Researchers find classical musical compositions adhere to power law
February 21, 2012 by Bob Yirka 
Image: Wikipedia.
(PhysOrg.com) -- A team of researchers, led by Daniel Levitin of McGill University, has found after analyzing over two thousand pieces of classical music that span four hundred years of history, that virtually all of them follow a one-over-f (1/f) power distribution equation. He and his team have published the results of their work in the Proceedings of the National Academy of Sciences.
One-over-f equations describe the relative frequency of things that happen over time and can be used to describe such naturally occurring events as annual river flooding or the beating of a human heart. They have been used to describe the way pitch is used in music as well, but until now, no one has thought to test the idea that they could be used to describe the rhythm of the music too.
To find out if this is the case, Levitin and his team analyzed (by measuring note length line by line) close to 2000 pieces of classical music from a wide group of noted composers. In so doing, they found that virtually every piece studied conformed to the power law. They also found that by adding another variable to the equation, called a beta, which was used to describe just how predictable a given piece was compared to other pieces, they could solve for beta and find a unique number of for each composer.
After looking at the results as a whole, they found that works written by some classical composers were far more predictable than others, and that certain genres in general were more predictable than others too. Beethoven was the most predictable of the group studied, while Mozart was the least of the bunch. And symphonies are generally far more predictable than Ragtimes with other types falling somewhere in-between. In solving for beta, the team discovered that they had inadvertently developed a means for calculating a composer’s unique individual rhythm signature. In speaking with the university news group at McGill, Levitin said, “this was one of the most unanticipated and exciting findings of our research.”
Another interesting aspect of the research is that because the patterns are based on the power law, the music the team studied shares the same sorts of patterns as fractals, i.e. elements in the rhythm that occur the second most often happen only half as often, the third, just a third as often and so forth. Thus, it’s not difficult to imagine music in a fractal patterns that are unique to individual composers.
More information: Musical rhythm spectra from Bach to Joplin obey a 1/f power law, by Daniel Levitin, Parag Chordia, and Vinod Menon, PNAS, 2012.
Journal reference:
Proceedings of the National Academy of Sciences
© 2011 PhysOrg.com
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Feb 21, 2012
Rank: 5 / 5 (4)
The last paragraph, however, is a classic example of utterly atrocious science writing. Bob Yirka, pay attemption, here. (1) power laws in the patterns do not mean the patterns are 'based' on power laws. This is a logical fallacy. Additionally, power laws are *statistical* relationships, and *could not* give rise to patterns. (2) Fractal patterns are *not* defined by their statistical relationships. Your explanation of what a fractal is is flat-out *wrong*. Fractal patterns are characterized by self-similarity across scales. This naturally leads to the existence of power laws relationships in fractals. (3) None of this even *hints* at fractal patterns unique to each composer's music.
Feb 21, 2012
Rank: 5 / 5 (3)
Feb 21, 2012
Rank: 5 / 5 (1)
@ Smellyhat: Indeed, statistician Shalizi has a paper where he makes a proper statistical test for power laws instead of eye balling fits, and finds that many of the claimed 'power laws' fits equally well or better with exponentials. The likelihood for this paper to use such a test remains nearly zero, unfortunately.
And the often contextual fractal woo that is a result of similar [self similar? sic] pattern search _is_ irritating.
Feb 21, 2012
Rank: 3 / 5 (2)
Feb 21, 2012
Rank: 5 / 5 (1)
couldn't find the original article at pnas.org
(not even a mention of the article)
you sure it is 'published' there?
Feb 21, 2012
Rank: 1 / 5 (1)
If there is a pattern, it's the Harmonic Sequence, which is not a fractal anyway. 1, 1/2, 1/3, 1/4,..., 1/n...
I'd be interested to see this study done on the music of Final Fantasy IV through VII and Chrono Trigger, and see if the "beta" variable can correctly discern the difference between the composers, including perhaps correctly identifying a few pieces which are not entirely original, but which are modifications of other works.
This "Beta" may or may not be a useful forensic tool for identifying counterfeit or plagiarized pieces in which a composer stole someone else's work.
If this unique mathematical signature appears in music, might a similar mathematical signature appear in poetry or prose?
Feb 21, 2012
Rank: 3 / 5 (2)
To drive a power law, you take the logaritm of values and see if they follow a straight line. But logarithms squash data into a small region. Log 10 =1, log 100 = 2, log 1000000 = 6. Any data can be fit in a small region with that!
Then researchers apply linear regression to calculate the line that best fits the data. Those who dont understand whats around them and the gullible think you can get a formula only if the data for a straight line! Thats not true! The formula gives the straight line that comes closest to all the data, even if they form a weird curve! In fact, a definition of random dats points can be that they have a linear regression with slope 0!
