[tuning-math] Monzo theory of sonance (was: unifying theory of interval "importance")

> ----- Original Message -----
> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, June 18, 2001 2:48 PM
> Subject: [tuning-math] unifying theory of interval "importance"
>

> Hey Monz . . . at one point, you showed me a section in your book
> where you equated dissonance with the area of a rectangle formed by
> the numerator on one side and the denominator on the other side. This
> area, of course, equals n*d. I think Dave, Graham, and I (and
> Benedetti, Tenney, Pierre Lamothe, and others) have settled on n*d or
> a monotonic function thereof as a measure of "complexity" -- which is
> one of the factors determining dissonance, along with "tolerance"
> and "span". So it seems there was something to the insight behind
> your "rectangle" construction. Care to fill us in on how you came up
> with that construction?

Sure thing. I'll give a brief overview first, then simply quote
an exchange between myself and Dave Keenan from 2 years ago (at the
height of the discussion on harmonic complexity) which explains it
in good detail.

Basically, I was trying to illustrate a foundation of my theory
of sonance, which I first found clearly explained by Ben Johnston,
but which actually AFAIK dates back to Euler. This is: that
absolute (or maximum) consonance is expressible, in prime-factor
terms, as n^0 (the 1:1 ratio) and dissonance increases directly as
*both* the number and size of the prime-factors and the absolute value
of the exponents get larger. The larger the area covered by the
graph - in all dimensions - the greater the dissonance of the interval.

If I had a good understanding of trigonometry, I'd probably
modify this now to reflect my (or any other) lattice formula.
Any ideas on that, Paul?

-monz

------- first explanatory email to Dave Keenan ---------

From: monz@juno.com
To: d.keenan@uq.net.au
Date: Thu, 8 Apr 1999 12:52:23 -0400
Subject: my sonance theory

Hey Dave,

I was just thinking about my sonance theory
and how it may be another type of complexity measure
you haven't considered yet.

You probably have - I just haven't yet grasped
(or spent the time trying to grasp) all the info that's
on your spreadsheet. (Plus I have to play with
the charts in some graphics program to reduce
their size so I can print them.)

This is it, in case you haven't covered it yet:

On plain square graph-paper, I drew a line representing
the cents value of the ratio, then a box the size of 1 cell
for each exponent in the numerator and denominator
on either side of the line respectively.

Boxes would be arranged according to size of prime-bases
and how exponents were stacked, or how prime-factors
combined in composites.

For example, 15:8 factors to 2^-3 * 3^1 * 5^1.
5 is the highest prime factor, so the length of the
line representing the note's pitch will be 5 units long
along the horizontal (x-) axis. The numerator factors
into 3 times 5, so there 3 groups of 5 boxes, stacked
3 high, above the line. Since the denominator factors
ultimately to 1 for octave-equivalent music-theory purposes,
the denominator is represented by only 1 box under the line.

The total number of boxes, and the length of both the
line horizontal and the height of the stacked boxes vertically,
describes the relative sonance (consonance/dissonance)
of the dyad.

Does it sound like something already discussed?

You're welcome to respond to this publicly on the List
if it merits further discussion.

-monz

------- Dave's response to my first explanatory email ----

From: Dave Keenan <d.keenan@uq.net.au>
To: monz@juno.com
Date: Fri, 09 Apr 1999 10:10:35 +1000
Subject: Re: my sonance theory

No, but your above description is really too ambiguous/contradictory and
vague for me to be sure. It sounds a bit like an octave-reduced n+d which I
would think would be useless.

It's octave equivalent so I assume we can cast out any 2's before we start
and there would always be at least one box on each side of the line. I'm
confused about what happens with odd-prime exponents greater than 1. What
does 9/8 look like, or 45/32?

How do "The total number of boxes, and the length of both the
line horizontal and the height of the stacked boxes vertically" go together
to produce a single complexity (= dissonance for simple ratios) number.

At first you say the (horiz?) line represents the cents value (how?) but in
the example the line length is given by the highest prime factor.

Yours confusedly,
-- Dave Keenan

------- second explanatory email to Dave Keenan ---------

From: monz@juno.com
To: d.keenan@uq.net.au
Date: Thu, 8 Apr 1999 22:21:48 -0400
Subject: Re: my sonance theory

Sorry about not being clear.

The cents value is represented along the y-axis
(vertical), but in truth, it's more or less
irrelevant to the sonance issue as being described.

I used the cents value only to have a value along
which to plot the different ratios and know how to
recognize them (by the pitch-height).

