Thursday, May 03, 2012

Surprise me - But not too much

Flute recording. Source.
A decades old paper made me philosophical.

In 1975, Voss and Clarke, two physicist from Berkeley, studied noise in electronic devices. For the fun of it, they also analyzed the spectra of different types of music. They found that the fluctuations in loudness and pitch decrease with the inverse of the frequency; they have a 1/f spectrum. This finding was basically independent of the type of music; Western, Oriental, Blues, Jazz, Classic all showed the same pattern. Voss and Clarke’s musical power spectrum made it into Nature.

A Fourier-transformation of a 1/f power spectrum leads to a power-law decay in the autocorrelation time of the fluctuations, meaning there are correlations over all times, rather than over a characteristic decay time as is most often the case. Physicists like power-law autocorrelated fluctuations because systems show them on a critical point, ie if something cool is going on, you get a power law. The opposite is not necessarily true though; there are more mundane ways to get a power law, but that hasn’t deterred enthusiasts. The 1/f spectrum is scale-invariant, so it has – theoretically – no preferred time scale or frequency to it, as one might have expected to be present in music.

In the 70s and 80s everything power-law was chique, and not all of that power-law-finding was very meaningful. To some approximation, in some parameter range, everything is a power-law. If you put your data on a log-log plot, you can make a linear fit, at least over some range. Yet, strictly speaking nothing really is a power law. And of course music doesn’t really have a 1/f spectrum either. To begin with, because it doesn’t use the full frequency spectrum, most of which we couldn’t hear. Also, Mozart hasn’t been composing since the Big Bang.

Scale-invariance is a property also shared by fractals. When I first heard about Voss and Clarke’s study, I jokingly asked when we’ll be listening to fractal music. Needless to say, I learned that had been said and done when I was still wearing diapers. Google “fractal music” to see where this thought leads you.

I’m not sure what the power-law finding means for the origin of music, but intuitively what it means for what you hear is that music (at least the type we find appealing) lives on the edge between predictability and unpredictability. White noise has a constant spectral density and no autocorrelation. A random walk moving a melody along adjacent pitches has a strong correlation and a 1/f2 spectrum. Somewhere in between are Bach and Adele.

When you turn on the radio, you want to be surprised – but not too much. Popular music today follows quite simple recipes. In most cases, you’ll be able to sing along when the chorus repeats. If you’ve heard a song a dozen times it gets dull though – it’s become too predictable. Symphonies are more complex, but they all have recurring motives and variations around that.

However, the musical edge must have a finite width. For some purposes, music can be more predictable than for others. What amount of predictability we find appealing doesn’t only depend on the occasion, it is also individually different. If you spend a lot of time analyzing pop songs, I suspect what’s in the charts today will sound very repetitive to you, though for the casual listener it arguably has an appeal.

It is tempting to extrapolate this to areas where autocorrelations are less easily measureable than pitch, for example to ideas in the written and spoken form. A scientific paper or a talk needs to strike a balance between the known and the unknown. Repeat too much common knowledge, and you’re obvious. Jump too far, and you’re crazy. The scientific pop stars are the ones on the edge. That also means the pop stars are the ones not too far ahead of their time.

It seems to me today the width of the scientific edge is very thin, maybe too thin. Sometimes, the obvious must be stated just so it remains in awareness. And sometimes the crazy starts making sense if you’ve listened to it often enough.

There’s another lesson. While fashions seem to come back, the cycle never is perfectly periodic, but always comes with a new twist. Thus, when the colors of the 70s will return to haunt us, maybe they’ll come with a metallic shine. And so, my impression that we’re having the same discussions over and over again must be wrong. They can’t be periodic, I am missing a change on longer time scales. History may be self-similar, but it’s not repeating. Though that's one of my all-time favorite songs.

22 comments:

Peter said...

Love the babies, Love the Physics, Love the less namable this.

Henrique said...

"The past does not repeat itself, but it rhymes." - Mark Twain

Robert L. Oldershaw said...

Two quick comments.

1. A discussion of Benford's law would fit well with your theme. This law shows that "logarithmic counting" and power laws that are both scale invariant and base invariant are ubiquitous in nature and society. Wikipedia gives some good starting information and sources.

2. Broken scale invariance [aka discrete scale invariance] is worth a very close look and long pondering. I believe that this is nature's most common form of scale invariance.

Robert L. Oldershaw
Discrete Scale Relativity
Discrete Fractal Cosmology

Matti Pitkanen said...

I have sometimes wondered what p-adic generalization of music would sound. Replace octaves by powers of 3,5,7,.. for small enough primes, and hear what it sounds.

Probably cacophony since p=2 has the special feature that octaves are equivalent harmonically (Why?: a fascinating problem for consciousness theorist). Someone should bother to do a simple experiment by transforming some simple tune from p=2 to p=3 by replacing 2^(1/12) with 3^(1/12) as a basic step in 12-note scale and therefore note 2^(m/12) by 3^(m/12).

Bee said...

Hi Peter,

I file it under "random thoughts" :o) Glad you like it. Best,

B.

