MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I've recently been studying Euler's theories on music, and I came across Euler's concept of gradus suavitatis or 'degree of pleasure' of a rational number representing the ratio of two tones. (I found this on

The formula is $G(p/q)=1+\Sigma e_i (p_i-1)$ where $p,q$ are relatively prime, the $p_i$ are the prime factors of $pq$ and $e_i$ is the multiplicity of $p_i$.

This formula seemed familiar. Is this formula used in number theory, and, if so, what is its mathematical significance?

share|cite|improve this question
$\sum e_ip_i$ is tabulated and discussed at $\sum e_i(p_i-1)$ seems to be, also – Gerry Myerson Mar 16 at 22:21
up vote 6 down vote accepted

This is not directly an answer to your question (graph theoretic relation, rather than number theoretic), but it's too long for a comment.

Euler's formula is a special case of a disharmonicity function

$$D(x)=\sum_i |e_i| g(p_i)$$

with $g(p_i)>0$ a function of the prime factors $p_i$ of a rational number $x=p_1^{e_1} p_2^{e_2}\cdots $ (each with positive or negative multiplicity $e_i$). Euler's disharmonicity has $g(p_i)=p_i-1$ (and adds unity to the sum over $i$). An alternative disharmonicity, due to Barlow, takes $g(p_i)=(2/p_i)(p_i-1)^2$.

Disharmonicity functions can be used to define the notion of a "harmonic distance" of two rational numbers, and to formulate the problem of the harmonization of a musical scale as a problem in graph theory.

See Musical scale rationalization – a graph-theoretic approach, by Albert Gräf.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.