prior paradigms
Paradigm-breaking developments
. Miracle temperament
The new paradigm
. Key ideas
. Incompatible ideas

-Carl

Graham Breed wrote:
> Herman Miller wrote:
> > >>One minor thing; Harry Partch's chromelodeon was tuned to the 43-note >>scale, not anything related to the 7-and-12 paradigm. He just retuned >>all the reeds and left the keyboard as it was (retuning the reeds alone >>would have been a pretty big task). Maybe you knew that, but someone >>could get the wrong impression from reading the page.
> > > Yes, but it's fiddly to explain. Perhaps I should say that none of the > instruments has anything to do with the 7-and-12 paradigm, and in a > footnote say that the chromelodeon still looks like it does. What about > the second chromelodeon?

That sounds like a better explanation. Both the original chromelodeon and the "New Chromelodeon II" have piano-style keyboards with labeled keys. The "Old Chromelodeon II" had extra keys added to the piano-style keyboard, but an octave of the 43-note scale still took up the space of two piano octaves.

>>You might want to mention what a generalized keyboard is before the >>comment about scales with the same mapping: "They'll work with the same >>generalized keyboard mappings". Someone reading this for the first time >>might wonder what a "keyboard mapping" is and what it would mean for one >>of those to be "generalized" (until they get down to the section on >>generalized keyboards later on).
> > > Yes, give them some idea and provide a link. I can't reach the > Anaphorian Embassy site to check my links. I have my own page talking > about them though.

If you want some links to the Anaphorian site, here's a link to the "Multi-Keyboard Gridiron" article. (It also has an illustration of the scale tree.)

http://www.anaphoria.com/keygrid.PDF

This is the article describing Larry Hanson's 53-tone keyboard:

http://www.anaphoria.com/hanson.PDF

And various articles referring to the Bosanquet keyboard:

http://www.anaphoria.com/xen1.PDF
http://www.anaphoria.com/xen2.PDF
http://www.anaphoria.com/xen3a.PDF
http://www.anaphoria.com/xen3b.PDF

>>You should also mention what the "scale tree" is (or link to Wilson's >>article describing it).
> > > Yes! Do you have a good link?

Well, I thought there was a description of scale trees somewhere, but I can't find anything quite suitable. The main Wilson archives page is here:

http://www.anaphoria.com/wilson.html

The scale tree is alluded to in this article, although not named as such:

http://www.anaphoria.com/xen1.PDF

This chart shows the scale tree in detail:

http://www.anaphoria.com/sctree.PDF

The problem with these articles is that you pretty much need to understand what they're about in the first place before they start to make sense. Many of them are little more than charts and diagrams.... It might be just as well to skip the scale tree reference (since the relationship between rank 2 temperaments and using the scale tree to find ET approximations isn't clear from any of these articles as far as I can tell). It might be enough just to say "you can find equal temperaments that get closer and closer to the tuning you're interested in". The details of how this works would take a good deal of explanation for the sort of reader this page seems to be aimed at, which would be good to have in a supplementary page for more advanced readers.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Paul G Hjelmstad wrote:
>
> > I like it, if anything, its a great refresher. I would be great
> > if you expanded the section on Group Theory. Also, now I'm really
> > curious about diatonic set theory. (I'm interested in enumerative
> > combinatorics, (Polya), but that is kind of separate from tuning
> > stuff, but maybe not?)
>
> I don't want to frighten off mathematicians, and anyway I don't
have my
> group theory textbook to check details. But what would you like to
see?
> I should probably mention Lindley and Turner-Smith's book (much
> derided by Gene) that does bring perceptual limits into group
theory.

I'd look at it, but why does Gene hate it?

> I don't know anything about enumerative combinatorics :P

It's pretty easy. It combines Group Theory and Combinatorics. It
shows, for example, the number of chords (0-, 1-, 2- triads,
tetrachords, etc) in 12t-ET in polynomial form.
You can also look at different symmetries like D4 X S3 for example.
(The "M5" symmetry).

>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> > I should probably mention Lindley and Turner-Smith's book (much
> > derided by Gene) that does bring perceptual limits into group
> theory.
>
> I'd look at it, but why does Gene hate it?

I've never even read it. That does not mean I may not have derided
something or other someone extracted from it.

Paul G Hjelmstad wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>>I don't want to frighten off mathematicians, and anyway I don't > > have my
>>group theory textbook to check details. But what would you like to > see? > >> I should probably mention Lindley and Turner-Smith's book (much >>derided by Gene) that does bring perceptual limits into group > theory.
> > I'd look at it, but why does Gene hate it?

There's an associated paper which Gene didn't like. IIRC because he couldn't see the point in using non-standard group theory.

http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley.art.html

There's also a review here:

http://mto.societymusictheory.org/issues/mto.95.1.4/mto.95.1.4.cuciurean.rev

>>I don't know anything about enumerative combinatorics :P
> > It's pretty easy. It combines Group Theory and Combinatorics. It > shows, for example, the number of chords (0-, 1-, 2- triads, > tetrachords, etc) in 12t-ET in polynomial form.
> You can also look at different symmetries like D4 X S3 for example.
> (The "M5" symmetry). Okay.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> There's an associated paper which Gene didn't like. IIRC because he
> couldn't see the point in using non-standard group theory.

Right. Never make things complicated unless there's a payoff, and
there didn't seem to be one. Anyway, if you must have fuzzy groups
then why don't have actual fuzzy groups?