Bee said...

Hi Matti,

An interesting thought. It makes me wonder, if we'll ever find extraterrestrial life, will they like the same music as we do? Best,

B.

Zephir said...

/*..sometimes the crazy starts making sense if you’ve listened to it often enough..*/

Repetition is the mother of wisdom. This is why I'm repeating myself across whole web. The people need to hear the very same ideas in many connections, before they will become able to work with it in creative way. The creative way of thinking is emergent, i.e. you'll need to see it as a product of interference of many longitudinal waves coming from many directions at the same moment.

The fractal character of observable reality follows from the fact, we are little piece of random space-time geometry. So we are perceive it random, but due the anthropocentric sampling not completely random. The number of possible states observable in our Universe is limited just with limited scope of us, human observers. Therefore even pure randomness has some rudimentary geometry hardwired into it for to remain observable.

http://www.guardian.co.uk/science/grrlscientist/2012/apr/30/1

The dispersive geometry of ripples at the multiparticle environment (like the water surface) shares most of signs with cyclic evolution. During it the transverse ripples disperse into longitudinal and back again. But a portion of energy is always dispersed during it into another dimensions. Therefore no generation of ripples is the same, like the another one.

Zephir said...

On the intrinsic structure of randomness the dense aether model is based. Dense aether model begins and ends with the insight, even the randomness has some rudimentary geometry hardwired into it. If we would generate the pixels of random color at the screen, then the result will not become quite structureless, because the colors are choosen from limited color gammut. It exhibits fuzzy patterns, which are similar to clouds or Perlin noise.

For example, for random datasets is typical, the extreme deviations are rare. You can throw the same number ten-times in random dice throws - but its improbable. In the group of random pixels the extreme pixel of extreme colour is improbable. As the result, the random pixels are forming clusters of clusters which are rather isolated each other. They're similar to material objects, composed of another smaller objects and particles, which we can observe inside of our Universe.

Zephir said...

OT post: Textbook Electrodynamics May Contradict Relativity (preprint).
The nature of this paradox could be understood with dense aether model easily.

The Lorentz force produces a drag, which transverse wave could never generate. Try to imagine this paradox with water surface model of dense aether theory: until its surface ripples are perfectly transverse (capillary waves), they don't have reference frame for their motion and we could say, they do follow the special relativity. But such waves cannot exert any force to the objects at the water surface - this force can be exerted only when these wave do have a longitudinal component. On the other hand, such a longitudinal component would introduce a frame drag, thus introducing an observable reference frame - which we don't observe at the vacuum.
The conceptual explanation of this paradox is simple though: the charge can never exist without heavily compacted/curved space-time inside of particle, where the transverse waves are becoming longitudinal - and vice-versa.

Eric said...
This comment has been removed by the author.
Eric said...

Zephir, I don't understand how the PDF cited by you makes makes one understand dense aether theory. Perhaps it's mostly because I can't visualize what you are saying in DAT. So far I haven't seen you relate it to standard mathematical equations in physics. While math isn't my specialty, ( I visualize also), when the final arguments come down you must relate it to with math. Otherwise one ends up relying on your salesmanship only.

However I glanced at the PDF and I do agree that it is illuminating a problem. An example I see myself in the standard interpretation is that QM spin , not orbital spin, is defined almost entirely as an abstract phenomena. Physicists say that classical spin defined by spin 1/2 fermions, hbar/2, equates to nonsense in terms of standard velocity for an electron. That is, it's speed would be about 1000 times faster than the speed of light. Nonsense right?

Well, that is an assumption everyone has pulled out of their butt. Why should we assume it is not a real world velocity just because it "seems" wrong. If the energy of the vacuum combined with what we call matter is finite and conserved, not at all a wild conjecture, then the flow of time itself would change as the universe has expanded. At the time of the electrons creation the energy of the vacuum was very hot and dense and it does not seem at all presumptuous to assume time was moving much more quickly then. We are now measuring the quantum mechanical spin angular momentum using our current rate of time flow in an expanded universe. I can't find anything wrong mathematically with how quantum mechanical spin is derived.

There is no reason in the world we shouldn't take that velocity literally using our current rate of time flow as the standard.

Zephir said...

/*...So far I haven't seen you relate it to standard mathematical equations in physics. While math isn't my specialty, ( I visualize also), when the final arguments come down you must relate it to with math. Otherwise one ends up relying on your salesmanship only...*/

It's like to say the Sherlock Holmes or Hercule Poirot after his verdict: it's all smart and nice my dear - but none of your deductions is supported with math. You cannot prove this murder guilty with such an arguments... How many times in your life you are forced to recognize the optimal solution without math?

For example, Bee argues here for insight, the art is balancing on the level of predictability and unpredictability. The most attractive people are balancing at the level of high genetic potential and mutagenity (which manifests with beauty mark). I know about these connections and I can agree with it in full depth - but where is the math to prove all these insight right?

Disclaimer: the PDF linked above was supplied to Bee for inspiration. It has no direct relation to AWT or article topic.

