Lattice visualization of music in real time

I think a wonderful utility would be one that displayed a piece of (up to)
7-limit music in real time as points lighting up at the vertices of a
lattice (the triangular lattice, of course). If two points that are
connected light up, the whole edge lights up. If three points form a utonal
triad (either a minor chord or another subset of a 7-limit utonality), all
three edges light up, but if three points form an otonal triad (either a
major chord or another subset of a 7-limit otonality), the entire triangle
lights up. A utonal tetrad would be a skeletal tetrahedron (just the edges),
but an otonal tetrad would be a tetrahedron of solid light (we need really
good graphics for this). In the case of tempered tunings, the same pattern
of lightenings would repeat itself, once in every periodicity block.

[Paul Erlich, as Brett Barbaro, TD 216.5]
>
> I think a wonderful utility would be one that displayed a piece
> of (up to) 7-limit music in real time as points lighting up at
> the vertices of a lattice (the triangular lattice, of course).
> If two points that are connected light up, the whole edge lights
> up. If three points form a utonal triad (either a minor chord or
> another subset of a 7-limit utonality), all three edges light up,
> but if three points form an otonal triad (either a major chord or
> another subset of a 7-limit otonality), the entire triangle lights
> up. A utonal tetrad would be a skeletal tetrahedron (just the
> edges), but an otonal tetrad would be a tetrahedron of solid light
> (we need really good graphics for this).

This is exactly one of the things my JustMusic software will do.

It would work wonderfully to illustrate, for example, the complex
analysis of what I did in my Beethoven tuning experiment.

And being able to animate Partch scores like this is going to
make it much easier to study and understand his harmonic practice.

Several different types of lattice plotting conventions will
be available, including Monzo, triangular, Planetary (circular),
Fokker, harmonic spiral, and hopefully various Wilson designs if
it's OK with Erv.

And really good graphics is the key element of my project.
Hopefully a version will be ready for release by some time
next year. Right now, it's able to read Scala files and
create a stationary Monzo or Planetary lattice of the pitches,
and vice versa: the user can create the lattice in the lattice
window and save it as a Scala file. Pitches can also be assigned
to keys on the computer keyboard.

Ken Fasano and I are hoping to present a piece at next year's
AFMM which will include an overhead projection of JustMusic
animation as described by Paul.

-monz

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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[Paul Erlich TD216.5]
>I think a wonderful utility would be one that displayed a piece of (up to)
>7-limit music in real time as points lighting up at the vertices of a
>lattice (the triangular lattice, of course). If two points that are
>connected light up, the whole edge lights up. If three points form a utonal
>triad (either a minor chord or another subset of a 7-limit utonality), all
>three edges light up, but if three points form an otonal triad (either a
>major chord or another subset of a 7-limit otonality), the entire triangle
>lights up. A utonal tetrad would be a skeletal tetrahedron (just the edges),
>but an otonal tetrad would be a tetrahedron of solid light (we need really
>good graphics for this). In the case of tempered tunings, the same pattern
>of lightenings would repeat itself, once in every periodicity block.

Absolutely!

Your different treatment of otonal vs. utonal is very good.

But why not make it playable too. Click on a vertex to play a single note,
an edge to play a dyad, inside a triangle to play a triad, but here it gets
difficult. In 7-limit (when projected into 2D), triangles overlap and some
are completely contained inside others, and how does one indicate a whole
tetrahedron with the mouse? Need a 3-axis input device or VR rig. Can
anyone think of a good way of doing it with just a mouse? Need to be able
to change octaves too. Hmm... Maybe it's best just for display after all.

This is the sort of thing Monz is interested in (playable lattices). But I
don't think he'll be using triangular somehow.

-- Dave Keenan
http://dkeenan.com

>> I think a wonderful utility would be one that displayed a piece
>> of (up to) 7-limit music in real time as points lighting up at
>> the vertices of a lattice (the triangular lattice, of course).
>> If two points that are connected light up, the whole edge lights
>> up. If three points form a utonal triad (either a minor chord or
>> another subset of a 7-limit utonality), all three edges light up,
>> but if three points form an otonal triad (either a major chord or
>> another subset of a 7-limit otonality), the entire triangle lights
>> up. A utonal tetrad would be a skeletal tetrahedron (just the
>> edges), but an otonal tetrad would be a tetrahedron of solid light
>> (we need really good graphics for this).

>This is exactly one of the things my JustMusic software will do.

Really? Wow. Did you come up with all that independently?

