Rhythm & Consonance

The coincidence of two periodic frequencies where the ratios are 1:1, 2:1, 3:1, etc. produce the unison and unison octaves. These are said to be the intervals of most consonance because the human ear is most sensitive to slight variation. For example, if you start with two sine waves tuned exactly together (a 1:1 ratio) and then slowly detune one, you will hear a slight ‘beating’ effect introduced as the ratio relationship drifts away from the perfectly consonant 1:1.

In the harmonic overtone series, the ratios of the first two partials (the fundamental tone and the 1st overtone) are 1:1 and then 2:1 – intervals of the unison and the unison octave. The ratio of the third partial is 3:2 and produces the interval of a fifth. Helmholtz conducted experiments using two sirens that spun at the rate of 3:2 and found if the frequency was decreased sufficiently, the pitch relationship remained the same but eventually all that is heard is the rhythmic beating of 3 against two.

If we skip over the fourth partial (3:1 – a unison two octaves above the fundamental) to the fifth partial, we find a ratio relationship of 5:4 – a ‘pure’ major third.

Henry Cowell, following from Helmholtz’s experiments, combined these intervals not just harmonically but rhythmically. Thus, a simple C major triad could become a rhythmically interesting composition from the simple materials of three notes regarded as consonantly related to each other and the polyrhymic nature that arises when combining their respective ratios. Example:

An audio example can be heard here: http://www.4shared.com/file/21153861/8befbb2f/Cowell1.html