Tymoczko

This page has some links to the work of Dmitri Tymoczko, a theorist, whose work echos that of many in the tuning community, with emphasis on the voice-leading implications of lattice techniques:

http://rogerbourland.com/redblackwindow/2006/07/08/dmitri-tymoc

There has been a general trend towards overlap, if not a reconciliation of sorts between academic theorists (especially those in the neo-Reimannian direction) and the mostly extra-academic tuning theorists. I expect to see a lot more in the future from academic theorists that looks like things tuning theorists have been doing. (The resemblances should be immediately apparent). It's a good thing.

DJW

--- In tuning@yahoogroups.com, Daniel Wolf <djwolf@...> wrote:
>
> This page has some links to the work of Dmitri Tymoczko,
> a theorist, whose work echos that of many in the tuning
> community, with emphasis on the voice-leading implications
> of lattice techniques:
>
> http://rogerbourland.com/redblackwindow/2006/07/08/dmitri-tymoc

I wrote to Tymoczko, giving him links to some of my stuff
about Tonescape lattices, and his response included this:

>> I think there's a fairly big difference between what
>> I'm doing and the more traditional "lattice-based"
>> approach commonly used by tuning enthusiasts. My spaces
>> are fundamentally continuous, which means they have a
>> very different geometry from the discrete, grid-based
>> spaces that derive from Euler, Riemann, and others.
>>
>> These discrete spaces (including, insofar as I can tell,
>> yours) can't really be used to represent voice leading.
>> The issue is that pitch classes that are arbitrarily
>> far apart on the lattice can be arbitrarily close in
>> pitch class space. (You can go up 1,000,000+ acoustically
>> pure perfect fifths, and end up very close to where you
>> started!)

So he's obviously familiar with what we tuning theorists
are doing with lattices.

Is it possible for anyone to post a copy of Tymoczko's
_Science_ article online?

-monz
http://tonalsoft.com
Tonescape microtonal music software

Daniel Wolf wrote:
> This page has some links to the work of Dmitri Tymoczko, a theorist, > whose work echos that of many in the tuning community, with emphasis on > the voice-leading implications of lattice techniques:
> > http://rogerbourland.com/redblackwindow/2006/07/08/dmitri-tymoc

make that

http://rogerbourland.com/redblackwindow/2006/07/08/dmitri-tymoczko/

We've looked at it on tuning-math, but not said much. The supporting material does mention alternative equal temperaments (or measuring scales).

> There has been a general trend towards overlap, if not a reconciliation > of sorts between academic theorists (especially those in the > neo-Reimannian direction) and the mostly extra-academic tuning > theorists. I expect to see a lot more in the future from academic > theorists that looks like things tuning theorists have been doing. (The > resemblances should be immediately apparent). It's a good thing.

We'll have to see where it goes. Currently tuning theory is one of those inter-disciplinary affairs that music-theory academics tend to stay away from. It'll also be interesting to see if more mathematicians get involved with it.

Note that Tymoczko's lattices show pitch distance rather than harmonic distance, so the overlap isn't that great. Perhaps applying the same geometric ideas to adaptive tuning will be the bridge.

Graham

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> I wrote to Tymoczko, giving him links to some of my stuff
> about Tonescape lattices, and his response included this:
>
> >> I think there's a fairly big difference between what
> >> I'm doing and the more traditional "lattice-based"
> >> approach commonly used by tuning enthusiasts. My spaces
> >> are fundamentally continuous, which means they have a
> >> very different geometry from the discrete, grid-based
> >> spaces that derive from Euler, Riemann, and others.
> >>
> >> These discrete spaces (including, insofar as I can tell,
> >> yours) can't really be used to represent voice leading.
> >> The issue is that pitch classes that are arbitrarily
> >> far apart on the lattice can be arbitrarily close in
> >> pitch class space. (You can go up 1,000,000+ acoustically
> >> pure perfect fifths, and end up very close to where you
> >> started!)

While we "tuners" usually collapse the powers-of-two axis, this isn't
always the case, as in Tenney's work, or when, for example, Wilson's
lattices are treated without octave equivalence, so he really has no
argument here.

