View Full Version : Mathematics and Music

Andrew D

2010-Jul-30, 09:24 PM

I've just finished a Calc I course and I'd like use my new toolbox to do some sort of application/ investigation for the rest of the summer. I'm really interested in the mathematical aspects of music and music theory, but I can't seem to find a book that feels right for what I'm looking for.

Does anyone know of any good books (text books are ok too) on the subject? Older is ok, I like buying used and cheap!

Thanks

grant hutchison

2010-Jul-30, 10:31 PM

Not sure if they'll meet your requirements, but I can recommend two books that seem to be in the ballpark of your interest:

Exploring Music: The Science And Technology of Tones and Tunes by Charles Taylor

The Science of Musical Sound by John R. Pierce

Both are more directly concerned with the physics rather than the mathematics, but they might generate ideas for you.

Grant Hutchison

kleindoofy

2010-Jul-30, 10:34 PM

The basic math in music is rather simple.

All you usually have to do is count to four and start over again: 1-2-3-4, 1-2-3-4 ;)

A waltz is even easier.

pzkpfw

2010-Jul-31, 12:40 AM

Not a book, but a mention of something you may find interesting:

A friend of my Wifes, at University, was doing a masters in maths.

His thesis project was a statistical analysis of what he termed "cowboy music".

Basically he was trying to analyse what notes tended to follow other notes, and in what combinations.

The idea was then to see if he could auto-generate acceptable music, by following "rules" developed out of the statistics.

This was in 1990 and I never met the guy again, no idea where it went.

dgavin

2010-Jul-31, 01:54 AM

You might start with the mathmatical formula behind fret postioning on guitars

Basically he was trying to analyse what notes tended to follow other notes, and in what combinations.

The idea was then to see if he could auto-generate acceptable music, by following "rules" developed out of the statistics.

I don't see that as being particularly improbable. It seems that (in western music at least) there are certain combinations and sequences of chords that just "work" together and as such are used really often. The classic example is the 'perfect cadence' which goes from the V to the I, and used most often at the end of songs. Here's a pretty good explanation of it: http://everything2.com/title/perfect+cadence

There are other cadences that don't sound quite so "final" and are often used in the middle of songs to transition to the next part of it.

kleindoofy

2010-Jul-31, 02:28 AM

... The classic example is the 'perfect cadence' which goes from the V to the I, and used most often at the end of songs. ...

Errr, V to I used used at the end of 99.99% of all Western music.

And with "Western" I don't mean Western as in "Country & Western" but as opposed to Oriental and others.

Andrew D

2010-Jul-31, 02:31 AM

Not sure if they'll meet your requirements, but I can recommend two books that seem to be in the ballpark of your interest:

Exploring Music: The Science And Technology of Tones and Tunes by Charles Taylor

The Science of Musical Sound by John R. Pierce

Both are more directly concerned with the physics rather than the mathematics, but they might generate ideas for you.

Grant Hutchison

Thank you, and physics is what I'm looking for, I suppose. I would hate to think about a mathematical examination of music that avoided physics. The result could no doubt be summarized by kleindoofy's reply above.

Not a book, but a mention of something you may find interesting:

A friend of my Wife's, at University, was doing a masters in maths.

His thesis project was a statistical analysis of what he termed "cowboy music".

Basically he was trying to analyze what notes tended to follow other notes, and in what combinations.

The idea was then to see if he could auto-generate acceptable music, by following "rules" developed out of the statistics.

This was in 1990 and I never met the guy again, no idea where it went.

I am aware of projects like this. A few modern composers (Brian Eno, for example) have done used algorithms to generate melodies. There is a website or two devoted to pieces of music that won't repeat for thousands of years, although their names escape me.

You might start with the mathematical formula behind fret positioning on guitars

This is kind of what I'm looking for. I'm not sure how familiar you are with music, but I really want a source that goes into depth with the relationships between wave frequency ratios and musical intervals, consonance and dissonance, acoustics and sound mechanics, and the like.

Andrew D

2010-Jul-31, 02:37 AM

Errr, V to I used used at the end of 99.99% of all Western music.

And with "Western" I don't mean Western as in "Country & Western" but as opposed to Oriental and others.

It's because on the western scale, 1=8, so 5 to 1 can also be 5 to 8. The ratios of the frequencies of the 8th tone and the 5th tone ≈ 1.618.

dgavin

2010-Jul-31, 03:19 AM

This is kind of what I'm looking for. I'm not sure how familiar you are with music, but I really want a source that goes into depth with the relationships between wave frequency ratios and musical intervals, consonance and dissonance, acoustics and sound mechanics, and the like.

This may be a place to start

http://en.wikipedia.org/wiki/Music_and_mathematics

parallaxicality

2010-Jul-31, 08:57 AM

one of the archives I work in has an air conditioner that plays a perfect fifth. I think there must be a 3:2 ratio in the ducts somewhere.

Hornblower

2010-Jul-31, 12:01 PM

one of the archives I work in has an air conditioner that plays a perfect fifth. I think there must be a 3:2 ratio in the ducts somewhere.

