The Matrix

Gentlepersons,

Andy Milne, Bill Sethares and I have recently published some papers
in the Computer Music Journal and the Journal of Music and
Mathematics that describe the use of isomorphic (generalized)
keyboards to control alternative tunings.

However, we think that this work can go much deeper -- right down to
the very souce of harmony. We've just completed a draft paper that
describes the Big Picture -- which we're calling the Matrix -- in
relatively accessible terms:
http://www.thummer.com/papers/Matrix.pdf

To the members of this group, many of the concepts of tuning and
temperament described therein will be entirely familiar. However, I
suspect that you will find them to be combined in a new way,
especially with regard to the tempering of timbres to match
continuously-variable tunings.

The odds are pretty good that we've got many of the details wrong,
are using non-standard terminology, etc. I'd appreciate any comments,
criticisms, or suggestions that you might have. I'm particularly
interested in adding references that give proper credit to people who
created the ideas on which the Matrix is based, wherever you can find
such an idea un-credited.

Thanks! :-)

Jim Plamondon
Austin, Texas
www.thummer.com

P.S.: I'm the primary author of this paper, so the mistakes are mine,
although Andy and Bill worked out the theory behind the Matrix.

I just took a look at it - seems interesting, I'll have to read the
whole thing when I have some time on hand.

As a side note, I've played around with your Thummer - a prototype, I
think it was. A professor at my school, Colby Leider from the
University of Miami, has two of them in his office. He asked me to see
if I wanted to use it to experiment with dynamic intonation or just
see what is possible with a microtonal MIDI controller in general... I
rigged one up to Scala, but I couldn't figure out how to get it to
control 19-tet or 31-tet or anything except 12-equal (which was
probably due to a lack of knowledge on my part). I did really like the
idea behind it though.

On Thu, Aug 7, 2008 at 5:06 PM, jimplamondon <jimplamondon@...> wrote:
> Gentlepersons,
>
> Andy Milne, Bill Sethares and I have recently published some papers
> in the Computer Music Journal and the Journal of Music and
> Mathematics that describe the use of isomorphic (generalized)
> keyboards to control alternative tunings.
>
> However, we think that this work can go much deeper -- right down to
> the very souce of harmony. We've just completed a draft paper that
> describes the Big Picture -- which we're calling the Matrix -- in
> relatively accessible terms:
> http://www.thummer.com/papers/Matrix.pdf
>
> To the members of this group, many of the concepts of tuning and
> temperament described therein will be entirely familiar. However, I
> suspect that you will find them to be combined in a new way,
> especially with regard to the tempering of timbres to match
> continuously-variable tunings.
>
> The odds are pretty good that we've got many of the details wrong,
> are using non-standard terminology, etc. I'd appreciate any comments,
> criticisms, or suggestions that you might have. I'm particularly
> interested in adding references that give proper credit to people who
> created the ideas on which the Matrix is based, wherever you can find
> such an idea un-credited.
>
> Thanks! :-)
>
> Jim Plamondon
> Austin, Texas
> www.thummer.com
>
> P.S.: I'm the primary author of this paper, so the mistakes are mine,
> although Andy and Bill worked out the theory behind the Matrix.
>
>

2008/8/7 jimplamondon <jimplamondon@...>:
> Gentlepersons,
>
> Andy Milne, Bill Sethares and I have recently published some papers
> in the Computer Music Journal and the Journal of Music and
> Mathematics that describe the use of isomorphic (generalized)
> keyboards to control alternative tunings.
>
> However, we think that this work can go much deeper -- right down to
> the very souce of harmony. We've just completed a draft paper that
> describes the Big Picture -- which we're calling the Matrix -- in
> relatively accessible terms:
> http://www.thummer.com/papers/Matrix.pdf

No more comments overnight -- is everybody on holiday?

> To the members of this group, many of the concepts of tuning and
> temperament described therein will be entirely familiar. However, I
> suspect that you will find them to be combined in a new way,
> especially with regard to the tempering of timbres to match
> continuously-variable tunings.

I've done continuously-variable tunings before, and even give instructions here:

http://x31eq.com/miracle/keyboard.html#kyma

Continuously-variable timbres are probably new.

About these temperament modulations. You talk about equal
temperaments being pivot tunings. You can also think of a higher-rank
system as being the pivot tuning between two temperaments. For
example, write something in meantone and then something in magic.
Make the transition work correctly in 5-limit just intonation. Then
you can modulate from meantone to JI to magic. From whatever
direction, Gene's talked about applying different unison vectors
(under whatever name) on tuning-math. I don't know how far he got in
actual music.

> The odds are pretty good that we've got many of the details wrong,
> are using non-standard terminology, etc. I'd appreciate any comments,
> criticisms, or suggestions that you might have. I'm particularly
> interested in adding references that give proper credit to people who
> created the ideas on which the Matrix is based, wherever you can find
> such an idea un-credited.

Your Matrix looks like a lattice. Generally lattices are
octave-equivalent but they don't have to be. See "The Pythagorean
Lattice" in Paul Erlich's A Middle Path...

O'Connell's (sp?) Xenharmonikon article may be worth a mention. It
describes a rank 2 system based on inharmonic timbres.

