> ----- 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

_________________________________________________________

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--- 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.