DJW

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>
> We'll have to see where it goes. Currently tuning theory is one of
> those inter-disciplinary affairs that music-theory academics tend to
> stay away from. It'll also be interesting to see if more
mathematicians
> get involved with it.
>
>

I've had a handful of encounters with academics that indicate that
they are, in fact, paying attention to us. The late John Clough, for
example, one of the leaders in the neo-Riemannian scene, was very
well-informed. We met in Budapest once, and he was full of admiration
for Paul Erlich, Erv Wilson, and Margo Schulter. In particular, he was
fascinated by Wilson's construction of MOS from any pair of relatively
prime numbers, applied not only to octave periods (contrary to popular
opinion around here, Wilson has applied MOS to periods other than
octaves, in my case, he demonstrated with fourths and twelfths), where
the melodic symmetry was absolutely clear in that with each additional
MOS, the generating intervals (+ one anomolous interval) were
subtended by the same number of tones, and the intervallic pattern of
the division of the generators was symmetrical.

DJW

monz wrote:

> So he's obviously familiar with what we tuning theorists
> are doing with lattices.

Nice!

> Is it possible for anyone to post a copy of Tymoczko's
> _Science_ article online?

You can get it here:

http://music.princeton.edu/~dmitri/ChordGeometries.html

The "here" linked paper is good background as well.

Graham

> While we "tuners" usually collapse the powers-of-two axis, this isn't
> always the case, as in Tenney's work, or when, for example, Wilson's
> lattices are treated without octave equivalence, so he really has no
> argument here.

Daniel,

Tymoczko's work is fundamentally different than the lattices
of any tuning theorist I'm familiar with.

-Carl

djwolf_frankfurt wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
> > >>I wrote to Tymoczko, giving him links to some of my stuff
>>about Tonescape lattices, and his response included this:
>>
>>
>>>>I think there's a fairly big difference between what
>>>>I'm doing and the more traditional "lattice-based"
>>>>approach commonly used by tuning enthusiasts. My spaces
>>>>are fundamentally continuous, which means they have a
>>>>very different geometry from the discrete, grid-based
>>>>spaces that derive from Euler, Riemann, and others.
>>>>
>>>>These discrete spaces (including, insofar as I can tell,
>>>>yours) can't really be used to represent voice leading.
>>>>The issue is that pitch classes that are arbitrarily
>>>>far apart on the lattice can be arbitrarily close in
>>>>pitch class space. (You can go up 1,000,000+ acoustically
>>>>pure perfect fifths, and end up very close to where you
>>>>started!)
> > > While we "tuners" usually collapse the powers-of-two axis, this isn't
> always the case, as in Tenney's work, or when, for example, Wilson's
> lattices are treated without octave equivalence, so he really has no
> argument here.

Octave equivalence is a red herring. You can go up 1,000,006 acoustically pure fifths and down 1,584,972 acoustically pure octaves and you'll still be a long way from where you started on the average tuning lattice. But less than 13 cents away in pitch space. So the geometries used for his orbifold and our lattices are fundamentally different. But surely reconcilable, because a theory that didn't map to pitch space wouldn't be much use for *tuning*.

I don't know if the discrete/continous distinction matters. As he measures everything to the nearest semitone a discrete geometry would probably give the same result. But what do I know? Monzo's lattices are continuous as well. If you want to associate a range of pitches with a point (or a number of points) on the lattice, that can be done. And you can use fuzzy logic if you don't want discontinuities. There are also plenty of continuous dissonance functions out there. One of them from Paul Erlich who thinks geometrically.

Dear old Rothenberg, who I'm still hoping to hear more from, may be relevant as well. He worked with ordered pitches in continuous space, with no need for prime factorization. Tymoczko's already noticed maximal evenness as a desirable property for chords and it guarantees propriety.

Graham

Hi Graham,

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

> I don't know if the discrete/continous distinction matters.
> As he [Tymoczko] measures everything to the nearest semitone
> a discrete geometry would probably give the same result.
> But what do I know? Monzo's lattices are continuous as well.

I just want to be totally clear on this. Do you say that
because i'm willing to use fractions to plot points between
the discrete points on a lattice? ... for example:

the graphic labeled "1/4-comma meantone rational implications" here:
http://tonalsoft.com/enc/m/meantone.aspx

or the graphic labeled "1/3-comma meantone, Salinas 1577" here:
http://tonalsoft.com/enc/number/19edo.aspx

On both of these, i'm plotting the meantone chain as a line
whose points fall at specific fractional exponents of 3 and 5,
so that, for example, on the 1/4-comma meantone lattice where
C=n^0 (i.e., 1/1) and E=5^1, i plot G=5^(1/4), D=5^(1/2), and
A=5^(3/4).

If this is not what you mean, then please explain further.
Thanks.

