# Science of music

1. Aug 22, 2013

### Avichal

I am trying to understand how our brain interprets sound and I am bumped on a few things.
We can differentiate between different frequencies of sound (athough if they are close enough we might get confused).
And there is something called pitch which is a subjective realisation of frequency of sound by our brain.

Now there is something called an octave which I don't understand. When you double the frequency you go an octave higher. For eg:- If the frequency is x Hz and you make it 2x Hz then it is an octave higher. But it turns out that 2x Hz and x Hz sound similar to us? Why is that?
Also why the factor two? Is it something related to music or does it relate to how our brain is structured?

2. Aug 22, 2013

### Pythagorean

I think its both physics and higher order processing. One the one hand, an octave contains the tonic in it every other peak, so there's some physics working for us. The 3x note, btw, happens to be called a perfect fifth because its also very pure with the tonic. It's also included in every nominal chord because of how well it harmonizes with the tonic.

On the other hand, most other animals don't seem to perceive octaves the way we do, so there's a processing aspect to it. Forgive my brevity, currently on the mobile.

3. Aug 22, 2013

### rbj

you are asking some very good questions that have been asked before. there is some pretty good stuff written. i don't know your level of math expertise nor how deep your pockets are but Hartmann: Signals, Sound, and Sensation
is pretty good. also Loy: Musimathics might be good.

i'm working now, but at a later time i might try to answer some of your questions expressed, but the complete answers really need more space than you get here (like in a book).

4. Aug 22, 2013

### Pythagorean

I would also like to note that there's an interesting contradiction between physics and music perception. In music, four fifths are equal to two octaves plus a major third (as music is handled in fractions, and the fractions add up nicely this way). Musically, they both add up to the same note.

However, physically (or mathematically) it depends on what tuning system you use. If you stick to the fractional view of music, you actually find that the two different intervals add up to a slightly different pitch. The ratio between the different pitches is called a syntonic comma.

It turns out there will always be imperfections, no matter what tuning you use, but the common modern approach is equal temperament, which basically spreads that error about all the notes in the 12-tone scale so that the two different intervals add up to exactly the same mathematical frequency. Before, people used just tuning (the fractional approach) which seemed to make more sense at the reductionist (single interval) level because each interval of notes were part of the harmonic series, but ended up producing that ugly syntonic comma in the bigger series.

5. Aug 22, 2013

### Greg Bernhardt

6. Aug 23, 2013

### Avichal

That was started by me too. There I was looking at a psychological level. In this thread I am looking for a more mathematical and deeper answer. Amazing how music brought me to math!

7. Aug 23, 2013

### atyy

I think it's a very interesting question. A few quick thoughts:

1) We only identify the octave as "the same" for frequencies above about 100 Hz. Metronomes ticking at 2 and 4 Hz sound "different" to us.

2) There are many other "identities" in our perception, of which it is tempting to think of the octave identity as one of them. For example, the pitch of complex tones in which eg. {600, 800, 1000 Hz} has the same pitch as a 200 Hz tone.

3) Some of these properties are found in nonlinear dynamical systems. For example

NONLINEAR DYNAMICS OF THE PERCEIVED PITCH OF COMPLEX SOUNDS
Cartwright J. H. E.; González, D. L.; Piro, O.
Physical Review Letters 82, 5389-5392 (1999)
http://ifisc.uib-csic.es/publications/publications.php?keyword=cartwright&research=&idioma=ENG

Cartwright's paper is beautiful, getting both perceptual effects known as the "first and second pitch shifts". However, we don't know if this is the mechanism in our brains.

4) There are intriguing hints in http://www.ncbi.nlm.nih.gov/pubmed/11073865 "The maximally enhanced response occurred when the first tone was either ~1 octave below or above the probe tone. This finding is in good agreement with a recent study on the frequency dependence of response enhancement in monkey auditory cortex (Brosch et al., 1999). The importance of octave relations in tone sequences for the induction of response enhancement is unclear but may relate to issues of spectral contrast enhancement in sequential stimuli beyond the range of critical bands."

5) Not free, and I haven't read it, but these guys are generally superb: http://www.ncbi.nlm.nih.gov/pubmed/23785145

Last edited: Aug 23, 2013
8. Aug 23, 2013

### Pythagorean

I think that's out of place though. You're talking about beat. At each beat, the metronome still makes higher frequency tones that are audibly perceptive (>20 Hz). The discussion of octaves pertains to tone, not beat.

9. Aug 23, 2013

### atyy

It's relevant in the sense that for a continuous change in physical parameter, we perceive a qualitative change in percept from pitch to rhythm. Presumably the biological mechanisms for pitch should show a similar change in output behaviour over the same range of input frequency.

10. Aug 23, 2013

### rbj

Avi,

i am still having some trouble finding time to respond well.

about our logarithmic perception of either amplitude or frequency, ask yourself what would happen if our perception was linear. if it was linear an "octave" (or whatever unit name you want to give it) would represent a fixed number of Hz. or a dB would represent a fixed amount of amplitude. that would lead to the inevitable "why is an octave = 100Hz? what's so special about 100Hz?" (to which the answer is that it ain't special because our perception of any physical quantity is not like that at all).

second, think about what makes two notes sound "consonant" (as opposed to "dissonant"). it has something to do with the frequency ratios (draw some conclusions from that).