This is evidently just another scam by the liars who make up the science community today, but whom so many outside the science inner circle do not know are working to deceive them.
Feb 21, 2012
Rank: 3 / 5 (2)
Clearly you're wrong.
The claim about the "beta" variable being consistent from one artist to the next is specific enough to be repeatable and testable by independent groups both for all of the artists in this original study, as well as additional composers.
There are also many other ways to do regressions. only an idiot would try "Just" a simple linear regression, or "just" a simply power or logarithm.
Of course you can try fitting transformations of trigonometric functions to the data as well, as most real datasets in the real world involve sin or cosine waves anyway.
You're being a little bit ridiculous.
Eureqa could process these data in a few seconds and find all possible curve families to fit it to and rank them.
Also, what are you talking about with logarithms?
If you plot curves for 1/f and Log f they are completely, unmistakably different.
Feb 21, 2012
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Oof, that reminded me when I wrecked the family piano by putting steel nuts and other junk between the piano wires. Darn your temptations, John Cage!
Feb 21, 2012
Rank: 1 / 5 (1)
Feb 21, 2012
Rank: 1 / 5 (1)
They could not have massaged or smoothed the data very much at all in this study, else the distinction of the "Beta" signature would have been destroyed, and would not have been recoverable on a per-composer basis.
It's kind of like, if you turn on the "smooth" thing on Eureqa, it's going to find a good function fit for basic data sets, but it would clearly destroy the statistical significance of something like this "beta" variable.
Feb 21, 2012
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Describing in more detail, to derive a "power law" relation between an independent variable, x, and a dependent variable, y, is to take log y and log x, then graph them. If y = a(x^n), the log-log graph wiill be the straight line log y = log a n(log x), which would be Y = A nX on the log-log graph. Even if you put the initial x and y values into a piece of software, that doesn't mean it's not performing that calculation behind the scenes!
Yes, you can try any of a number of formulas, indeed, theory says you can always find a polynomial that fits the data exactly, but, if you're a "neurobiology" hack, you won't hnecessarily be aware of that and, scammers know that anything that fits too well many disbelieve and, if it's too complicated, many ignore it.
Feb 21, 2012
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Feb 21, 2012
Rank: 1 / 5 (1)
Feb 21, 2012
Rank: 1 / 5 (1)
Classical music is definitely not a randomly constructed data set, so you blew your own argument.
Upon further research into this subject, I found all of my intuitions to be correct, including about poetry and prose, although I hadn't previously studied that.
Julianpenrod:
I tend to forget mathematical formulas, so I re-derive them at a later time. I have always been a much more intuitive problem solver. My intuitions and basic statements were correct, though my terminology was not.
Feb 21, 2012
Rank: 1 / 5 (1)
It isn't even a matter of fancy mathematics, as you can count discrete objects, such as words in a book, or notes in a piece of music, and put them in order of most common to least common.
if you have data,
1 G 15
2 J 8
3 E 6
4 D 4
5 P 3
6 F 2
7 G 2
8 H 2
.
.
.
14 X 1
15 W 1
Then while this is definitely not perfect, it clearly follows the basic notion of a harmonic, so a regression is not crankery.
Now if the deviations were huge compared to what you'd expect them to be, then a regression might not be appropriate, or you might need to account for hidden variables due to a bad correlation.
Feb 21, 2012
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Feb 21, 2012
Rank: 1 / 5 (1)
Well, that remains to be seen, for example, it should be applied to many composers in different eras and different styles of music, including pop and country and themes from video games, etc, to see if they all obey this law or if it's a coincidence.
Then, if only some music obeys this law, you would need to come up with some test to see whether it's by coincidence, or whether there is something unique to the process of the composer's mind which produces this result.
I admit that there are many things in different data sets which appear to follow a harmonic or a power law simply due to an artifact in the numbering system, rather than any special relationship in the data.
i.e. the whole "Log(1 (1/n))" thing with digits in addresses is not special. It's an artifact of the minimum digits needed to service the population.
Feb 21, 2012
Rank: 1 / 5 (1)
Like for example, if I took the reciprocal of fibonacci numbers, 0,1,1,2,3...
Compare that to 1/f, then if you disregard 1/0, since it's undefined in both cases, then all fibonacci numbers' reciprocals appear in both the range and domain of 1/f.
If you plotted the points with respect to their position in the sequence, it would even be very similar to 1/f (disregarding the 0 and the first 1).
Does that mean anything special? Probably not.
Hey, maybe I'm wrong.
Proving the difference between a coincidence and a special relationship might not even be possible in some cases.
I don't think that's a contradiction. It rather points out a difficulty in defining what is coincidence and what is relationship when coincidence appears to be related, or relation appears to be coincidence.