>Several different types of lattice plotting conventions will
>be available, including Monzo, triangular, Planetary (circular),
>Fokker, harmonic spiral, and hopefully various Wilson designs if
>it's OK with Erv.

The triangular lattice of course corresponds with Erv Wilson's favorite
design.

>Ken Fasano and I are hoping to present a piece at next year's
>AFMM which will include an overhead projection of JustMusic
>animation as described by Paul.

Amazing!

[Paul Erlich, TD 219.13]
>
> [Paul, in a previous post]
>>> I think a wonderful utility would be one that displayed a piece
>>> of (up to) 7-limit music in real time as points lighting up at
>>> the vertices of a lattice <etc. - snip>
>>
>> [me, monz]
>> This is exactly one of the things my JustMusic software will do.
>
> Really? Wow. Did you come up with all that independently?

Yep. I first started thinking about this kind of thing about
11 years ago, long before I 'discovered' prime-factorization of
ratios and lattice diagrams.

I was drawing diagrams that were altered versions of Ellis's
Harmonic Cell and Harmonic Decad [Helmholtz, p 458-459], which
placed Ellis's ratios in a scheme somewhat like Partch's Tonality
Diamond, and then I outlined the harmonic cells in the decad, and
realized that doing this in real-time with computer software would
be a great way to study harmonic movement.

I've been expanding and refining these ideas ever since. In
the early 1990s I wrote several little BASIC programs that did
various aspects of calculating or graphing ratios, but it's
only very recently that development on my JustMusic project
has really taken off. Ken Fasano is doing the actual coding
based on my ideas, and John Chalmers helps out quite a bit
with plotting methods.

I should also note that there will be no arbitrarily-defined
limits on my software's lattices, at least in my own lattice
design, other than those imposed by the user. Right now it can
handle up to 13-limit, but we are working on increasing the
prime/odd-limit and also including capability to diagram n-tETs
(which is easy) as well as CETs (which I need to think about
some more).

Scales can be analyzed and modified either in the lattice
window or in a database table format. But more to the point
of Paul's post, real-time sequencing and analysis are the most
important design objectives of the project.

> [me, monz]
>> Several different types of lattice plotting conventions will
>> be available, including Monzo, triangular, Planetary (circular),
>> Fokker, harmonic spiral, and hopefully various Wilson designs if
>> it's OK with Erv.
>
> The triangular lattice of course corresponds with Erv Wilson's
> favorite design.

This is true, but Erv has used *many* other geometrical
configurations, and one that I particularly like (and have posted
on before) is his Logarithmic Harmonic Spiral. The Penrose
tilings are also very interesting. (I've always been a big
fan of Buckminster Fuller's work, and I can see ways to apply
his ideas to tonal lattices too.)

-monz

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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I wrote,

>> The triangular lattice of course corresponds with Erv Wilson's
>> favorite design.

Joe Monzo wrote,

>This is true, but Erv has used *many* other geometrical
>configurations, and one that I particularly like (and have posted
>on before) is his Logarithmic Harmonic Spiral. The Penrose
>tilings are also very interesting.

The Penrose tilings arise in the triangular lattice (What Erv calls the
"pentagon" and "centered pentagon" lattices are but types of triangular
lattice as Paul Hahn and I see the latter).

>(I've always been a big
>fan of Buckminster Fuller's work, and I can see ways to apply
>his ideas to tonal lattices too.)

Cool! Such as?

I wrote,

>> Really? Wow. Did you come up with all that independently?

Monz wrote,

>Yep. I first started thinking about this kind of thing about
>11 years ago, long before I 'discovered' prime-factorization of
>ratios and lattice diagrams.

>I was drawing diagrams that were altered versions of Ellis's
>Harmonic Cell and Harmonic Decad [Helmholtz, p 458-459], which
>placed Ellis's ratios in a scheme somewhat like Partch's Tonality
>Diamond, and then I outlined the harmonic cells in the decad, and
>realized that doing this in real-time with computer software would
>be a great way to study harmonic movement.

So you came up with the tetrahedral-octahedral lattice around this time too?
I thought you went to parallelogram lattices before or at the time you went
beyond 5-limit.

[Paul Erlich, TD 221.8]
>
> [me, monz]
>> [11 years ago] I was drawing diagrams that were altered
>> versions of Ellis's Harmonic Cell and Harmonic Decad
>> [Helmholtz, p 458-459], which placed Ellis's ratios in a
>> scheme somewhat like Partch's Tonality Diamond, and then
>> I outlined the harmonic cells in the decad, and realized
>> that doing this in real-time with computer software would
>> be a great way to study harmonic movement.
>
> [Paul]
> So you came up with the tetrahedral-octahedral lattice around
> this time too? I thought you went to parallelogram lattices
> before or at the time you went beyond 5-limit.

