Yahoo Tuning Groups Ultimate Backup tuning Dave Keenan's "musical complexity"

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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🔗Dave Keenan <d.keenan@xx.xxx.xxx>

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

🔗Joseph L Monzo <monz@xxxx.xxxx>

[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
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🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

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!

Dave Keenan's "musical complexity"

🔗Joseph L Monzo <monz@xxxx.xxxx>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗Joseph L Monzo <monz@xxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Patrick Pagano <ppagano@xxxxxxxxx.xxxx>

🔗Gary Morrison <mr88cet@xxxxx.xxxx>