Tuning perfect fifths

Accomplished amateur and professional violinists do not usually have any trouble tuning the perfect fifths between the strings of the violin. In fact, they usually make as good a job of tuning the fifths as would a piano tuner with years of training and experience in tuning, using advanced techniques. How do musicians know when a perfect fifth is well tuned? What are violinists listening to when they are playing two strings at the same time and tuning them apart by a perfect fifth?

The general answer is consonance. More specifically, it is acoustical consonance. But what is this consonance, and what makes the well tuned perfect fifth between two strings, consonant?

The answer lies in the tone structure of the strings. The tone of the string has a structure in a way analogous to light. White light can be diffracted by a prism or a raindrop into the colours of the rainbow, and similarly, when the colours of the rainbow are added together, they create white light. By analogy, the tone of the tensioned musical string can be 'split up' into a set of partial tones. Just as white light is not normally directly perceived as a mixture of rainbow colours, so the string tone is not normally perceived as a mixture of partial tones, but the partial tones are nonetheless there. In fact, with training, the human ear-brain system can be used as a kind of acoustical 'prism' so that the partial tones can be heard. Even when they are not consciously heard in this way, the partial tones play an important role in creating consonance.

The partial tones all have different musical pitches and are related in a special way. If, for example, we were to represent these pitches in staff notation, then for a note C28, we would write the first 10 partials, numbered from the lowest pitch (partial number 1) upward  as:

Here, the unusual accidental sign on the B in the treble clef stave, is a contemporary notation sign indicating that the 'B flat' is a little flatter than usual. The lowest partial here is at the pitch of the note C28 itself. Thus, the flattened 'B flat' represents the seventh partial. The lower partials have musical intervals between them - the first 10 partials form, more or less, a dominant seventh chord with the beginnings of a tone cluster at the top of the chord. 

The musical intervals between the partials, consecutively, from the bottom upwards, are octave, fifth, fourth, major  third, minor third, etc.

Similarly, the first 6 partials for a note G35, tuned a fifth above the C28, would be: 

This 'chord' produced by the strings' partials is sometimes called the chord of nature. (More on the chord of nature can be opened in a new window by clicking here).

So when the two strings are sounding together, we get, putting the 'chords of nature' for the strings side by side:

The pitches in blue are the same. A little consideration will show that if the interval tuned between the lowest partials of the two strings is a perfect fifth, tuned precisely the same as the interval between the 2nd and 3rd partials in the chord of nature itself, then the coincidence of pitch at the blue coloured partials will be exact. Some partials will coincide exactly whenever the two lowest partials are tuned apart by one of the intervals in the chord of nature itself.

If the tuning between the two lowest partials is slightly inaccurate, however, then the pitches of these coincident partials will differ slightly. If they differ slightly, the discrepancy becomes obvious, and the musical interval sounds "out of tune". It is the slightly different partials mixing together, that gives the musical interval a strong "out of tune" tone quality, in addition to any sense that the notes are simply not at the right pitch. 

To understand in more detail why this is so, we must look more closely at what these individual partials are, and what happens when two slightly different, but coinciding partials, are mixed together.

First, listen to the tone by clicking the button below. It is called a pure tone. A pure tone can be at any pitch. This one happens to be mid-range pitch.

(Pure tones from piano strings also die away, which is called decay). 

Now listen again to the next pure tone below, which changes, as it is tuned up a little in pitch, then down, then back up again to where it started. 

Finally, listen to what happens when these two tones coincide, or are mixed simultaneously.

When the two pure tones are slightly different, they beat. The beating gets faster, the further apart the two tone are. 

Each individual partial that each note in the chord shown above represents, is either a 'pure tone', or somewhat similar to a 'pure tone' (a partial may be a small group of closely related pure tones). 

The musician does not necessarily consciously hear individual partials generated by a musical string, but when two strings are tuned so that some of their partials  are close in pitch, yet still in slight disagreement with each other, the musician can hear that discrepancy between the strings easily. This is because of the 'clues' provided by beating in the tone of the interval. 

By accurately tuning the strings, the discrepancies and beating can be reduced, or removed from the overall sound.  Then when both strings are played together, fewer mis-matched pairs of pure tones are generated, and the tuning sounds more consonant. 

In fact, each chord's partials or "notes" in the "chord of nature", continue upwards, in theory to infinity, and do not stop at partial number 10. In practice, for the fifth illustrated, it would be the blue notes on the G line in the treble clef, that would determine the strongest 'clue' to consonance. Above the 10th partial the partials are known as upper or higher partials. These form something like a continuous tone cluster, and higher still a semitone cluster, and yet higher still a microtone cluster, and so on. 

Really, all the partials together constitute a spectrum of partials, rather than a musical 'chord', but the lower part of the spectrum 'expands' into a recognizable musical 'chord', with a seventh that is slightly flatter than most Western musicians would expect to hear.

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