Most of known musical systems contain relations 2:1 (octave) and 3:2 (fifth).
The ratios 1/1, 2/1, 3/2, 4/3 contain only two prime characteristic, 2 and 3.
Such ratios were used from the age of the Ancients and they prevailed still in the
age of early polyphony (9.-11. century).
In this era it probably was not possible to recognize the prime characteristic 5.
(Pythagoras and Aristotle assumed thirds are dissonant.)
Intervals of thirds and sexts have been gradually asserted (from the 12.century).
Theoretically were acknowledged as consonant in work of J.P.Rameau (18. century).
Name | Ratio | 2^i | 3^j | 5^k |
octave | 2/1 | 2^1 |
||
fifth | 3/2 | 2^-1 | 3^1 | |
fourth | 4/3 | 2^2 | 3^-1 | |
major third | 5/4 | 2^-2 | 5^1 | |
minor sext | 8/5 | 2^3 | 5^-1 | |
minor third | 6/5 | 2^1 | 3^1 | 5^-1 |
major sext | 5/3 | 3^-1 | 5^1 |
At present all the ratios assumed to be consonant fit the formula:
i = 2^c * p/q, where p,q ε {1,3,5} and c ε {-2,-1,0,1,2,3} .