Acoustical consonance

The sound of a musical interval on a stringed instrument, and on many other instruments, is typically a complex recipe (usually a changing recipe) of many different acoustical ingredients called partials. A partial can be "steady" and unfluctuating, or it can be fluctuating, often producing a fluctuation pattern called beating. The amount of fluctuation or beating in the recipe, caused by the partials in the tone recipe, depends amongst other things, on the tuning between the two notes. In general, there will always be some partials in the recipe that beat or fluctuate, but an interval will be acoustically consonant when the two notes are tuned such that the amount of fluctuation in the recipe is minimised. This is also known as tuning the interval to Just Intonation.

If the interval between the two notes is increased or decreased from its acoustically consonant size, the amount of fluctuation or beating steadily increases with this change. This happens until the interval size starts to become another musical interval altogether, when the fluctuations or beating then begin to decrease. For example, if we increase the size of a major third, eventually it will become a perfect fourth. (An explanation of musical intervals is here). When an interval is tuned, say, as an acoustically consonant major third, fluctuation will be at a minimum. Somewhere between a major third and a perfect fourth, the fluctuations in the interval will be at a maximum. When the interval is tuned to an acoustically consonant perfect fourth, fluctuation will be at a minimum again.

If we were to gradually "stretch" the musical distance between two notes from nothing (a unison) to an octave, we would pass through all the recognised musical intervals on the way. At each one, would occur a minimum of fluctuations in the overall tone.

This effect only occurs with musical intervals between tones that have recipes rich in partials. The tones of instruments with tensioned strings, including the piano and the guitar, are rich in partials.

Partials in detail :

You will need Windows Media Player installed or another MP3 player.

Broadband is recommended

The root cause of acoustical consonance is the behaviour of partials. Partials themselves can be "broken down" into still simpler components called pure tones. It is mostly the natural behaviour of pure tones that causes the fluctuations and beats in partials. When an interval is close to acoustical consonance, many of the partials in the interval's tone recipe contain pure tones that are close in frequency (this can be thought of as close in pitch). When pure tones are close in frequency they produce the fluctuation called beating. As the difference in their frequency increases, the beating fluctuation becomes faster. 

Click on the button to hear a constant pure tone.

Click on this button to hear a pure tone that starts the same pitch and frequency as the one above, but then rises above it in pitch slightly, then falls below it, and then rises back up to where it started.

Click on this button to hear what happens when the two pure tones above are mixed as one partial.

The beating is slowest when the two pure tones are the same, at the beginning, the middle, and the end. The beating is fastest when there is maximum difference between the two pure tones.

Out of acoustical consonance, the tone recipe of the interval contains much beating of the kind heard in the last partial example above. In acoustical consonance, the beating is reduced as far as it can be.