My hunch would be the hum of a 3-phase motor. There are three 60Hz alternating currents, a third of a cycle out of phase with each other, causing the field assembly of the motor to vibrate. This could generate a 360Hz ripple on top of the familiar alternating current hum, which would be consistent with the faint F#, first space in the treble clef, that I can hear from the circulating pump in the heating system at my church.

loglo

2010-Aug-01, 07:19 AM

One of the links in the Wiki would seem to address the OP pretty thoroughly, it is a 530 page PDF file though:-

Music: A Mathematical Offering (http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf)

Chapter 1. Waves and harmonics

Chapter 2. Fourier theory

Chapter 3. A mathematician’s guide to the orchestra

Chapter 4. Consonance and dissonance

Chapter 5. Scales and temperaments: the fivefold way

etc etc etc

Andrew D

2010-Aug-01, 06:06 PM

One of the links in the Wiki would seem to address the OP pretty thoroughly, it is a 530 page PDF file though:-

Music: A Mathematical Offering (http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf)

Chapter 1. Waves and harmonics

Chapter 2. Fourier theory

Chapter 3. A mathematician’s guide to the orchestra

Chapter 4. Consonance and dissonance

Chapter 5. Scales and temperaments: the fivefold way

etc etc etc

I just came back to report that I had found exactly what I was looking for... just to find that you had already found it! Thanks

mugaliens

2010-Aug-02, 02:54 AM

You might start with the mathmatical formula behind fret postioning on guitars

Is there one? I'd thought its design was meant to make the more common notes easier to play.

Jens

2010-Aug-02, 02:56 AM

Is there one? I'd thought its design was meant to make the more common notes easier to play.

I think what was meant is that the frets are longer at the base of the neck and get smaller as they go toward the bridge. So there is a mathematical formula on how long the frets have to be. Not whether it's tuned in fourths or fifths or stuff like that.

kleindoofy

2010-Aug-02, 03:08 AM

... So there is a mathematical formula on how long the frets have to be. ...

Yes, but it's secondary.

Frets have been around since Christ was a corporal. I doubt the instrument designers used or even thought about math back in the old, old days. But, they did think about intervals and how they sounded. Those intervals correspond to mathmetical formulae, even if they didn't know or care about it.

Btw, the holes in flutes, recorders, oboes, clarinets, bassoons, saxophones, even ancient ocarinas correspond to mathematical formulae.

Trial and error will get you pretty far without realizing that, once you get it right, you've adhered to the math.

Jens

2010-Aug-02, 03:33 AM

Yes, but it's secondary.

I didn't mean to imply that you need to understand the mathematical formula to make a guitar. Obviously, people built them empirically without understanding the formulas. I just meant to say that as with flutes and other instruments, there are mathematical formulas that describe the placement.

Frets have been around since Christ was a corporal.

That's the first time I've ever heard that. Interesting phrase. Just out of curiosity, is that originally a pun meaning "since Christ was a-corporal"?

clop

2010-Aug-02, 04:57 AM

Oooh goody, do we get to discuss again the fact that a perfect fifth, with a ratio of 1.5, does not exactly equal an interval of 7 semi-tones, with a semi-tone being the twelth root of an octave, i.e. the twelth root of 2?

clop

GeorgeLeRoyTirebiter

2010-Aug-02, 08:47 PM

I've built guitars, and positioning the frets is not very difficult. Guitars use equal temperament, which means the ratio of the distance between any two adjacent frets is constant. This is most commonly expressed as the "rule of eighteen," where the distance to the next fret is approximately equal to the distance between the previous fret and the bridge divided by 17.817 (derivation available upon request).

kleindoofy

2010-Aug-02, 08:59 PM

... is that originally a pun meaning "since Christ was a-corporal"?

Oh great. We go from math and music to a theological discussion of divine pre-existence. ;)

No, just a metaphor for "a long time."

Jens

2010-Aug-03, 02:43 AM

Oh great. We go from math and music to a theological discussion of divine pre-existence. ;)

No, just a metaphor for "a long time."

I wasn't meaning to get into a theological discussion. I was just wondering out loud (and hijacking in the process, I admit) about where the expression came from. It seems odd, because the choice of "corporal" rather than "lieutenant" or whatever seems too coincidental.

kleindoofy

2010-Aug-03, 03:12 AM

... the choice of "corporal" rather than "lieutenant" ...

No, it's a combination of an alliteration (http://en.wikipedia.org/wiki/Alliteration) and extremes. "Private" might be better, but that wouldn't be an alliteration. "Lieutenant" is an officer, so it starts too high in rank.

I first heard the expression when I was, oh, in the sixth grade, over fourty years ago.

It's similar to expressing that a joke one hears is old by saying "the last time I heard that joke, I laughed so hard that I fell off my dinosaur."

Sometimes a cigar is just a cigar.

grant hutchison

2010-Aug-03, 10:42 AM

Sometimes a cigar is just a cigar.... but a good cigar is a smoke. :)

Grant Hutchison

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