Hi all,

From: Graham Breed on Fri May 26, 2006:
>
> I've written a web page about how some of us think about tuning theory
> now. Here it is:
>
> http://x31eq.com/paradigm.html
>
> I welcome comments. Anybody who things it applies to them, or doesn't
> and should. Or general suggestions for improvements.
>
> Once it's been through a bit of peer review I'll tell the main list
> about it.

Possibly the most illuminating thing I've read in a year ...!

Thank you, Graham.

Yes, I do have quibbles and questions. ;-)

The quibbles are -

1) some typos

2) saying you don't really get much from group theory.
That, I suggest, depends on who you are.

The questions are -

3) Who is your intended audience? In the main,
I think this is a populist exposition of a new set
of ideas, that might perhaps appear in a monthly
magazine for music-lovers. But that impression
is countered by the fact that it is necessary to
understand several theoretical terms that you
do not explain, eg ...

4) Why talk about regular temperaments before
describing their place in the paradigm?

5) Alternatively, why not make it easier to under-
stand (by definition or example) what is AND is
not a regular temperament?

Enough for now. As you see, I'm a bit behind
in my reading ...

Regards,
Yahya

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Yahya Abdal-Aziz wrote:

> Possibly the most illuminating thing I've read in a year ...!
> > Thank you, Graham.

Oh, super!

> Yes, I do have quibbles and questions. ;-)
> > The quibbles are -
> > 1) some typos

Can you list them?

> 2) saying you don't really get much from group theory.
> That, I suggest, depends on who you are.

What can we do with group theory that we couldn't do without it?

> The questions are -
> > 3) Who is your intended audience? In the main,
> I think this is a populist exposition of a new set
> of ideas, that might perhaps appear in a monthly > magazine for music-lovers. But that impression > is countered by the fact that it is necessary to > understand several theoretical terms that you > do not explain, eg ...

It's aimed at anybody who might be interested in what we're doing but isn't going to struggle through the full explanations. It's part of the realization that this constitutes a new paradigm. It means people who don't understand the paradigm can't appreciate the work that's carried out within it.

I found at the MicroFest that it wasn't easy to explain what we'd been doing. Even though everybody there was interested in microtonality, and some of them had a mathematical background. There were also professors of music who didn't have much knowledge at all of our kind of microtonal theory. One target audience is definitely professors of music.

It's also a first draft to get my ideas done and share with people here. I need to work out what terminology to improve and when to introduce it.

> 4) Why talk about regular temperaments before > describing their place in the paradigm? Regular temperaments existed before the paradigm took shape. And I explain the paradigm itself after the background because I think people will understand it better when they know the reasons for it.

> 5) Alternatively, why not make it easier to under-
> stand (by definition or example) what is AND is > not a regular temperament?

Because it's going to bore the readers with technical details before they know why they need to know them. Really I need a pseudo-definition that lets them know we're playing with some things called regular temperaments without knowing what they really are. Like "elephants are big animals that live in hot countries" rather than a full anatomical description.

> Enough for now. As you see, I'm a bit behind > in my reading ...

That's no problem, I'm a bit behind in my redrafting :P

Graham

From: Carl Lumma on Fri May 26, 2006:
>
> >> It seems kinda randomly organized and has lots of undefined
> >> terms (rank 2, to give one example, seems to come out of nowhere),
> >> but it is certainly better than nothing. I like the idea of an
> >> exposition on this stuff, and I like the name
> >> "Regular Mapping Paradigm" combined with a disclaimer about
> >> the term "paradigm".
> >
> >The organization is:
> >
> >1) describe the existing paradigms
> >
> >2) show why they proved inadequate
> >
> >3) describe the new paradigm
> >
> >4) discuss its implications
>
> About half way through I saw a control statement alluding to
> this. It wasn't completely clear. Maybe you should put the
> above 4 headings into the document.

Graham,

This was the sort of structure I thought
I had discerned. I agree with Carl about
the value of including these 4 headings.

IMO, the weakest part of the exposition
was in part 2). You could user stronger
assertions, eg "The <X> paradigm fails to
explain <property Y> of <entity Z>" - with
suitable substitutions, of course! ;-)

Regards,
Yahya

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I've updated the page to reflect comments I've had up to but not including this message. So if anybody wants to check:

http://x31eq.com/paradigm.html

Yahya Abdal-Aziz wrote:
> From: Carl Lumma on Fri May 26, 2006: > >>>>It seems kinda randomly organized and has lots of undefined
>>>>terms (rank 2, to give one example, seems to come out of nowhere),
>>>>but it is certainly better than nothing. I like the idea of an
>>>>exposition on this stuff, and I like the name
>>>>"Regular Mapping Paradigm" combined with a disclaimer about
>>>>the term "paradigm".
>>>
>>>The organization is:
>>>
>>>1) describe the existing paradigms
>>>
>>>2) show why they proved inadequate
>>>
>>>3) describe the new paradigm
>>>
>>>4) discuss its implications
>>
>>About half way through I saw a control statement alluding to
>>this. It wasn't completely clear. Maybe you should put the
>>above 4 headings into the document.
> > > Graham,
> > This was the sort of structure I thought
> I had discerned. I agree with Carl about > the value of including these 4 headings.

Yes, I've tweaked it so that it's nearer to what Carl suggested.

> IMO, the weakest part of the exposition > was in part 2). You could user stronger
> assertions, eg "The <X> paradigm fails to
> explain <property Y> of <entity Z>" - with
> suitable substitutions, of course! ;-)

Okay, I'll think about this. But how am I supposed to say that it fails to explain a property that *I* can't explain at that point because I haven't introduced a suitable paradigm?

Graham