The original drawings I made of Ellis's 5-limit systems were
tetrahedral, akin to Partch's Diamonds.

At the time, I was interested in constructing a more-or-less-fixed
19-limit system, and the problem I was having was where to put
the 7-, 11-, 13-, 17-, and 19-limit ratios.

I could see that it was fairly easy to incorporate 7, because it
gave a 3-dimensional layout similar to Fokker's (with which I was
unfamiliar at the time). But including 11 and the rest was
perplexing.

It was when I independently stumbled upon the realization that
Ellis's Duodenarium illustrated prime-factorization (of 3 and 5)
that I began to prefer the simplicity and logic of the
parallelogram design.

I stayed with this design, using a 3x5 grid or matrix layout
(think of it as a plane), which could be reproduced like rungs
of a ladder for all the higher primes, until a year ago.

Even tho I used it for a few years, I still wasn't entirely
pleased with this type of design, because it was difficult to
portray prime-factors higher than 5 which were raised to powers
higher than +/- 1.

In May of 1998, when I 'broke thru the lattice barrier', I
realized that giving each vector a unique angle would allow me
to incorporate any prime at any power. Of course, this design
had been used and published, in a somewhat less complex fashion,
by Erv Wilson and John Chalmers as long ago as the late 1960s
(see the Wilson Archives at www.anaphoria.com), and has already
been incorporated in software on the Mac - I believe it's
JICalc (? - I'm a PC user) - but I wasn't aware of any of this
until late last year, after I had thought of it independently.

As I've said here before, I don't have any serious reservations
against using tetrahedral lattices, and the reason I generally
don't show the triangular connections on my lattices is simply
because they clutter it up too much.

The reason I prefer the parallelogram design is because I find
it just a tad bit more logical and simple than the tetradhedral,
altho it has its drawbacks too, as Paul has pointed out. Chief
among them is the implications of sonance or complexity given
by the spatial distance between certain ratio points.

The beautiful thing about the JustMusic software which Ken Fasano
and I are developing is that it will allow the user to change
the view of a scale or system to a number of different lattice
designs, so that the unique information provided by all the
different designs can be viewed at once and compared.

The two designs already implemented are my 'Monzo' lattice
design, with which most of you should be familiar and which
can be seen on my website, and my 'Planetary' graph, which
places the 'octave'-reduced ratios around a circle representing
the 'octave'. Thus I can see the prime-factor layout with
the 'Monzo' lattice, as well as the logarithmic intervallic
distance between the pitches with the 'Planetary' graph.

And it can read any rational Scala file. :)
(and eventually, the irrational ones too)

And of course the most powerful aspect of it will be sequencer
(i.e., real-time) capability.

-monz

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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Monz wrote,

>>> [11 years ago] I was drawing diagrams that were altered
>>> versions of Ellis's Harmonic Cell and Harmonic Decad
>>> [Helmholtz, p 458-459], which placed Ellis's ratios in a
>>> scheme somewhat like Partch's Tonality Diamond, and then
>>> I outlined the harmonic cells in the decad, and realized
>>> that doing this in real-time with computer software would
>>> be a great way to study harmonic movement.
>
>> [Paul]
>> So you came up with the tetrahedral-octahedral lattice around
>> this time too? I thought you went to parallelogram lattices
>> before or at the time you went beyond 5-limit.

>The original drawings I made of Ellis's 5-limit systems were
>tetrahedral, akin to Partch's Diamonds.

So it's not the same as my idea. I depict 5-limit systems in two dimensions
only, obtaining tetrahedra only at the 7-limit. So where did your tetrahedra
come from?

[Paul Erlich, TD 223.17]
>
> [me, monz]
>> The original drawings I made of Ellis's 5-limit systems were
>> tetrahedral, akin to Partch's Diamonds.
>
> [Paul]
> So it's not the same as my idea. I depict 5-limit systems in
> two dimensions only, obtaining tetrahedra only at the 7-limit.
> So where did your tetrahedra come from?

Oops!... I should have said that the 5-limit diagrams were
triangular. My design, like yours, became tetrahedral when
I added 7-limit limit ratios. The main reason I abandoned
it was because of the difficulty/clutter of representing
7^2 or 7^-2, or other higher exponents of 7, or other higher
prime-factors.

Also, as I said, the parallelogram layout has a logical
simplicity that really appeals to me, and that's a bit more
complicated in the tetrahedral layout.

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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