Agreeable consonances are pairs of tones which strike the ear with a certain regularity; this regularity consists in the fact that the pulses delivered by the two tones, in the same interval of time, shall be commensurable in number.
No music system, no matter how ingenious, will have any lasting validity if it does not take into account the way it is first perceived by the ear and then analysed by the brain.
It comes as no surprise to discover that sound is registered in a similiarly proportional manner to how it is produced. I talked above about how instrumental tone can be analysed in terms of its constituent partials, its pattern of overtones. This is called Fourier analysis, and is basically the same process as that at work in the cochlea of the human ear; for the nerve cells on the basilar membrane within the cochlea act as harmonic resonators, registering frequencies proportionally.
The perceived consonance of two tones played together is therefore dependent upon the relative simplicity of the ratio between the two frequencies - and, by implication, between the two string lengths which produced them . So if we hear two vibrating strings of equal mass and under equal tension, one of them half the length of the other, they will be experienced as more harmonious than two strings, one being four fifths the length of the other.
This natural preference for simple numerical proportion in pitch can also be demonstrated from the point of view of rhythm. Below about 26Hz vibration ceases to be perceived as tone but rather as a periodic pulse, as rhythm. It would seem logical therefore that the principles of proportion in tone are reflected in those of rhythm. Following the previous example it is clearly easier to beat two simultaneous pulses in a relation of 1 against 2 than to maintain a cross-rhythm of 4 against 5. Consonance derives from the perception of order.