I have no idea how to tell you to get one complexity
number out of the two or three different measurements
I described. That's why I use a graph :)
and asked you to do the numbers :)

On the graph, no matter how complex the formula is,
you can see at a glance exactly how much 'space'
the ratio takes up (on paper), and how it's arranged,
which I think also carries important harmonic
or sonance information.

One thing I forgot to mention: with this kind
of plotting, all complementary intervals have
an equal 'value'.

It's a graph made of all squares, so compared to
some of the ASCII lattices I've drawn, it's a piece
of cake. Here are some examples, we'll just
disregard the whole issue of cents value:
_
|_| 1:1 and 2:1
|_|

_ _ _
|_|_|_| 3:2 (invert for 4:3)
|_|

_ _ _ _ _
|_|_|_|_|_| 5:4 (invert for 8:5)
|_|

_ _ _
|_|_|_|_ _ 6:5 (invert for 5:3)
|_|_|_|_|_|

_ _ _ _ _ _ _
|_|_|_|_|_|_|_| 7:4 (inv. 8:7)
|_|

_ _ _ _ _ _ _
|_|_|_|_|_|_|_| 7:6 (inv. 12:7)
|_|_|_|

_ _ _
|_|_|_|
|_|_|_|
|_|_|_| 9:8 (inv. 16:9)
|_|

_ _ _ _ _
|_|_|_|_|_| 10:9 (inv. 9:5)
|_|_|_|
|_|_|_|
|_|_|_|

I think this visual model represents well what
I've read of the most commonly-accepted sonance
theories (Helmholtz, Ellis, Ben Johnston, Partch,
lots of people), and what my own ears inform me of.

I was trying to find a way to model how both
increasing size of prime-base *and* increasing size
of exponents leads to increasing dissonance,
but still take account of both the uniqueness
of the primes themselves, and the multi-dimensionality
of their combination.

So now we're up to a point where several
different ratios can have the same 'box count' value,
thus the same sonance
(= level of relative consonance/dissonance).

For example, 7:6 and 9:8 both have a total
of 10 boxes. So I would say that they have
approximately the same consonance or dissonance,
but with two different *qualities*, described
by the distibution of primes and exponents
in the ratio, and by the different shapes
on the graph (even tho they take up the same
area or volume).

Now, ideally, this would be a 3-or-more-dimensional graph,
so that complicated multiples could be portrayed simply
by using a different dimension for each prime.

This would probably be kind of confusing for this type
of graph, but it's exactly what worked for me in my
lattice diagrams. Even plotting them 2-dimensionally,
using unique angles and vector-lengths for the different
primes gives the lattice a multi-dimensional aspect.

So anyway, to graph 45:32, it would be best to stack
a horizontal layer of 5 both 3 high and 3 deep.
To me, this is the best way to portray 5-limit
composite factoring.

I'll try my best to show you in ASCII what
it would look like, in a view from the top front:

_ _ _ _ _
/_/_/_/_/_/|
/_/_/_/_/_/|/ 45:32
/_/_/_/_/_/|//
|_|_|_|_|_|/// <-- just use your imagination
|_|_|_|_|_|// for this part
|_|_|_|_|_|/
|_|/ <-- that's just one tiny little
block on the bottom

You can easily see it as 15*3 or 9*5.
I think this 'double meaning' is what makes
the composite ratios so interesting, and
probably why they're important enough for
some people to like odd-limit.

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Basically, I was trying to illustrate a foundation of my theory
> of sonance, which I first found clearly explained by Ben Johnston,
> but which actually AFAIK dates back to Euler.

Benedetti comes closer and goes back even further.

> This is: that
> absolute (or maximum) consonance is expressible, in prime-factor
> terms, as n^0 (the 1:1 ratio) and dissonance increases directly as
> *both* the number and size of the prime-factors and the absolute
value
> of the exponents get larger. The larger the area covered by the
> graph - in all dimensions - the greater the dissonance of the
interval.
>
> If I had a good understanding of trigonometry, I'd probably
> modify this now to reflect my (or any other) lattice formula.
> Any ideas on that, Paul?

I don't think you should! I think n*d is a much better measure of
complexity than one you'd derive from your lattice formula. For
simple ratios, harmonic entropy is proportional to log(n*d). And log
(n*d) is known as Tenney's Harmonic Distance, since it's the city
block distance in his lattice. Thus I think his (octave-specific)
lattice is much better than yours for depicting dissonance.