You're not the first to talk about paradigm shifts, of course, remembering

http://x31eq.com/paradigm.htm

That does mention inharmonic timbres so are paradigms are the same,
but with different approaches. There's also a discussion of prime
intervals independent of prime numbers in

http://x31eq.com/primerr.pdf

Graham

Graham Breed wrote:
> 2008/8/7 jimplamondon <jimplamondon@...>:
>> Gentlepersons,
>>
>> Andy Milne, Bill Sethares and I have recently published some papers
>> in the Computer Music Journal and the Journal of Music and
>> Mathematics that describe the use of isomorphic (generalized)
>> keyboards to control alternative tunings.

...

>> The odds are pretty good that we've got many of the details wrong,
>> are using non-standard terminology, etc. I'd appreciate any comments,
>> criticisms, or suggestions that you might have. I'm particularly
>> interested in adding references that give proper credit to people who
>> created the ideas on which the Matrix is based, wherever you can find
>> such an idea un-credited.

The usage of the word "comma" for any small interval is (or was) common on the tuning-math list, but some prefer to reserve "comma" for intervals around the size of the syntonic and Pythagorean commas. Alternative terms include "unison vector" (Fokker), but that's also a disputed term (some prefer to use "unison" only for an exact 1:1).

> Your Matrix looks like a lattice. Generally lattices are
> octave-equivalent but they don't have to be. See "The Pythagorean
> Lattice" in Paul Erlich's A Middle Path...

Later on in the paper he also indicates that the same lattice can represent meantone tuning. Magic temperament is also mentioned in the "Middle Path" paper, although the earliest mention of magic is probably on one of Graham Breed's web pages. http://x31eq.com/catalog.htm

Paul's paper is:
A Middle Path Between Just Intonation and the Equal Temperaments
Xenharmonikôn 18, pp. 159-199, 2006.
http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

The idea of tempering out the starling comma, although not mentioned by that name (the page calls it "small septimal comma" is described on one of Dave Keenan's pages.

A method for optimally distributing any comma
David C Keenan, 25-Nov-1999
http://dkeenan.com/Music/DistributingCommas.htm

I believe that Shohe Tanaka was the originator of the term "schismatic temperament". Does anyone have a reference?

> O'Connell's (sp?) Xenharmonikon article may be worth a mention. It
> describes a rank 2 system based on inharmonic timbres.
> > You're not the first to talk about paradigm shifts, of course, remembering
> > http://x31eq.com/paradigm.htm
> > That does mention inharmonic timbres so are paradigms are the same,
> but with different approaches. There's also a discussion of prime
> intervals independent of prime numbers in
> > http://x31eq.com/primerr.pdf

> I believe that Shohe Tanaka was the originator of the term
> "schismatic temperament". Does anyone have a reference?

Ellis uses "schismic". If you have a Tanaka reference I'll
tell you which came first.

-Carl

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> The usage of the word "comma" for any small interval is (or was) common
> on the tuning-math list, but some prefer to reserve "comma" for
> intervals around the size of the syntonic and Pythagorean commas.
> Alternative terms include "unison vector" (Fokker), but that's also a
> disputed term (some prefer to use "unison" only for an exact 1:1).

With the indefinite article, "a comma" is typically any rational
interval smaller than a scale step. But with the definite article,
"the <something> comma" is typically in the range from about 12 to 33
cents. Other size ranges are called schismas, kleismas, and dieses. It
is usually clear from the context whether the term is being used
generically or specifically. See
http://tonalsoft.com/enc/c/comma.aspx

A vector is only one possible mathematical representation of a comma
and so "unison vector" or any other qualification of "vector" cannot
replace the generic usage of the musical term "comma".

> The idea of tempering out the starling comma, although not mentioned by
> that name (the page calls it "small septimal comma" is described on one
> of Dave Keenan's pages.
>
> A method for optimally distributing any comma
> David C Keenan, 25-Nov-1999
> http://dkeenan.com/Music/DistributingCommas.htm

Herman is being modest. I mention it, yes, but credit the idea to him.
See http://www.io.com/~hmiller/music/marrgarrel.html

-- Dave Keenan

2008/8/10 Herman Miller <hmiller@...>:

> Later on in the paper he also indicates that the same lattice can
> represent meantone tuning. Magic temperament is also mentioned in the
> "Middle Path" paper, although the earliest mention of magic is probably
> on one of Graham Breed's web pages. http://x31eq.com/catalog.htm

There's nothing to say a lattice can't be tempered.

It's become traditional not to give a citation for magic temperament
as it's a fairly obvious idea. But note that George Secor's recent
Xenharmonikon article on miracle temperament says that he looked at it
(without a name) in 1974.

> I believe that Shohe Tanaka was the originator of the term "schismatic
> temperament". Does anyone have a reference?

Helmholtz used the term "schismatische Verwechselung". While German
speakers tell us this doesn't translate as "schismatic temperament" I
don't know what would. Liberty Manik wrote an obscure book that goes
through the history. Helmholtz didn't invent the term -- maybe
Riemann?

Tanaka also used the term "kleismatische Verwechselung" for what we
call hanson temperament. In so far as he talked about it as a
temperament class (he identified the generator at least) I believe he
was the first. The paper's on the Anaphorian site.

Graham