# Math project on music...

Discussion in 'Miscellaneous [BG]' started by jenderfazz, Sep 25, 2003.

1. ### jenderfazz

Apr 17, 2003
So in Calculus today we had to choose a topic for a math project, "How Math is used in daily life" and obviously I'm doing it on music. We'll do some things like the pitches in Hz of different strings on instruments and the sine functions that accompany them, and some other things. Any suggestions/info/tips?

2. ### jenderfazz

Apr 17, 2003
Sounds like a very good idea. Were gonna bring in our instruments and show them anyways, so I can show the harmonics at the 12th, 7th, 5th, 4th, etc. Heck I can make an exponential function out of it or something.

3. ### jenderfazz

Apr 17, 2003
I'm guessing artificial harmonics work like the regular ones? Example, the 2/6. The string is shortened a little, and the 6 is the new halfway/quarter of the scale, no? That would be a cool concept. Any other suggestions?

4. ### jenderfazz

Apr 17, 2003
Any other suggestions?

5. ### Johnalex

Jul 20, 2001
South Carolina
you can do either algorythms wich is realyl broad and I know nothing about or you can do the logarthmic ideas involved with decibals and frequency. You know how your frets get closer the higher the tones go?...That is expressed in log when you do "music math".

http://www.tonmeister.ca/main/textbook/electronics/03.html

6. ### JoseNeville

There are music about just 5 note, I think India have that kind,

There are music about 12 note, the one we know.

And you can can make music with 41 diferent note.

This is all about a natural logarimic funtion, we just use the aproximation, remember the number e is irrational so he have infinite decimal number. I read this on a paper, I will try to find out.

better aproximation, you will have more different notes.

7. ### jondog

Mar 14, 2002
NYC metro area
One thing that interests me is where math *stops* being the best way to describe music. Yes, I know that math applies to frequency and rhythm and loudness, and that all music *can* be measured with numbers, but at some point the numbers start telling you the least interesting or important things about the music. For instance, in written music they give you metronome markings in numbers, but as soon as the composer writes a part where they want the tempo to "live" rather than beat like a clock, they resort to language and write "rallentando" or "accelerando" or whatever over the passage and rely upon the artistic interpretation of the players and conductor to make it happen. Numbers are powerful symbols, but they aren't the only or best ones.

8. ### ARA punk

Jul 11, 2001
USA, Shelby, NC
Sounds interesting man. I haven't gotten around to Calculus yet, but i'd be interested in reading that report (if its going to be a typed one) if you wouldn't mind posting it. Good luck.

9. ### JoseNeville

Never!!

The history is that math and music are strong related. Until the 17 century the music deparment was on the math deparment. The only way to study music was studing math. How do you think Mozart make those increible composition. Was using combinatories and permutations.

10. ### JoseNeville

I seem to have answered several times the question "why 12 tones per octave?"
Here are the posts, with some email and followups.
dave
==============================================================================

Date: Wed, 28 Jul 93 16:06:05 CDT
Wrom: UIVOTQNQEMSFDULHPQ
To: bert@netcom.com
Subject: log3/log2

I use log3/log to explain the importance of the 12-note scale of Western music.
Here is my reasoning.

One note does not music make (the "One Note Samba"
notwithstanding). Now, which notes sound best with a note of a given
frequency? The ancient greeks more or less decided it was those whose
frequencies were integer multiples of the first (Indeed, those other
frequencies are present in practice because the Fourier expansion of a single
note on an instrument includes those other frequencies with small but not
tiny coefficients).

Now, unless you play nothing but octaves, say, you have two frequencies which
are multiples of the first but not of each other. The most audible will be
the lowest two, which are in a 3:2 ratio. This is the pure fifth, and still
makes a pleasing chord. Other ratios can be tried but as a rule the larger the
integers necessary to describe the ratio, the worse the sound.
If we build fifths above the fifths, we get more tones in the scale (typically
we reduce by an octave, i.e., a factor of 2, whenever producing a tone
of more than twice the original frequency). This of course is the construction
of the circle of fifths. I have gotten pretty good at using this to tune
pianos.

Unfortunately, the process never terminates: no power of (3/2) is ever a
whole number of octaves (or indeed any integer multiple of the first frequency).
I make this observation whenever teaching about Unique Factorization. Thus,
we introduce more and more tones describing more complex ratios which, as I
noted above, sound worse and worse.

So we fudge the fifth to make the equal-tempered scale: find a ratio r
roughly equal to 3/2 so that some small power of r is a power of 2. This
amounts to finding good integer approximations for the solutions of
(3/2)^x = 2^y, which we rewrite as 3^k=2^l, or l/k=log3/log2=1.584962501...

