Why There are 12 Tones in a Scale

Date: 12/28/2000 at 18:52:43
From: Ashley
Subject: Why 12 tones in a scale

I've written to you before, but I need help again. Your last answer 
was very helpful and I'm grateful for it!

I have a question about some information I got from another source as 
to why there are 12 tones in a scale. We choose an r, which is some 
number that is close to 1.5. To get the 12 notes of the scale, 
multiply one frequency by r and keep multiplying what you get by r. 
Then, bring the frequencies down into the appropriate octave by 
dividing by 2 as many times as needed. This r should be close to 3/2 
(1.5) since this is a perfect fifth. However, a perfect fifth doesn't 
work because you never get an octave that gives a 2:1 ratio with the 
note that you started with after you put your ending note down into 
the appropriate octave.

I got the equation r^n = 2^m, where n = the number of fifths needed to 
complete the progression (which is also the number of notes in the 
scale), and m = the number of octaves needed to get to the end of the 
progression.

David Rusin's Web page says that the trick is to get an r that is 
close to 1.5. 

  Mathematics and Music - David Rusin
  http://www.math.niu.edu/~rusin/uses-math/music/   

He says that there is no better value for n than 12 until you get to 
29. To fill in the equation and get the value of r, you need to have 
the value for m, the number of octaves used.

Is there an easier way to get the number of octaves? One that can be 
done without multiplying a frequency by 1.5 many times and then 
dividing repeatedly by 2? Also, can you explain why r^n = 2^m? Why 
should this be? What's the reasoning behind it? Also, what's so 
special about a fifth? Why should the scale be based on the fraction 
3/2? I'm doing a project for school that's due when I come back from 
Christmas break and I have to teach it to the class, so I'm trying to 
understand as much about this as I can. Just answer all of the 
questions that you can and it will be greatly appreciated - you're a 
great help!

Date: 12/29/2000 at 16:51:37
From: Doctor Rick
Subject: Re: Why 12 tones in a scale

Hi, Ashley. I think I can answer some of your questions.

One question you asked is why the ratio 1.5 is important. It was 
Pythagoras, I believe, who noticed that two vibrating strings make a 
harmonious sound together if the ratio of their lengths is a ratio of 
small whole numbers. The frequencies of the resulting sounds are 
inversely proportional to the lengths of the strings, so these ratios 
will again be ratios of small whole numbers.

The simplest such ratio between different tones is 2:1, which is the 
ratio of tones an octave apart. The next simplest is 3:1. Here the 
second tone is more than an octave above the first; if we divide it by 
2 to bring it down an octave, we get the ratio 3:2, or 1.5:1. 
Recognize the number? This second-simplest ratio is what we call a 
fifth.

You ask where the equation comes from. Because it is ratios that 
matter in musical pitches, we'd like to build a scale as a geometric 
sequence of frequencies, in which the ratio between successive 
elements of the sequence is a constant. The octave of every tone (the 
tone with twice the frequency) must be in the scale, because that's 
the simplest, best-sounding ratio. Thus we'll have n tones in an 
octave, and the tones in any other octave will be just these n tones 
multiplied by a power of 2.

We'd like to be able to build the scale in such a way that, for any 
tone in the scale, its fifth (that is, 1.5 times its frequency) is 
also in the sequence. This allows you to make pleasant-sounding chords 
in any key - that is, you can build a chord starting with any tone in 
the scale.

We can build this scale by starting with the tonic (ratio 1) and 
finding its fifth (1.5 * 1 = 1.5), then the fifth of this tone (1.5 * 
1.5 = 2.25), then the fifth of this tone (2.25 * 1.5 = 3.375), etc. 
For each of these tones, its octaves will also be in the scale: you 
can multiply or divide the ratio by 2 to get other tones in the scale, 
and particularly one that is in the range 1 to 2. The fifth of the 
fifth is thus one octave above 2.25/2 = 1.125, and the fifth of this 
is an octave above 3.375/2 = 1.6875.

We want this process to terminate sooner or later, otherwise we'll 
have an infinite number of tones in our scale. How can it terminate? 
When we get to a tone that we have already found. This will happen 
when we get a tone that is a power of two: this tone is the same as 
the starting tone, some number m of octaves up. If it's the nth tone 
we've found, then we have:

     1.5^n = 2^m

Unfortunately, there is no solution to this equation with whole 
numbers for n and m. Instead, we look for a ratio r that is close to a 
fifth (1.5) but is compatible with having the octaves in the scale. 
This is where we get the equation:

     r^n = 2^m

The other question you're asking is about the math: How can you find 
an r that makes this work, with relatively small numbers n and m? You 
don't have to use trial and error, at least not so much of it. You can 
solve the equation by taking the log of both sides:

     log(r^n) = log(2^m)

     n*log(r) = m*log(2)

          m/n = log(r)/log(2)

Put r = 1.5, and the right side evaluates to 0.584962500... What we 
need to do is to find a number m/n that is close to this number, where 
m and n are the smallest possible whole numbers. You can try different 
values of n (the number of tones in the scale), making a table of:

     m = n*log(1.5)/log(2)

       = r*0.584962500

The values of m will not be whole numbers. Round each m value to the 
nearest whole number, then use this rounded value to compute:

     r = 2^(m/n)

You will find that r is close to 1.5 when n = 12, and again when n = 
24. A 24-tone scale is just the 12-tone scale with 12 more 
quarter-tones in between. The next value for n that gives a better 
approximation to the fifth is 29 tones.

