Euler's gradus suavitatis

Euler suggested a way of measuring the cosonance of an interval in terms of a "degree of pleasure" he called (in Latin) gradus suavitatis. It is defined for all rational frequency ratios \(\frac{p}{q}\) as \(G : \mathbb{N} \times \mathbb{N} \times \mathbb{N}\) with \(G(p, q) = 1 + \prod (r_i - 1)\) where \(g_i\) are all prime factors of the least common multiple (\(\mathop{lcm}\)) of \(p\) and \(q\) so that \(\prod r_i = \mathop{lcm}(p, q)\). The larger the value the more dissonant the interval is.

The lower limit for musically relevant intervals is the value for an octave (the doubling of a frequency) \(G(1, 2) = 1 + (2 - 1) = 2\).

The pure fifth with frequency ratio 2:3 has a gradus suavitatis of \(G(2, 3) = 1 + (2-1)(3-1) = 4\).