The theory of continued fractions tells us how to do this: Form the continued
fraction expansion of this real number, stop at certain points, and reevaluate
process, but two facts are useful here: The fractions so attained are better
approximants than any others with smaller denominators; and the approximations
are unreasonably good iff we stop just before a big term shows up in the cont.
frac. expansion.

So here it is: log3/log2= cont.frac[1,1,1,2,2,3,1,5,2,23,...], which gives the
following optimal approximations:
1/1, 2/1, 3/2, 8/5, 19/12, 65/41, 84/53. 485/306, 1054/667,...
(I stop here because the next term to use, 23, is really large, so that
1054/667 is a much better appproximation than the 667 leads you to expect;
the next best approximation has a huge denominator).
Musically, these numbers tell us that by building a pure fifth repeatedly
we get closer and closer to a real octave if we use 1, 2, 5, 12, 41, 53, ...
fifths; other numbers of fifths offer no particular advantage.

I find it a curious twist of nature that the highly-divisible number 12 shows
up here; had it been 11 or 13, music would have developed a lot differently
as you noted in your news post. Also I think it is interesting that the 12
shows up corresponding to the [1,1,1,2,2] part of the fraction; the next
number (3) is larger so 19/12 is a pretty good approximant to log3/log2,
considering the size of its terms; the 65/41, by contrast, is less impressively good. I guess nature just provided for us well.

I think this explains the prevalence of the 5-note and 12-note scales. I
have heard compositions written in n-note scales where n was in the range of 10
to 20; all were (necessarily?) Bach-like and kept suggesting 12-note music
which was "slipping" from time to time. From the mathematics of it I would
expect that no really good new music would result until we tried a 41-note
scale. Just offhand I would expect a keyboard with 3 or 4 levels of blackness
of keys instead of the current 2 (white and black), which would be used with
decreasing frequency (no pun intended) to correspond to the fact that they
corresponded to the higher powers of 2^(24/41) ("higher" meaning mod 41 I guess)

As my electronic music equipment is limited to the world-class IBM PC speaker,
I am not in a position to try this out, but I think it would be a real
kicker to turn a 5-row computer keyboard into this kind of musical instrument.

I am a better mathematician than musician, but I have a good time linkning the

dave rusin@math.niu.edu

11. ### jenderfazz

Apr 17, 2003
Any last suggestions? I'm making the introduction/description for Monday and I need to know exactly what I'm gonna build upon.

12. ### JoseNeville

Hi!!

I don,t know if you are asking me. I'm a math professor so I teach calculus a lot. What you have to do?

1. Go to the library and find the history of eastern music, 5 note music. Find a CD and put a example in the class.

2. Go to the library and find the history of weastern music, 12 note music. Find a CD and put a example in the class.

3. Go to the internet and find a music example about the 41 note music. Burn onto a CD and put in class.

4. Read the post that I put before. Explain how log3/log2 give the diferent numbers of note. For example 5, 12, 41.

This is the part:
"Unfortunately, the process never terminates: no power of (3/2) is ever a
whole number of octaves (or indeed any integer multiple of the first frequency).
I make this observation whenever teaching about Unique Factorization. Thus,
we introduce more and more tones describing more complex ratios which, as I
noted above, sound worse and worse.

So we fudge the fifth to make the equal-tempered scale: find a ratio r
roughly equal to 3/2 so that some small power of r is a power of 2. This
amounts to finding good integer approximations for the solutions of
(3/2)^x = 2^y, which we rewrite as 3^k=2^l, or l/k=log3/log2=1.584962501...

The theory of continued fractions tells us how to do this: Form the continued
fraction expansion of this real number, stop at certain points, and reevaluate
process, but two facts are useful here: The fractions so attained are better
approximants than any others with smaller denominators; and the approximations
are unreasonably good iff we stop just before a big term shows up in the cont.
frac. expansion.

So here it is: log3/log2= cont.frac[1,1,1,2,2,3,1,5,2,23,...], which gives the
following optimal approximations:
1/1, 2/1, 3/2, 8/5, 19/12, 65/41, 84/53. 485/306, 1054/667,...
(I stop here because the next term to use, 23, is really large, so that
1054/667 is a much better appproximation than the 667 leads you to expect;
the next best approximation has a huge denominator).
Musically, these numbers tell us that by building a pure fifth repeatedly
we get closer and closer to a real octave if we use 1, 2, 5, 12, 41, 53, ...
fifths; other numbers of fifths offer no particular advantage. "

Good luck,
Jose Neville