> Since the denominator factors
> ultimately to 1 for octave-equivalent music-theory purposes,
> the denominator is represented by only 1 box under the line.

That's where I disagree with what you're doing. It's a very subtle
point, but even when you're dealing with octave-equivalent music
theory, you can't simply ignore the factors of 2 in this
calculation . . . they'll contribute to the dissonance of various
intervals in different amounts no matter which particular range of
sizes you're focusing on.
>
> _ _ _
> |_|_|_| 3:2 (invert for 4:3)
> |_|
>
> _ _ _ _ _
> |_|_|_|_|_| 5:4 (invert for 8:5)
> |_|
>
> _ _ _
> |_|_|_|_ _ 6:5 (invert for 5:3)
> |_|_|_|_|_|
>
>
> _ _ _ _ _ _ _
> |_|_|_|_|_|_|_| 7:4 (inv. 8:7)
> |_|
>

This is not at all what I remembered. I guess we can drop the whole
thing.

I did a search on this, and found this old posting, to which I
respond.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I don't think you should! I think n*d is a much better measure of
> complexity than one you'd derive from your lattice formula. For
> simple ratios, harmonic entropy is proportional to log(n*d). And log
> (n*d) is known as Tenney's Harmonic Distance, since it's the city
> block distance in his lattice. Thus I think his (octave-specific)
> lattice is much better than yours for depicting dissonance.

Let's see what happens if we try to make this work in a Euclidean
framework. In the 5-limit, we want the following distances from the
origin: d(2) = ln(2), d(3) = ln(3), d(5) = ln(5), d(3/2) = ln(6),
d(5/2) = ln(10) and d(5/3) = ln(15). The corresponding quadratic form
is

u^2 + v^2 + w^2 - 2uv - 2uw - 2vw,

where u = ln(2)x, v = ln(3)y, w = ln(5)z.

The matrix for the corresponding bilinear form is

[ 1 -1 -1]
[-1 1 -1]
[-1 -1 1],

which is not positive definite, having eigenvalues of -1, 2, and 2.
This is therefore a Lorentzian metric, like the geometry of space-
time, which does seem a little goofy--should the consonance of 30
really be imaginary? You can pick eigenvalue coordinates, and
collapse the -1 part belonging to [1 1 1] out of the picture and get
something positive definite in two dimensions, but this collapses 30
down to 1, which doesn't seem any better.

On the other hand if you stick with the obvious, namely u^2+v^2+w^2
then you get 5/3 the same size as 15, which is what the taxicab
metric gave you, and a measurement for consonance which seems
generally in line with what I think you want:

d(2) = ln(2), d(3) = ln(3), d(5) = ln(5),
d(3/2) = sqrt(ln(2)^2 + ln(3)^2) = d(6),
d(5/2) = sqrt(ln(2)^2 + ln(5)^2) = d(10),
d(5/3) = sqrt(ln(2)^2 + ln(5)^2) = d(15).

Is there some reason not to use this?

--- In tuning-math@y..., genewardsmith@j... wrote:
> I did a search on this, and found this old posting, to which I
> respond.
>
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > I don't think you should! I think n*d is a much better measure of
> > complexity than one you'd derive from your lattice formula. For
> > simple ratios, harmonic entropy is proportional to log(n*d). And
log
> > (n*d) is known as Tenney's Harmonic Distance, since it's the city
> > block distance in his lattice. Thus I think his (octave-specific)
> > lattice is much better than yours for depicting dissonance.
>
> Let's see what happens if we try to make this work in a Euclidean
> framework. In the 5-limit, we want the following distances from the
> origin: d(2) = ln(2), d(3) = ln(3), d(5) = ln(5), d(3/2) = ln(6),
> d(5/2) = ln(10) and d(5/3) = ln(15). The corresponding quadratic
form
> is
>
> u^2 + v^2 + w^2 - 2uv - 2uw - 2vw,
>
> where u = ln(2)x, v = ln(3)y, w = ln(5)z.
>
> The matrix for the corresponding bilinear form is
>
> [ 1 -1 -1]
> [-1 1 -1]
> [-1 -1 1],
>
> which is not positive definite, having eigenvalues of -1, 2, and 2.
> This is therefore a Lorentzian metric, like the geometry of space-
> time, which does seem a little goofy--should the consonance of 30
> really be imaginary? You can pick eigenvalue coordinates, and
> collapse the -1 part belonging to [1 1 1] out of the picture and
get
> something positive definite in two dimensions, but this collapses
30
> down to 1, which doesn't seem any better.
>
> On the other hand if you stick with the obvious, namely u^2+v^2+w^2
> then you get 5/3 the same size as 15, which is what the taxicab
> metric gave you, and a measurement for consonance which seems
> generally in line with what I think you want:
>
> d(2) = ln(2), d(3) = ln(3), d(5) = ln(5),
> d(3/2) = sqrt(ln(2)^2 + ln(3)^2) = d(6),
> d(5/2) = sqrt(ln(2)^2 + ln(5)^2) = d(10),
> d(5/3) = sqrt(ln(2)^2 + ln(5)^2) = d(15).
>
> Is there some reason not to use this?

