To find how many tones there are in an octave we are looking for the value of x such that (9/8)x = 2 then
x log 9
8
 = log2, x = log2
log9/8
 = 5.8849492
to 8 significant figures.

To find the number of thirds in an octave we are looking for the value of y such that (5/4)y = 2. Observe that:
æ
ç
è 		5
4
 ö
÷
ø 		2

 
 = 1.5625,  æ
ç
è 		5
4
 ö
÷
ø 		3

 
 = 1.953125,  æ
ç
è 		5
4
 ö
÷
ø 		4

 
 = 2.4414063
and hence 3 < y < 4 and y » 3.1. Using logs
ylog 5
4
 = log2, y = log2
log5/4
 = 3.1062837
to 8 significant figures.

The comparison of scales is given in the following table:
C D E F G A B C
Equal tempered scale 0 200 400 500 700 900 1100 1200
Pythagorean scale 0
1200*(log9/8)/log 2
204
1200*(log81/64)/log 2
408
498 702 906 1110 1200
Just intonation 0
1200*(log9/8)/log 2
204
1200*(log5/4)/log 2
386
498 702 884 1088 1200