Feb 21, 2012
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As you approach great distance then red shift becomes infinite, such that the wavelength of light is longer and longer, and wavelength is of course the reciprocal of frequency, i.e. 1/f.
Therefore, Dark Energy is related to the wavelength of light by a power rule.
Ironically, it's very Newtonian, when you think about it, particularly since an object can also be heavily red shifted or blue shifted due to local motion as well.
Lots of stuff obeys is 1/r^2
Wavelength of light obeys 1/r for cosmological red shift.
and so on.
Now why does harmonics appear in nature? Is it an artifact of our mathematics, or is there an underlying principle which produces harmonics in all aspects of physical laws?
Feb 21, 2012
Rank: 3 / 5 (2)
299,792.458^2 equal 89,875,517,873.7
46,000,000,000 * 2 = 92,000,000,000
Is that a coincidence?
The speed of light squared produces an integer equal to twice the radius of the universe in light years (being the diameter of a sphere expanding at the...speed of light...)
Which suggests that the speed of light and the age of the universe are related by a power rule, since this is statistically within margin of error of measurement.
Why should the age of the CMB equal half the square of the speed of light in km/s?
i.e. CMB age looks like this:
t = (1/2)c^2 (time is in years, may need a scalar constant here).
coincidence? I think not.
look where else we see this.
Ek = (1/2)mv^2
d = (1/2)at^2
e = mc^2
Strange how music is related to Dark Energy...somehow...
Feb 21, 2012
Rank: 3 / 5 (2)
5771.5(Jewish years) = (1/3)*c^(1/2)
or
299792458 = ~3^2 * 5771.5^2 = 17,314.52^2
c = 299,792,458m/s = speed of light
Watch THIS...
5772*365*24*3600 = 182,010,190,275s = t in seconds
299,792.458^2 = 179,751,035,747
(1/2) t = v^2, v ~ 301,670.5 (very close to light speed in km/s)
t = 2v^2
Which shows a very close power law relationship between the speed of light in km/s and the age of the universe in seconds, according to the Jewish calendar.
It also shows a statistically close power law relationship between the Jewish calendar in years and the speed of light.
Feb 22, 2012
Rank: 3 / 5 (2)
c = (3 * 5771.50527255yrs)^2 = 299792458m/s
or to use this year exactly:
c ~ = 9 * (5772yrs)^2 = (3 * 5772yrs)^2 = 299,843,856 m/s ~ =c
Is this also a coincidence?
You may find this to be an "odd" calculation since the units don't match up, but you could introduce a scalar constant multiplied by necessary units, (similar to the gravitational constant,) to fix that problem.
Feb 22, 2012
Rank: 3 / 5 (2)
You can't prove a lack of special relationship between numbers, but you can prove it correct theoretically, given the right data and evidence.
Feb 22, 2012
Rank: 5 / 5 (1)
It can be said that, compared to mechanical repetitveness, creativity is random.
Feb 22, 2012
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Feb 22, 2012
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Feb 22, 2012
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Feb 22, 2012
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All it shows is that good music happens to follow closely a power law relationship, but so does good poetry and even prose.
It means good music follows some hidden laws of syntax and grammar, even if the composer isn't consciously aware of that fact.
Do a little research on this. Organized systems of languages and numbering systems repeatedly produce similarly peculiar relationships.
Feb 25, 2012
Rank: 5 / 5 (1)
0
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
10946
17711
28657
46368
75025
So that's the first 26 Fibonacci numbers. I didn't want to go to the next one because the leading 1 would bias it.
Leading digits are close to prediction of "Log(1 plus 1/n)".
1 8 30.77%
2 5 19.23%
3 3 11.54%
4 2 7.69%
5 2
6 2
7 1
8 2
9 1
If you run enough digits it will ultimately approach the same average limits.
Now more interesting...
73 digits total.
1 13
5 9
7 9
3 7
4 7
6 7
8 7
2 6
9 4
0 4
1 is twice as common as the mean would be expected to be for a random sequence, and there are seven more "1's" in the next three terms...
121393
196418
317811
Just looking only at these
1 7
3 3 article predicted 7/2.
9 2 predicted 7/3, round down.
8 2 p. 7/4 1.75 r up
2 1 p. 7/5 1.4 r down
4 1 p 7/6 r. down
6 1 p. 7/7
7 1 p. 7/8 round up
The trend suggest Fibonacci numbers probably follow this rule over the long term.
see below.
Feb 25, 2012
Rank: 5 / 5 (1)
It seems that if you skip the first several terms, then a random sub-set of the Fibonacci numbers probably obeys a harmonic law nearly perfectly, both in leading digits and in total appearances of any one digit, assuming you take enough terms.