To find the number of thirds in an octave we are looking for the value of $y$ such that ${\left(\frac{5}{4}\right)}^y=2$. Observe that:

${\displaystyle {\left(\frac{5}{4}\right)}^2=1.5625,\hspace{0.5em}{\left(\frac{5}{4}\right)}^3=1.953125,\hspace{0.5em}{\left(\frac{5}{4}\right)}^4=2.4414063}$

and hence $3<y<4$ and $y\approx 3.1$. Using logs

${\displaystyle y\log \frac{5}{4}=\log 2,\hspace{0.5em}y=\frac{\log 2}{\log 5/4}=3.1062837}$

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