Dave,

I find your ideas on "musical complexity"

very interesting, but I don't understand it all.

The significant thing I get out of it is that you're

looking for one operation to replace the dichotomy

of prime-vs-odd.

What *exactly* is the cost factor measuring?

Why is it significant that the natural logarithm

is involved? (Forgive my mathematical ignorance)

Why do you choose to call this operation

"musical complexity"? What exactly are your

spreadsheet charts explaining?

Given the significance with which I regard

prime-factor musical notation, I'd really like to

get more out of what you're doing here - I think

it may be pretty important.

- Monzo

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I (Dave Keenan) wrote,

>>Surely someone has proposed something like this before?

"Paul H. Erlich" <PErlich@Acadian-Asset.com> replied:

>You betcha! That's Barlow's Indigestibility Function! You can read

>about it in Georg Hajdu's paper, to which you provide a link from your

>own web page. Hajdu doesn't go into the justification for Barlow's

>function, and I find odd-limit to be much more reliable.

Thanks for that. Barlow/Shepard's Indigestibility does not allow a choice

of weights for the different prime exponents (my k_p's). It just says

k_p = (p-1)^2/p

note that this is roughly proportional to p like my proposed approximation

to prime limit. So it's another logarithmic complexity measure. And indeed

Barlows Indigestibility is almost a constant times the prime limit. It

seems to rank the numbers in the same order anyway. No wonder you don't

like it Paul.

So Barlow's is yet another that I can approximate by a choice of weights.

>([your web page] seems inaccesible these days -- is it offline?)

Shouldn't be. I moved some stuff into a subdirectory (but I thought I left

a redirection in place). What URL did you use? You can get to my equal

temperaments article now from http://dkeenan.com/Music/

George Hadju's paper is at

http://www.uni-muenster.de/Musikhochschule/Dozenten/Hajdu/Articles/LowEnergy

.pdf

>>Maybe even determined the factors by experiment?

>

>Why don't you propose one?

I guess the main thing would be to ask many people to rank various

intervals by perceived consonance. Surely this has been done. Can someone

report some results? I bet there's no real consensus.

>Joseph L Monzo <monz@juno.com> wrote:

>I find your ideas on "musical complexity"

>very interesting, but I don't understand it all.

>The significant thing I get out of it is that you're

>looking for one operation to replace the dichotomy

>of prime-vs-odd.

Yes. And also integer limit and now Barlow's Indigestibility.

>What *exactly* is the cost factor measuring?

Excellent question. I've switched to referring to them as "prime exponent

weights" now. I suppose they represent something about how the human brain

processes combinations of tones. The relative importance human evolution

gas given to the various primes. I don't see why we should expect them to

be any simple smooth function of the primes themselves as in Barlow's

indigestibility or my approximation to prime-limits. But nor would I expect

them to be equal (as in integer limit) or leap suddenly from zero to 1

between 2 and 3 (as in odd limit). I *would* however expect it never to

decrease with increasing prime (as with all these proposals).

>Why is it significant that the natural logarithm

>is involved? (Forgive my mathematical ignorance)

Not significant. Any log base would do, and then change a multiplicative

factor. But note that although there is no simple function to generate

primes, their distribution is approximately logarithmic. The higher you go,

the further apart they are. The nth prime is somewhere around e^n (I think).

>Why do you choose to call this operation

>"musical complexity"?

Well it just seemed like a kind of complexity that relates to musical

consonnance. But indigestibility is good too. I couldn't think of anything

else at the time that hadn't already been used (or has it?). Give me a

better term.

>What exactly are your

>spreadsheet charts explaining?

It was intended to let you try different weights ("cost factors") for the

prime exponents and then look at the chart to see it it ranks various numbers.

9 seems to be the main bone of contention. Integer and odd limits rank it

as more complex than 7. They give too much weight to 3. Prime limit and

Barlow's indigestibility rank it as less complex than 5. They give 3 too

little weight. I personally would like to put 9's complexity *between*

those of 5 and 7. What do you think? That's why I chose my weights as I did

(0.3, 0.8, 0.9, 1.0, 1.0, ...).

I've updated the spreadsheet to include Barlows indigestibility for

comparison. And to show both linear complexity (for comparison to integer

and odd) and log complexity (for comparison to prime and Barlow).

http://dkeenan.com/Music/MusicalComplexity.xls

Regards,

-- Dave Keenan

http://dkeenan.com

[Monzo:]

> What *exactly* is the cost factor measuring?

[Keenan:]

> Excellent question. I've switched to referring

> to them as "prime exponent weights" now.