d(3/2) is between ln(3) and ln(4)! Not good.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> d(3/2) is between ln(3) and ln(4)! Not good.

You probably want to stick with your taxicab if you don't like this;
but remember that coordinate transformations transform taxicab spaces
to other taxicab spaces, you can't treat any of them like Euclidean
spaces.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > d(3/2) is between ln(3) and ln(4)! Not good.
>
> You probably want to stick with your taxicab if you don't like
this;
> but remember that coordinate transformations transform taxicab
spaces
> to other taxicab spaces, you can't treat any of them like Euclidean
> spaces.

You better believe it! So, any comments on the questions I asked?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> You better believe it! So, any comments on the questions I asked?

I'll look at it again, but I have some questions also:

(1) Can you define harmonic entropy in terms of your taxicab metric,
or if not in any terms you like?

(2) Do you know how to retune a midi file in such a way that the
pitches are set to anything you choose?

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > You better believe it! So, any comments on the questions I asked?
>
> I'll look at it again, but I have some questions also:
>
> (1) Can you define harmonic entropy in terms of your taxicab
metric,
> or if not in any terms you like?

Look over the harmonic entropy list (are you subscribed yet)? It's a
continuous function of interval size, has local minima at the simple
ratios, is conceptually very simple . . .
>
> (2) Do you know how to retune a midi file in such a way that the
> pitches are set to anything you choose?

Lots of people should be able to help you with this. Herman Miller?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Look over the harmonic entropy list (are you subscribed yet)? It's
a
> continuous function of interval size, has local minima at the
simple
> ratios, is conceptually very simple . . .

I looked at it; I think you need to upload something to the files
which defines what the group is about by defining harmonic entropy.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Look over the harmonic entropy list (are you subscribed yet)?
It's
> a
> > continuous function of interval size, has local minima at the
> simple
> > ratios, is conceptually very simple . . .
>
> I looked at it; I think you need to upload something to the files
> which defines what the group is about by defining harmonic entropy.

There are a couple of posts which tell you how to calculate harmonic
entropy . . . let me know if those are clear.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> There are a couple of posts which tell you how to calculate
harmonic
> entropy . . . let me know if those are clear.

Which posts?

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > There are a couple of posts which tell you how to calculate
> harmonic
> > entropy . . . let me know if those are clear.
>
> Which posts?

350 . . . follow the thread from there.

On Thu, 06 Sep 2001 20:46:10 -0000, "Paul Erlich" <paul@stretch-music.com>
wrote:

>> (2) Do you know how to retune a midi file in such a way that the
>> pitches are set to anything you choose?
>
>Lots of people should be able to help you with this. Herman Miller?

I used to do it by hand, setting the pitch bend in Cakewalk, but now I
mainly use Graham Breed's Midiconv program, which puts in the pitch bends
according to a scale file you can edit. With pitch bends you have to be
careful not to have overlapping notes, and I found while doing the Warped
Canon project that certain timbres respond to pitch bend messages even
after the note off, so I had to alternate channels. And sometimes, as in
"Transformation" for instance (17-TET), I'll use Midiconv to tune the 12
most common notes and then tune the others by hand.

You can get Midiconv from this link on Graham Breed's page:

http://x31eq.com/progs/Midiconv.zip

Midiconv can also split one channel into multiple channels to get around
the problem with overlapping notes.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

--- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:

> I used to do it by hand, setting the pitch bend in Cakewalk, but
now I
> mainly use Graham Breed's Midiconv program, which puts in the pitch
bends
> according to a scale file you can edit.

How do get Midiconv to input a midi file and output an ascii file of
pitch values which I can edit?

In-Reply-To: <9n9a0e+cngv@eGroups.com>
In article <9n9a0e+cngv@eGroups.com>, genewardsmith@juno.com () wrote:

> How do get Midiconv to input a midi file and output an ascii file of
> pitch values which I can edit?

You can't.