# Researchers find classical musical compositions adhere to power law



## doctorGwiz (Sep 25, 2011)

This article talks about a paper published in PNAS that found composers' music adheres to a specific rhythmic "pattern" that can be described mathematically. Unfortunately I couldn't find the article at the PNAS website (I have access to PNAS at school); it must not be published yet. Interestingly the researchers found that Beethoven was the most predictable while Mozart was the least predictable. At first I thought this blasphemy (if you haven't noticed I'm a fan of Beethoven) but the scientist in me wants to find reason in it. I don't have difficulty seeing Beethoven as predictable in some respects, but the most predictable?

The team analyzed about 2000 pieces of music from a variety of composers so it would be interesting to see how they chose what they did. The PhysOrg article says they sampled from a "wide variety" of composers. If we assume that to mean about 40 (just a guess), thats 50 pieces per composer, which I suppose could attempt to represent a composers entire body of work. I'm definitely going to look out for when it's published.


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## Manxfeeder (Oct 19, 2010)

I don't know what they're referring to specifically, but Beethoven did like the trochaic (long-short) rhythmic combination, and he used it different ways.


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## Amfibius (Jul 19, 2006)

I have a science background as well, but unfortunately I don't have the mathematics to understand what they are talking about. The scientist in me wonders if the study suffered from sample bias? How did they choose the compositions they used to represent each composer? Did they pick the pieces at random, or did they choose the most well known pieces?


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## doctorGwiz (Sep 25, 2011)

Amfibius said:


> I have a science background as well, but unfortunately I don't have the mathematics to understand what they are talking about. The scientist in me wonders if the study suffered from sample bias? How did they choose the compositions they used to represent each composer? Did they pick the pieces at random, or did they choose the most well known pieces?


Yeah, there are a lot of questions regarding the details of the study that could be resolved by reading the actual paper. I'm interested but skeptical.


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## kv466 (May 18, 2011)

I completely see the association and would be interested to have it shown in detail.


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## Ukko (Jun 4, 2010)

I have read that Mozart was the most likely of the big three to adhere to the Classical forms. Maybe the researchers were looking at something else?


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## mmsbls (Mar 6, 2011)

I could not find the paper but would love to read it. Apparently pitch, loudness, and rhythm all follow a similar mathematical law (1/f) in music. The details these researchers study would be impossible to "hear" when listening to works and certainly would not be obvious.


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## Polednice (Sep 13, 2009)

Could somebody explain to me what it actually means for these different musical attributes to follow the power law? Is it about how repetitive it is? Or about how predictable the next notes coming up are? I don't get it.


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## Polednice (Sep 13, 2009)

Btw guys, I've found and downloaded the paper, so if anyone wants a copy, just let me know (can I send it somehow on here? If not, I don't mind emailing). Alternatively, if you've got institutional access to the journal, this is the link: CLICK ME.


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## Polednice (Sep 13, 2009)

I've read the article now and am none the wiser because it assumes significant prior understanding which I don't have. Could someone explain to me what a power law in terms of rhythm means? I've tried looking up the significance of power laws elsewhere, but the information is all so vague. Is the suggestion that the length of a rhythm corresponds either positively or negatively with the amount it occurs? Because either correlation seems silly to me.


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## Lunasong (Mar 15, 2011)

I think the comments posted below the story are rather telling. They are from individuals who seem to know what they are talking about, and most are crying bunk.

My favorite:


> ...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!


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## brianwalker (Dec 9, 2011)

I read the paper. 

"Symphony" is "most predictable". 
"Ragtime" is "least predictable". 

The "spectral range" for the "scherzo" is the longest. 

Well, now I know.

I wish they had a 1. a list of pieces analyzed and 2. divide composers into different periods and 3. have more composers. I want to see how Stravinsky and Wagner and Mahler and Ravel and Debussy fared. 4. I'd like to see a similar thing done for conductors.


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## Couchie (Dec 9, 2010)

So this is how mathematicians spend their time when they give up on real problems.


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## Sid James (Feb 7, 2009)

The whole thing is above my head, but with regards to this -



doctorGwiz said:


> ...Interestingly the researchers found that Beethoven was the most predictable while Mozart was the least predictable...


...i'll make the general comment that I can understand that conclusion in terms of just my listening to the music of these two composers. Eg. listening to Mozart's string quartets, he goes off in so many different directions, kind of spontaneously. Sometimes you think he will do something, but he interrupts it with something else and only then comes back to what he was doing.

Of course, Beethoven does this too, but in Mozart it comes across for me as more pronounced. This "surprise" element of Mozart was also discussed at a pre-concert talk I attended a while back, by a professor of music, no less.

Of course, going beyond these composers, Haydn's _Surprise_ symphony kind of relates to this too, I think.

BTW, Beethoven is a big favourite of mine as well, in terms of listening I listen more often to him than Mozart...