Zephir said...

/*..It seems to me today the width of the scientific edge is very thin, maybe too thin...*/

This width is just defines with determinism of formal math, which must remain low-dimensional and relatively simple for to remain maintainable with rigor, well conditioned and predictable - as the string theory memo learns us.

The following of formal math is just like the following the water surface - which works well, until this surface remains smooth and flat. At the moment, when this surface becomes foamy it's just the strictly formal approach, what increases the number of formal theories and general level of confusion.

Of course, the formally thinking people itself have no problem with it, because the more formal theories we will maintain, the more theorists could keep their jobs. But is such approach really advantageous for the rest of people?

Somewhat surprisingly, the modern physical theories don't tend to be very consistent and logical, despite the pile of formal math, because they're based on rather counterintuitive (if not unphysical) assumptions and postulates. We should balance the formal and nonformal approach, while maintaining its determinism with adherence to formal logics.

Bee said...

Zephir: I don't need your inspiration. Please omit off-topic posts. Thanks,

B.

Plato Hagel said...

Hi Bee,

Wasn't quite sure I understood the whole context of your post, but as I go through it I see where you might want to be surprised while assuredly their is really nothing of significance here?

In this example I’m going to map speed to the pitch of the note, length/postion to the duration of the note and number of turns/legs/puffs to the loudness of the note.See: How to make sound out of anything.

Why the transference then to sound based simulated algorithmic fabrications of natural features in the production of "pitch" as something explaining the natural functions of the terrain?

How one might look at the cosmos? Wayne Hu.

Then there is of course the tone as a residual expression of the thought about things while it most commonly amounts to the trends in science what may be sounding like a choir most pleasing or not?

Relevant or not? Being Able to See?

Best,

Plato Hagel said...

Red Noise?

In science, Brownian noise, also known as Brown noise or red noise, is the kind of signal noise produced by Brownian motion, hence its alternative name of random walk noise. The term "Brown noise" comes not from the color, but after Robert Brown, the discoverer of Brownian motion.

Zephir said...

The golden mean ratio (1.618) serves as an aesthetic criterion of proportions in many areas of art. This is how the dense aether model explains it - as a ratio of particle packing maximizing the energy transfer in form of transverse and longitudinal waves waves in particle environment. The spreading of information in longitudinal and transverse waves correspond the roles of intuitive and deterministic thinking, the surprise and expectation, etc..

Kaleberg said...

Human communications are a combination of carrier and signal.

Text, for example, has a repeated vertical carrier with the information conveyed by height variation and horizontal linkages. That's why we can recognize italics or compressed text in Latin, Hindi or Chinese writing. That's why you can read crossed text with the part of the message horizontal and part vertical. The eye can recognize the carrier and extract each message independently.

You get the same thing in music, a carrier, the beat and measure structure, with the melody superimposed on it and higher order structure, like verses and choruses, imposed on the melody. The carrier imposes an ordering that lets one follow the song.

(I suspect that this combination of carrier and signal is also part of dance and massage, though both pass messages less precisely.)

There are probably good sound physiological and informatic reasons for this. If nothing else, the carrier provides a base for paying attention to the message.

Plato Hagel said...

Wassily Kandinsky-Yellow, Red, Blue
1925; Oil on canvas, 127x200cm; Centre Georges Pompidou, Paris


Kandinsky, himself an accomplished musician, once said Color is the keyboard, the eyes are the harmonies, the soul is the piano with many strings. The artist is the hand that plays, touching one key or another, to cause vibrations in the soul. The concept that color and musical harmony are linked has a long history, intriguing scientists such as Sir Isaac Newton. Kandinsky used color in a highly theoretical way associating tone with timbre (the sound's character), hue with pitch, and saturation with the volume of sound. He even claimed that when he saw color he heard music.

Zephir said...

Intrinsic perspective corresponds the deterministic, curvature geometry based view of general relativity, where the space-time is considered deformed, whereas the path of light is considered straight due Lorentz invariance. Extrinsic perspective corresponds the perspective of quantum mechanics, based on fuzzy reality composed of overlapped particles in Hilbert space, where the path of light is deformed heavily. At the human distance scale (CMBR wavelenght scale ~ 1.7 cm) these perspectives become reversed.

In general, impressionism is earlier movement, related to rather optimistic illustrations of real exteriors oriented to future - whereas the expressionism is more recent movement, connected more to internal, abstract feeling of the past. Abstract or psychical impressionism is based on intuitive preference of overlapping spots, where positive curvatures and altitude prevails (tachism). Whereas the expressionism is introvert art, based on curves and lines with zero or negative curvature and pessimistic connotations (informel).

In this connection it's interesting, the subjective value (and objective price) is highest for J. Pollock's abstract oil drip "paintings", which are of fractal nature and where the spot and lines feature of 2D curvature remains balanced in it.

Andrew said...

I think it's "Voss & Clarke".

Bee said...

Hi Andrew,

You're right of course. Thanks, I've fixed that. Best,

B.