-monz
http://tonalsoft.com
Tonescape microtonal music software

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

> >
> > While we "tuners" usually collapse the powers-of-two axis, this isn't
> > always the case, as in Tenney's work, or when, for example, Wilson's
> > lattices are treated without octave equivalence, so he really has no
> > argument here.
>
> Octave equivalence is a red herring. You can go up 1,000,006
> acoustically pure fifths and down 1,584,972 acoustically pure octaves
> and you'll still be a long way from where you started on the average
> tuning lattice. But less than 13 cents away in pitch space. So the
> geometries used for his orbifold and our lattices are fundamentally
> different. But surely reconcilable, because a theory that didn't
map to
> pitch space wouldn't be much use for *tuning*.
>
> I don't know if the discrete/continous distinction matters. As he
> measures everything to the nearest semitone a discrete geometry would
> probably give the same result. But what do I know? Monzo's lattices
> are continuous as well. If you want to associate a range of pitches
> with a point (or a number of points) on the lattice, that can be done.
> And you can use fuzzy logic if you don't want discontinuities. There
> are also plenty of continuous dissonance functions out there. One of
> them from Paul Erlich who thinks geometrically.
>
> Dear old Rothenberg, who I'm still hoping to hear more from, may be
> relevant as well. He worked with ordered pitches in continuous space,
> with no need for prime factorization. Tymoczko's already noticed
> maximal evenness as a desirable property for chords and it guarantees
> propriety.
>
>
> Graham
>

Graham --

I am thinking of Tenneys lattice of "harmonic space" with a central,
vertical pitch-height projection axis. All other axes are alligned so
that the coordinate of any tone on the pitch-height projection axis is
pitch space. On such a lattice all close voice-leading moves are close
in terms of that central axis.

I've always thought of voice leading as simultaneous moves in discrete
steps in harmonic space, described by a lattice of n-dimensions, and
the coordinate moves on the pitch-height axis. From that premise, the
idea that good voice leading is economical in distance and patterns
seems obvious.

I've scanned Tenney's lattice and will put it up online.

DJW

Here is Tenney's two-dimensional (2,3) lattice in harmonic space,
showing at centre the pitch-height projection axis:

http://home.snafu.de/djwolf/Tenney.JPG

> Here is Tenney's two-dimensional (2,3) lattice in harmonic space,
> showing at centre the pitch-height projection axis:
>
> http://home.snafu.de/djwolf/Tenney.JPG

I didn't know Tenney did this! What year?
It's the same as David Canright's "Harmonic Melodic" diagrams

http://www.redshift.com/~dcanright/eps4ji/

but still not quite the same as Tymoczko's.

-Carl

monz wrote:
> Hi Graham,
> > > --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > >>I don't know if the discrete/continous distinction matters.
>>As he [Tymoczko] measures everything to the nearest semitone
>>a discrete geometry would probably give the same result.
>>But what do I know? Monzo's lattices are continuous as well.
> > > > I just want to be totally clear on this. Do you say that
> because i'm willing to use fractions to plot points between
> the discrete points on a lattice? ... for example:
> > the graphic labeled "1/4-comma meantone rational implications" here:
> http://tonalsoft.com/enc/m/meantone.aspx

That looks like it, yes. Not particularly relevant to Tymoczko's work but makes the point that continuous/discrete isn't the problem.

Graham

Hi Graham,

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> monz wrote:
> >
> > --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> >
> > > Monzo's lattices are continuous as well.
> >
> >
> >
> > I just want to be totally clear on this. Do you say that
> > because i'm willing to use fractions to plot points between
> > the discrete points on a lattice? ... for example:
> >
> > the graphic labeled "1/4-comma meantone rational implications" here:
> > http://tonalsoft.com/enc/m/meantone.aspx
>
> That looks like it, yes. Not particularly relevant to
> Tymoczko's work but makes the point that continuous/discrete
> isn't the problem.

But maybe i should remind everyone following this thread
that i've taken a lot of flak from Paul Erlich for doing
this.

I'm still not clear why he argues against it, something
having to do with the fact that the prime-factorization is
therefore not unique, as it is with discrete integer exponents.

But my rebuttal is always that yes, he's correct that the
fractional points make the plotting non-unique, but the
linear plotting of fractional points on the lattice
still shows exactly how that particular tuning works ...
actually i'm really talking more about the way the
fractional points spiral around the twisted helical lattice
(for a rank-2 temperament example).