I hope this information helps you. 

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   
    

Date: 03/13/2006 at 22:08:34
From: Gin
Subject: why 12 tones in a scale

Doctor Rick said that "the next value for n that gives a better 
approximation to the fifth is 29 tones."  Please explain.  I thought 
I understood why 12 and 24 tones gives an approximation to the fifth, 
but 29 has me wondering as I assumed the next number would be 48.

Date: 03/14/2006 at 18:07:11
From: Doctor Rick
Subject: Re: why 12 tones in a scale

Hi, Gin.

The link that Ashley gave still works, and contains more information 
on this topic than you probably want to know! A brief summary is 
found here:

    

Let's do some of what David Rusin said to do. I just made a 
spreadsheet to look at the ratios for 5 through 42 tones per octave. 
Here is how it starts:

 5 1 1.1486 1.3195 1.5157 1.7411 2

 6 1 1.1224 1.2599 1.4142 1.5874 1.7817 2

 7 1 1.1040 1.2190 1.3459 1.4859 1.6406 1.8114 2

 8 1 1.0905 1.1892 1.2968 1.4142 1.5422 1.6817 1.8340 2

 9 1 1.0800 1.1665 1.2599 1.3607 1.4697 1.5874 1.7144 1.8517 2

10 1 1.0717 1.1486 1.2311 1.3195 1.4142 1.5157 1.6245 1.7411 1.8660 2

11 1 1.0650 1.1343 1.2080 1.2866 1.3703 1.4594 1.5544 1.6555 1.7631 
1.8778 2

12 1 1.0594 1.1224 1.1892 1.2599 1.3348 1.4142 1.4983 1.5874 1.6817 
1.7817 1.8877 2


Then I looked for the tone in each scale that comes closest to 1.5, 
which is in the second column below:

   5   1.515716567   0.015716567
   6   1.587401052   0.087401052
   7   1.485994289   0.014005711
   8   1.542210825   0.042210825
   9   1.469734492   0.030265508
  10   1.515716567   0.015716567
  11   1.459480106   0.040519894
  12   1.498307077   0.001692923
  13   1.531966357   0.031966357
  14   1.485994289   0.014005711
  15   1.515716567   0.015716567
  16   1.476826146   0.023173854
  17   1.503406654   0.003406654
  18   1.527435131   0.027435131
  19   1.493758962   0.006241038
  20   1.515716567   0.015716567
  21   1.485994289   0.014005711
  22   1.506195553   0.006195553
  23   1.479610446   0.020389554
  24   1.498307077   0.001692923
  25   1.515716567   0.015716567
  26   1.49166449    0.00833551
  27   1.50795418    0.00795418
  28   1.523239565   0.023239565
  29   1.501294382   0.001294382
  30   1.515716567   0.015716567
  31   1.495517882   0.004482118
  32   1.509164428   0.009164428
  33   1.490459915   0.009540085
  34   1.503406654   0.003406654
  35   1.485994289   0.014005711
  36   1.498307077   0.001692923
  37   1.510048194   0.010048194
  38   1.493758962   0.006241038
  39   1.504979244   0.004979244
  40   1.489677463   0.010322537
  41   1.500419433   0.000419433
  42   1.510721887   0.010721887

The third column shows the difference between this ratio and 1.5. If 
you look down the list, you'll see that multiples often have the same
closest ratio to 1.5: 5, 10, 15, 20, 25, and 30 all have 1.515716567. 
In a 35-tone scale, the same value is there, but another becomes 
closer to 1.5 than this one.

In the same way, 12, 24, and 36 all have the same closest ratio to 
1.5, namely 1.498307077. This is closer to 1.5 than any value before 
12, *or* any value after 12 until you get to 29. As I said, it is 
the same as the ratio for a 24-tone scale, but the 29-tone scale is 
the first that has a ratio *closer* to 1.5 than this (namely, 
1.501294382). That's what I meant.

You'll see also that the 29-tone scale is only a little better than 
the 12-tone (or 24-tone) scale. However, when you get to a 41-tone 
scale, you get a much closer ratio to 1.5, namely 1.500419433.

Mr. Rusin goes into detail elsewhere about how this result can be 
found by use of continued fractions, so that it is not necessary to 
generate this entire chart. I think that is beyond your need, 
however.

I must tell you that I am not far from being out of my depth here; I 
am mostly reporting what I read. I'm standing on tiptoe to keep my 
nose above water, so don't hold on to me too tightly or I'll go 
under. But we can have some fun swimming together if you have more 
questions, and there is probably a Math Doctor with more knowledge 
of music, continued fractions or whatever, who can assist us if 
necessary.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/