> I suppose they represent something about

> how the human brain processes combinations

> of tones. The relative importance human evolution

> has given to the various primes. I don't see why

> we should expect them to be any simple smooth

> function of the primes themselves as in Barlow's

> indigestibility or my approximation to prime-limits.

> But nor would I expect them to be equal (as in

> integer limit) or leap suddenly from zero to 1

> between 2 and 3 (as in odd limit). I *would*

> however expect it never to decrease with increasing

> prime (as with all these proposals).

This is very interesting to me. I've already concluded

regarding the study of meters that we break all meters

down into simpler and simpler subdivisions, until

ultimately everything can be expressed by combinations

of 2s and 3s.

A simple example: 4/4 is a group of two 2s:

4 = [ 2 x 2].

A complex example: I wrote a tune in 14/8

subdivided as follows: 14 = 2 x 7. The first

group of 7 is subdivided into 4 [= 2 x 2] + 3,

the second into 3 + [2 x 2], forming a

symmetrical pattern of accents for each bar.

I've long thought that probably something

like this was at work in our harmonic perception

of pitch. I allude to it in the last chapter of

my book, but was afraid to say anything really

detailed about it, as I had no way to quantify it.

With your original approximated weights

(0.3, 0.8, 0.9, 1.0, 1.0, ...), which I thought were

OK at first, it's evident that there is a sharp

increase in complexity after 2. Perhaps with

a different weighting your formula is a mathematical

validation and explanation of my idea.

I'm certain that our difficulty of understanding

more complex meters as anything but 2s and 3s

has a lot to do with "The relative importance human

evolution has given to the various primes". We

can comprehend 1, 2 or 3 of anything *right away*,

and from my knowledge of how evolution works,

speed of recognition or comprehension ranks

near the top of the list of importance.

>> Why do you choose to call this operation

>> "musical complexity"?

> Well it just seemed like a kind of complexity that

> relates to musical consonnance. But indigestibility

> is good too. I couldn't think of anything else at the

> time that hadn't already been used (or has it?).

> Give me a better term.

"Prime importance"?

> 9 seems to be the main bone of contention.

> Integer and odd limits rank it as more complex

> than 7. They give too much weight to 3. Prime

> limit and Barlow's indigestibility rank it as less

> complex than 5. They give 3 too little weight.

> I personally would like to put 9's complexity

> *between* those of 5 and 7. What do you think?

> That's why I chose my weights as I did

> (0.3, 0.8, 0.9, 1.0, 1.0, ...).

It's always been my feeling that the main problem

with my prime-limit idea was that 9 should be

ranked higher in some way than 3. And after

trying a few other possibilities in your spreadsheet,

I'd thought that you gave too much emphasis to 2.

I gave it a weight of as low as .05 before it looked

like something I agreed with, but the weight of 3

was much lower also - I don't remember what now.

As I said above, my inclination would be to have

low weights for 2 and 3 and then increase sharply

after that. What do you think? How about the

question of 15? I've always thought that 15/8

is a pretty consonance in a chord - is it more,

or less, consonant than 7/4?

- Monzo

|\=/|.-"""-. Joseph L. Monzo......................monz@juno.com

/6 6\ \ http://www.ixpres.com/interval/monzo/homepage.html

=\_Y_/= (_ ;\ c/o Sonic Arts, PO Box 620027, San Diego, CA, USA

_U//_/-/__/// || "The ability of the human ear is ||

/monz\ ((jgs; || vastly underestimated" - Harry Partch ||

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Joe Monzo wrote,

>How about the

>question of 15? I've always thought that 15/8

>is a pretty consonance in a chord - is it more,

>or less, consonant than 7/4?

The phrase "in a chord" makes all the difference here. Presumably Joe

means "on top of a major triad". Looking at the odd limits, a 15/8 forms

a 3-limit interval, a 5-limit interval, and a 15-limit interval with

notes of the major triad. A 7/4 forms a 7-limit interval, a 7-limit

interval, and a 7-limit interval. So on the basis of odd limit alone,

you could argue that a dominant seventh chord could "resolve" to a major

seventh chord because two intervals become more consonant while one

becomes more dissonant.

This is an example of what I was talking to Dante Rosati about -- people

think about prime limit because they're tacitly assuming the presence of

other notes. When you really break it down, odd limit works better.

A 15/8 alone vs. a 7/4 alone? You better believe the 7/4 is more

consonant!

I must agree 7/4 is like buttah

> A 15/8 alone vs. a 7/4 alone? You better believe the 7/4 is more

> consonant!

>

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> I must agree 7/4 is like buttah

Me too. It's one of the most intuitively obvious and easy-to-hear

non-12TET intervals out there.