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Here is Tenney's two-dimensional (2,3) lattice in harmonic space,
> > showing at centre the pitch-height projection axis:
> >
> > http://home.snafu.de/djwolf/Tenney.JPG
>
> I didn't know Tenney did this! What year?
> It's the same as David Canright's "Harmonic Melodic" diagrams
>
> http://www.redshift.com/~dcanright/eps4ji/
>

It's in Tenney's 1983 paper "John Cage and the Theory of Harmony",
which is in the Soundings Issue devoted to Tenney.

> but still not quite the same as Tymoczko's.
>
> -Carl
>

If you line up your lattice with the pitch-height projection, you're
in real pitch space, so all voice-leading are close moves along this
axis. Tymoczko add some formal description of symmetries and repeated
patterns among these motions, add some nice animation (a la Dave
Keener's tumbling dekany), and it's fundamentally the same project.

DJW

--- In tuning@yahoogroups.com, "djwolf_frankfurt" <djwolf@...> wrote:

> Tymoczko add some formal description of symmetries and
> repeated patterns among these motions, add some nice animation
> (a la Dave Keener's tumbling dekany), and it's fundamentally
> the same project.

Correct spelling is Dave Keenan.

(Most people don't do things the way Clarence
Barlow / Bahlough / Barlo / etc. does) ;-)

-monz
http://tonalsoft.com
Tonescape microtonal music software

is this process and examples providing us with unique answers that we have not seen before?
and is it the simplest process to do so?
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

to answer my own question.
this way of mapping would be useful in say working with a harmonic construct and one wanted to know where one might add chords in order to fill MOS/ constant structure space.
Also i imagine one would also be able to spot and harmonize the implying microtonal scalesteps in ones material.

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> I'm still not clear why he argues against it, something
> having to do with the fact that the prime-factorization is
> therefore not unique, as it is with discrete integer exponents.

As I've pointed out on a few occasions, it remains unique so long as
you confine yourself to rational exponents. And quite a lot can be
done that way; optimized tunings are usually of that character.

--- In tuning@yahoogroups.com, "djwolf_frankfurt" <djwolf@...> wrote:

> While we "tuners" usually collapse the powers-of-two axis, this isn't
> always the case, as in Tenney's work, or when, for example, Wilson's
> lattices are treated without octave equivalence, so he really has no
> argument here.

His point was that his spaces are continuous--they are real manifolds,
in fact. They are therefore of a fundamentally different type than a
lattice, which is embedded in a real space, but discretely.

I also tried writing him, but it bounced, BTW. But I didn't really
have all that much to say, precisely because the kinds of models are
so different.

I posted something to tuning-math this morning with
a link to a page of mine which is another good example
(similar to the Salinas one in my quote) of my linear plots
of meantones using fractional prime-factor (3 and 5) exponents,
because the graphics are simply placed inside the text of
Zarlino's classic treatise, which gives a step-by-step
explanation

http://tonalsoft.com/monzo/zarlino/1558/zarlino1558-2.htm

on this page i made a lattice with a linear plot
of 2/7-comma meantone ... it shows the whole tuning unfolding
exactly as Zarlino describes it.

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Graham,
>
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
>
> > I don't know if the discrete/continous distinction matters.
> > As he [Tymoczko] measures everything to the nearest semitone
> > a discrete geometry would probably give the same result.
> > But what do I know? Monzo's lattices are continuous as well.
>
>
> I just want to be totally clear on this. Do you say that
> because i'm willing to use fractions to plot points between
> the discrete points on a lattice? ... for example:
>
> the graphic labeled "1/4-comma meantone rational implications" here:
> http://tonalsoft.com/enc/m/meantone.aspx
>
> or the graphic labeled "1/3-comma meantone, Salinas 1577" here:
> http://tonalsoft.com/enc/number/19edo.aspx
>
>
> On both of these, i'm plotting the meantone chain as a line
> whose points fall at specific fractional exponents of 3 and 5,
> so that, for example, on the 1/4-comma meantone lattice where
> C=n^0 (i.e., 1/1) and E=5^1, i plot G=5^(1/4), D=5^(1/2), and
> A=5^(3/4).

-monz
http://tonalsoft.com
Tonescape microtonal music software

Tymoczko is referred to in an article by Steven K. Blau in the October 2006
edition of Physics Today. Titled "Good music unfolds in small steps", the
article demonstrates the relavance of the M�bius strip (made up of tertial
dyads in a 12-equal framework) to voice leading. Blau says that this musical
smorgasbord comprises other tunings also.

Oz.