# Mathematics behind music.



## KaerbEmEvig (Dec 15, 2009)

Recently I've stumbled across a very interesting article, which explains how, at its core, music is bound to mathematics:

http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.1873v2.pdf

I think you might find it interesting as I did.


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## Ukko (Jun 4, 2010)

Fairly amusing. Music predates mathematics. These guys, most of them, don't recognize that mathematics is a method of codifying nature, and that music is an earlier -and more precise - method of codifying the part of nature that is the brain's response to sounds.


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## emiellucifuge (May 26, 2009)

Music is hardly more precise than mathematics. When we talk about notes we are really referring to a rough group of frequencies that were defined only recently. For example, the Wiener Philharmoniker tunes a few Hz differently than most other orchestras yet they play the same notes.

There is hardly anything more precise and true than x=x or [1/3 x 2 = 2/3]


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## Argus (Oct 16, 2009)

This seems like a perfect thread to ask some mathematical music questions that I've been pondering over recently.

They will probably be pretty simple for anyone who has studied higher maths but they are a bit confusing for myself.

Here's the first question.

When c = cents and n/p = the interval ratio between tones, then

c = log(n/p) x 1200/log2

Therefore,

a perfect fifth of 3/2 = log(3/2) x 1200/log2 = 701.955 cents

That makes sense to me. But how would I reverse the equation so if I knew the cent number and wanted to find the interval ratio (frequency ratio)?

My second question is more Pythagorean.

If I have a monochord, or simply a string, and place bridges between the regular nut and bridge, at simple divisions of the string starting from the nut (1/2 the whole string length, 1/3, 1/4, 1/5 etc), can anyone work out a way of finding the interval ratio between various bridges, assuming all bridges are present along the string simultaneously .

So the ratio between the string from the centre (1/2) bridge to the nut and the whole string is 2/1, or an octave. The ratio of the string between the 1/3 bridge and the 1/2 bridge in relation to the whole string is 6/1, which when brought into the range of an octave is 3/2. This is where I get a bit stuck. I believe the length of string between the 1/4 bridge and the 1/3 bridge to be 12/1, which again in the span of an octave becomes 3/2. My instinct expected 4/3 but I'm not sure why.

Can anyone figure out a formula for working these relationships out?

Here's a picture to give you an idea, but imagine the added bridges are permanant and continuing infinitely into smaller ratios of the string. Instead of measuring from the nut to the new ridge I am interested in the length from the new bridge to the second newest bridge if that makes sense.(1/5 to 1/6 compared to 1/1, 1/7 to 1/8 compared to 1/1 etc)










I'll have a look at that article when I get chance, but it seems more like a mathematical explanation for existing music. I'm more interested in basing musical theories off mathematics. When letters like A, Bb, C etc are used instead of numbers it seems biased towards a system of tradition and not a more sensible scientific view. Also, I'm not sure my maths is up to scratch to understand a lot of it.


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## Huilunsoittaja (Apr 6, 2010)

880 Hz - A6
440 Hz - A5 (in staff) (that is, if you're American  )
220 Hz - A4
etc.

The phenomena of Enharmonics is also very interesting. It's when the sound waves 2 notes (wind instruments only) played by 2 instruments combines together, and makes a 3rd note. There are calculations to figure out what the note is, I think it's the Hz of higher note minus Hz of lower note.


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## Argus (Oct 16, 2009)

KaerbEmEvig said:


> Recently I've stumbled across a very interesting article, which explains how, at its core, music is bound to mathematics:
> 
> http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.1873v2.pdf
> 
> I think you might find it interesting as I did.


I had a read but didn't find it relevant to my interests. Seemed more aimed at mathematicians interested in music than vice versa.

I've worked out that theoretical monochord division procedure. The string lengths decrease from 1 by the amounts 1/2, 1/6, 1/12, 1/20, 1/30, 1/42, 1/56 etc. Then from this it's clear to see the denominators increase in 2's by the series 4, 6, 8, 10, 12, 14 etc. Transferring that data into musical intervals is a bit trickier because of inversions and octaves but I should be able to work out a simple formula now.