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## doctorGwiz (Sep 25, 2011)

Here is the abstract:

Much of our enjoyment of music comes from its balance of
predictability and surprise. Musical pitch fluctuations follow a 1/f
power law that precisely achieves this balance. Musical rhythms,
especially those of Western classical music, are considered highly
regular and predictable, and this predictability has been hypothesized
to underlie rhythm’s contribution to our enjoyment of music.
Are musical rhythms indeed entirely predictable and how do they
vary with genre and composer? To answer this question, we analyzed
the rhythm spectra of 1,788 movements from 558 compositions
of Western classical music. We found that an overwhelming
majority of rhythms obeyed a 1/f β power law across 16 subgenres
and 40 composers, with β ranging from ∼0.5–1. Notably, classical
composers, whose compositions are known to exhibit nearly identical
1/f pitch spectra, demonstrated distinctive 1/f rhythm spectra:
Beethoven’s rhythms were among the most predictable, and
Mozart’s among the least. Our finding of the ubiquity of 1/f
rhythm spectra in compositions spanning nearly four centuries
demonstrates that, as with musical pitch, musical rhythms also
exhibit a balance of predictability and surprise that could contribute
in a fundamental way to our aesthetic experience of music.
Although music compositions are intended to be performed, the
fact that the notated rhythms follow a 1/f spectrum indicates that
such structure is no mere artifact of performance or perception,
but rather, exists within the written composition before the music
is performed. Furthermore, composers systematically manipulate
(consciously or otherwise) the predictability in 1/f rhythms to give
their compositions unique identities.


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## doctorGwiz (Sep 25, 2011)

Sid James said:


> BTW, Beethoven is a big favourite of mine as well, in terms of listening I listen more often to him than Mozart...


Agreed, although I've been listening to the Great Mass in C minor a lot recently and it's brilliant.


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## doctorGwiz (Sep 25, 2011)

Couchie said:


> So this is how mathematicians spend their time when they give up on real problems.


Scientific discoveries are made not only through searching for answers to real-world problems, but also through curiosity. Did 17th century science need to know why the orbits of the planets were elliptical? No, but Newton had to invent calculus to figure out why. While I think that, for instance, cancer and HIV/AIDS research or alternative energy research is more important, I fully support curiosity-based science. 





BTW Neil deGrasse Tyson is the man.
And BTW you totally just googled "problems in mathematics." It's the 4th result.


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## Polednice (Sep 13, 2009)

OK, I think I've figured out what they're saying. Their thesis is that the frequency of rhythms in a piece of music conform to the formula 1/n x f^β, where 'n' is the rank of the rhythm in commonality, 'β' is between 0.48 and 1, and 'f' is the frequency of the most common note. A concrete example:

Let's say we have a piece where the most common rhythm is a crotchet. Throughout the piece, there are 100 distinct rhythms, and 40 of them are crotchets, therefore the frequency of crotchets is 0.4. The next most common note in the piece is a quaver, and, according to the formula, you can predict how many times a quaver should occur, as n = 2 (the note is 2nd most common), f = 0.4 (as established by the most common note, a crotchet), and β is there pretty much to allow for a little variation, so, for the sake of easy argument, we'll call it 1 for now.

Therefore, the frequency of a quaver ought to be 1/2 x 0.4^1 (or, in other words, half of 0.4), which is 0.2. Quavers ought to make up 20 out of the 100 notes. The 3rd most common rhythm, whatever it is, would then be 1/3 x 0.4^1 = 0.1333... Etc.

This all seems very interesting, but where it gets less interesting for me is in the arbitrary application of β. If β was always 1, then I'd be impressed. Instead, β allows a degree of variation. In this example, where crotchets occur with f = 0.4, let's take the lower boundary of β instead, which is 0.48. Then the frequency of quavers (2nd most common) ought to be 1/2 x 0.4^0.48 = 0.32. In other words, the formula that they have devised allows quavers to occur in our hypothetical piece anywhere between 20 and 32 times, which allows a variation of 12%. Actual pieces of music will obviously have a far greater number of rhythms, and, therefore, much more room for variation. For me, the amount of variation allowed makes the finding pretty pointless, but maybe I've got all this maths wrong!


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## poconoron (Oct 26, 2011)

An interesting aspect of this Study is that apparently the "resulting differences in rhythmic predictability" exhibited by various composers might permit the identification of works which are questionable as to authorship. Perhaps a "signature" of sorts?


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## doctorGwiz (Sep 25, 2011)

poconoron said:


> An interesting aspect of this Study is that apparently the "resulting differences in rhythmic predictability" exhibited by various composers might permit the identification of works which are questionable as to authorship. Perhaps a "signature" of sorts?


Yeah I hadn't thought of that. It would perhaps be useful for composers such as Beethoven, Vivaldi, Bach, Haydn, and Mozart who had relatively narrow spectral ranges, but would likely be less useful for say Schubert, Scriabin, or Buxtehude, who had wider ranges. Since this study focused only on rhythm specra, frequency specra data could also be incorporated to increase identification capabilities.


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## poconoron (Oct 26, 2011)

Another article on this:

http://www.canada.com/entertainment...n+used+classical+composers/6187887/story.html


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## Vaneyes (May 11, 2010)

Another "least predictable' found...

http://www.bbc.co.uk/news/entertainment-arts-17490070


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