Dissonance, sound spectrum and musical scale for ancient idiophones and aerophones

We do not know how the earliest musical instruments—such as idiophones and aerophones—were played, but their acoustic properties can provide valuable clues. As a first step, we present here the concept of dissonance curves for a sound of a given spectrum. These curves show the relative dissonance that results for all intervals of a given musical instrument. This idea leads to the association of spectra and scales, which are related because the dissonance curve has minima in the intervals that define the scale. A computational method for calculating dissonance curves is presented and several examples of its use in practical cases, both for Western and Eastern musical instruments, are given and interpreted. These results allow us to explain from a physical point of view the reason of existence of well-known modern 12-note scales, as well as some uncommon but documented scales for various instruments in early musical history.