Still can't figure out the logarithmic equation though.



Huilunsoittaja said:


> 880 Hz - A6
> 440 Hz - A5 (in staff) (that is, if you're American )
> 220 Hz - A4
> etc.
> ...


I think you're either referring to combination (or Tartini) tones (sum and difference) which arise when any two sound waves are sounded together or multiphonics, which is when more than one tone is sounded simultaneously on a wind instrument. Enharmonics are when the same note is written differently and also an ancient Greek genus/tetrachord.

Not sure why you listed the current concert pitch frequencies of A at the start of your post.


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## emiellucifuge (May 26, 2009)

Argus said:


> When c = cents and n/p = the interval ratio between tones, then
> 
> c = log(n/p) x 1200/log2
> 
> ...


Im not too familiar with the whole cents and interval ratio thing, and im assuming that Youre using log to the base of 10.

Then simply rearrange the equation to give:

(Clog2)/1200 = log(n/p)

Do you understand logarithms enough to work it out from this?

[it seemed pretty simple to me, maybe too simple, so excuse me if ive missed the point]


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## Argus (Oct 16, 2009)

emiellucifuge said:


> Im not too familiar with the whole cents and interval ratio thing, and im assuming that Youre using log to the base of 10.
> 
> Then simply rearrange the equation to give:
> 
> ...


Log to the base of 2. I think. It's been a few years since I did maths at school. Even then I can't remember ever doing logarithms. That's why I only got a B at GCSE.

Using log 10 would either be,

c = log (n/p) x 3986.3137

or

c = 1200 × 3.322038403 log10 (n/p)
1/log 2 = 1/0.301029995 = 3.322038403

Using your equation when I input the 701.955 for c, I get -0.754 instead of 1.5 (3/2) I'd expect. I think I must be doing something wrong.


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## Comus (Sep 20, 2010)

Hey Argus,

Your initial equation ( "=701.955 cents) shows that the log is base 10 so:

n/p=10^[ (c*log 2) /1200 ]


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## Huilunsoittaja (Apr 6, 2010)

Argus said:


> I think you're either referring to combination (or Tartini) tones (sum and difference) which arise when any two sound waves are sounded together or multiphonics, which is when more than one tone is sounded simultaneously on a wind instrument. Enharmonics are when the same note is written differently and also an ancient Greek genus/tetrachord.
> 
> Not sure why you listed the current concert pitch frequencies of A at the start of your post.


That's right, I termed it wrong, it's called Harmonics.

The Hz for the pitch of A shows a pattern that the octave above is always double the A below. A pattern, if not mathematical.


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## Great Gate (Oct 17, 2010)

*J. S. Bach was the most mathematical Composer.*

In his Brandenburg Concerti, Bach's phrasing represents a precise mathematical subdivision of his measures, everythng being exact multiples (or sub-multiples) of the tempo.

Atomic Clocks can be set to Bach's precise continuity of tempo.

Great Gate


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## Argus (Oct 16, 2009)

Comus said:


> Hey Argus,
> 
> Your initial equation ( "=701.955 cents) shows that the log is base 10 so:
> 
> n/p=10^[ (c*log 2) /1200 ]


Cheers. That seems to work for me. It's a lot simpler than Ellis' method in the appendices of On the Sensation of Tone. No tables required.:tiphat:


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## emiellucifuge (May 26, 2009)

When you write 'log x' it is assumed you mean to the base 10, but no problems!


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## Argus (Oct 16, 2009)

emiellucifuge said:


> When you write 'log x' it is assumed you mean to the base 10, but no problems!


Correct. I redid your equation afterwards with 10 to the power of the answer and it worked fine. I just needed a bit of a clear up on what log to the base of 10 meant.


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