Researchers find classical musical compositions adhere to power law

Credit: Wikipedia.

( -- 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 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 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 from a wide group of noted composers. In so doing, they found that virtually every piece studied conformed to the . 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. 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.

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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.

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Citation: Researchers find classical musical compositions adhere to power law (2012, February 21) retrieved 21 July 2019 from
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Feb 21, 2012
In general, research discovering power laws in everything has become ubiquitous enough that the existence of a 'power law' no longer seems an indicator of any sort of special relationship or causality; one suspect it rather has more to do with the mathematics involved than the subject matter.

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
It's called style. I am a musician and an arranger/composer. We all have our "resonant frequencies". I arrange music for every style includng jazz, blues, rock & roll, C&W. I even a piece for classical voice. The key to productive songwriting is to first master the instrument. Doing so takes the composer on a course of discovery and learning that is an extension of the root, the place from where he begins to formulate arrangements. From this is developed a musical style that is inherent to the character and tastes of the musician, and which root belongs to a unique, yet shared path for all future compositions. All my individua compositions are very different, yet a savvy analyst can identify every piece as coming from the work of a single composer. I understand that.

Feb 21, 2012
Shot noise obeys 1/f spectra, so why wouldn't rhythms? =D

@ 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
a clear candidate for non-science of the week, and more media grandstanding by Levitin (did anyone catch his "Your Brain on Beethoven" tour? No? Me neither) but what might be interesting is if their method had found fractal relations in the intentionally fractal-based works of Udo Kasemets, or if they found power law expressed by the later works of John Cage. As it stands, the quined use of "classical" also worries me, as if they are trying to say somehow that there's a 'better' foundation to the western european conservatory school, despite this supremist notion having been definitively trounced by Charles Burney and his son back in 1750.

Feb 21, 2012
Question for the author of this news item.
couldn't find the original article at
(not even a mention of the article)
you sure it is 'published' there?

Feb 21, 2012
What the author is describing isn't even a fractal.

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
The "power law fraud is a scam based on a mathematical trick.
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

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
...if they found power law expressed by the later works of John Cage...

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
What's the math look like predicting the probability of at least 65% of the comments attacking something related to the article itself?

Feb 21, 2012
Sorry to beat a dead horse, but there's more.

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
Lurker2358 comparing the graphs of 1/f and log f demonstrates they know so little of the subject they cannot necessarily be expected to make an informed comment.
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
As for the "beta" variable, note how vague they are describing what specifically it is. They give it daunting claims, that Lurker2358, seems to consider compelling, without Lurker2358 even knowing what it is, but they never define it. In terms of frequency analysis, there is a beta which is related to, basically, the inverse of the average power of the "power law". For a 1/f or f^(-1) distribution, beta is 1. For a completely random sequence, which has slope 0, the beta is -0 or 0. Since the "research" refers to beta being between about .5 and 1, it seems this is the definition of beta they were using. But note, by definition, in frequency analysis, of beta is any number other than 1, the distribution is not 1/f!

Feb 21, 2012
Levitin is such an ignorant grandstanding hack. He never mentions "Zipf's Law", which is what this 1/f distribution is. I presume that's because then people would be able to easily look up what's known about this law, and realize that under certain conditions it emerges from randomly constructed datasets and so it is probably meaningless in his study. My guess is he is as poor at stats as he is at understanding music. Damn good at getting himself in the media though.

Feb 21, 2012
Levitin is such an ignorant grandstanding hack. He never mentions "Zipf's Law", which is what this 1/f distribution is. I presume that's because then people would be able to easily look up what's known about this law, and realize that under certain conditions it emerges from randomly constructed datasets and so it is probably meaningless in his study. My guess is he is as poor at stats as he is at understanding music. Damn good at getting himself in the media though.

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.

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
they didn't massage the data, some things just automatically follow a power regression, or simple harmonic, or a very close approximation of it.

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
Lurker, I didn't say classical music was randomly constructed, I said that *even* a randomly constructed dataset, when interpreted the proper way, will obey Zipf's law. Look up Zipf's law. So, big freaking deal that this set of music obeyed it. By the way you mentioned the example of counting notes in a piece of music. Well the statistical distribution of notes in a given key is well-known (look up Krumhansl key profile). Do you seriously believe Levitin has discovered something worthy of this press attention, or are you just trolling?

Feb 21, 2012
Do you seriously believe Levitin has discovered something worthy of this press attention, or are you just trolling?

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
Some relationships are purely coincidental, and don't necessarily mean anything.

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
And for another example, Dark Energy and Red Shift obey power laws.

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
The cosmic microwave background has a redshift of z = 1089, corresponding to an age of approximately 379,000 years after the Big Bang and a current comoving distance of more than 46 billion light years

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
The present counting method for years use the Anno Mundi epoch (Latin for "in the year of the world", in Hebrew , "from the creation of the world"), abbreviated AM or A.M. and also referred to as the Hebrew era. Hebrew year 5771 (a leap year) began on 9 September 2010 and ended on 28 September 2011. Hebrew year 5772 began at sunset on 28 September 2011 and will end on 16 September 2012.

5771.5(Jewish years) = (1/3)*c^(1/2)

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
This means that the speed of light in m/s is EXACTLY the square of thrice Jewish time in years.

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
Anyway, I made my point.

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
Lurker2358's lunatic japery is only spitting in everyone else's face, and a sign of what contempt Lurker23258 has for everyone.
It can be said that, compared to mechanical repetitveness, creativity is random.

Feb 22, 2012
the classical period is predictable, because the composers were trying to write as natural as possible.

Feb 22, 2012
My math skills are poor, so i dont fully get it, but does this mean that they can do more to help me find music i like? I am so sick of the music i have, but it is not a priority to find more music. i would rather something else did the work for me so i can get back to work.

Feb 22, 2012
My math skills are poor, so i dont fully get it, but does this mean that they can do more to help me find music i like? I am so sick of the music i have, but it is not a priority to find more music. i would rather something else did the work for me so i can get back to work.

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
Let's see:


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...


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
Strangely, when I looked at the last 3 terms shown in the sequence, then the 3 most common digits, "1,3,9," obey both the harmonic law mentioned in this article, AND the "Log(1 plus 1/n)" law to the nearest digit.

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.

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