The Pythagorean and Just Intonation systems are derived from the ratios and or mathematical formulas in relation to a fundamental tone. When the diatonic scales are generated for the above systems the intervals are not uniform throughout. For example if we’re in the Key of C the major second would be a D with a 9/8 ratio and the major sixth from Just Intonation would have the ratio of 5/3. If we then wanted to build a major triad for the D we would multiply 9/8 X 3/2 and get the ratio of 27/16 which would be a different note then the 5/3 sixth from the C generated from the C’s overtone series. The above truth makes harmonizing on an instrument such as the piano or guitar very difficult.

First let’s compare ratios of the different tuning systems below:

C Scale	Interval	ET Ratio	Pythagorean Ratio	JI ratio
C	Fundamental	1/1	1/1	1/1
C#	m2	7893/7450	256/243	16/15
D	M2	5252/4679	9/8	9/8
D#	m3	10754/9043	32/27	6/5
E	M3	6064/4813	81/64	5/4
F	P4	6793/5089	4/3	4/3
F#	A4	11482/8119	729/512	45/32
G	P5	10178/6793	3/2	3/2
G#	m6	4813/3032	128/81	8/5
A	M6	9043/5377	27/16	5/3
A#	m7	17189/9647	16/9	9/5
B	M7	17843/9452	243/128	15/8
C	Octave	2/1	2/1	2/1

 
Let’s look at a chart measured in cents of the differences in interval size for the three systems. As you can see there is a great discrepancy in the size of a number of the intervals.

C Scale	Interval	Equal Temperament	Pythagorean	Just Intonation
C	Fundamental	0	0	0
C#	m2	100	114	111.7
D	M2	200	204	203.9
D#	m3	300	294	315.6
E	M3	400	408	386.3
F	P4	500	498	498
F#	A4	600	612	590.2
G	P5	700	702	702
G#	m6	800	816	813.7
A	M6	900	906	884.4
A#	m7	1000	996	1017.6
B	M7	1100	1110	1088.3
C	Octave	1200	1200	1200

 
Note the discrepancies in Cent values.