Music | Investigations in Forms

Given a musical sound $X$ , its pitch is characterized by the frequency $f(X)$ of its sound wave. For example, if we regard the pitch as a musical note, then the pitch of frequency $2\times\mathrm{freq}(X)$ corresponds to what is called an octave higher above $X$ ; so we have had this feeling of consonance in between two frequencies differing by a multiple of $2$ . Now a thesis here is that our minds’ ears are musically interested in the ratios of frequencies, and this is the basis of the usual octave division in western music: if you are going to divide pitches between the note $X$ and its higher octave into $12$ notes, then it might be better to put this notes in a geometric progression, i.e. the $k$ -th note has frequency $\mathrm{freq}(X) \times 2^{\frac{k-1}{12}}$ . In this manner, the ratio between any two consecutive notes is equal to a universal amount, $2^{1/12}$ . Pianos, classical guitars, saxophones, and other instruments all obey this tuning, known as 12 equal temperament (12-ET) tuning.

Yet, in the meantime you may have heard of the consonance quality of the perfect fifth interval, e.g. when you play the notes $G$ and $C$ of an octave, that the note $G$ in an octave has the frequency $\frac{3}{2}\times \mathrm{freq}(C)$ for the note $C$ in the same octave, and it’s said that it is because of this simplest ratio of $\frac{3}{2}$ in between them that the two notes sound so resonant to our ears (I checked with Google Gemini now, it said it too). But is the ratio actually equal to $\frac{3}{2}$ ? Doing the math from the above paragraph, we have that the perfect fifth interval $G$ is $8$ divisions (called semitones officially) higher than $C$ , we have $k=8$ in the above formula and