Continued ...

Tuning a whole scale

We must now consider what happens when we tune more than one perfect fifth, or tune a whole scale, bearing in mind that we want our musical intervals to be acoustically consonant, as described on the previous page. When an interval is a long way from being acoustically consonant, it is acoustically dissonant.

If we had a keyboard instrument with one string per note, we would need 13 strings for a complete chromatic scale, including the two notes an octave apart. How could we tune this whole scale, starting from one note tuned to a tuning fork, and then tuning the rest of the scale by tuning only fifths?

A little thought will show that you can't do it. Rather, we would have to 'invert' some of the fifths, and tune fourths (inverted fifths), instead. In other words, we could use a sequence like:

We could finish by tuning the upper C, by tuning an octave up from the lower C.

Tuning the fourths could be done by taking the note we are tuning from, tuning an octave above or below it, and then tuning a fifth from this, to the required note. That way, we would tune by using fifths only (and octaves, of course). But it is possible to tune the fourths directly, using consonance, in the same way as the fifths. The partials that coincide in the 'chords of nature' will however be different for the fourths. Try writing the 'chords' for two notes a fourth apart - which are the lowest partials that coincide? [Answer is here]

The Pythagorean comma

Either way, we would end up with the same result. If we did a good job, all the fifths we tuned would be consonant and all the fourths we tuned would be consonant. However, let's now consider tuning the upper C by tuning it an octave above the lower C. Having done this, we complete the scale. So is the scale now perfectly in tune?

We did not actually tune the interval from the E flat to G sharp. Would this just 'turn out' to be an acoustically consonant interval ? 

The answer, perhaps surprisingly, is 'no'. In musical grammar, the interval is not a fourth, but an augmented third, even though on the keyboard it occurs between the two black keys we might otherwise call E flat and A flat, which would be, grammatically, a perfect fourth. The interval would in fact sound grossly mistuned, and dissonant. Compared to the E flat, the G sharp would sound far too flat to be a perfect fourth above the E flat.

Following correct musical grammar, if we had tuned an A flat rather than a G sharp, say by tuning down from E flat to A flat below our scale area, and then tuning up an octave from the A flat to put an A flat in our scale area, we would find that the interval between the same two black notes, now called E flat and A flat, was indeed a consonant perfect fourth. However, we would then find that the interval between the black keys C sharp and A flat in the scale area, was dissonant. Whatever we do, if any octaves we tune are in tune, there will always be one bad interval between keys we would normally expect to be a fifth apart, or one bad interval between keys we would normally expect to be a fourth apart.

These dissonant intervals are musically unusable, and are called wolf intervals.

Why doesn't the scale just work out?

Many text books answer this question by proving mathematically, that it doesn't work out. If you want the mathematical argument, you can branch off now and find it here. But in fact, it's not strictly necessary in order to understand why. All it requires, is to do what is now often called 'thinking outside the box'.

If you are surprised to find that a scale tuned by tuning a sequence of good, acoustically consonant, perfect fifths or fourths, cannot be achieved, it is only because you were thinking inside a box. The thinking 'box' in any situation, is a set of unquestioned assumptions. The assumption here, was that there is such a naturally occurring thing as a musical scale, made entirely of acoustically consonant musical intervals. Probably this idea was reinforced by seeing  keyboard instruments, which have a scale of notes, that seem to be in tune.

The kind of acoustical consonance on which, in our example, violinists rely in order to fine tune perfect fifths, arises in nature from the natural acoustical behaviour of musical strings. It is the natural phenomena that creates the 'chord of nature'. However, the truth is that the musical scale, is a 'man-made' artifact'. There is actually no 'philosophical' reason to assume in the first instance, that what we have invented and called the musical scale, should be comprised entirely of naturally, acoustically consonant, intervals. The fact is, that there is no arrangement of these intervals that can be 'put together', to make either a diatonic or a chromatic scale, in which all the intervals between the scale's notes, are acoustically consonant.

The difference between the size of the interval E flat up to G sharp in our sequence, and a consonant perfect fourth, is a small interval known as Pythagorean comma. Similarly, if we tuned up an octave from the E flat, the difference between the interval G sharp up to the new E flat, and a consonant perfect fifth, is also the Pythagorean comma.

The Pythagorean comma is roughly a quarter of a semitone.

Equal Temperament

On a properly tuned piano, none of the intervals apart from octaves, are acoustically consonant. The scale used, is called the equally tempered scale. Like all scales, this is a man-made invention, or artifact. In principle, the octave is divided into 12 equally sized semitones. This leaves the thirds, sixths, fourths and fifths, etc., all tuned to a condition that is not perfectly, acoustically consonant.  

This is why some people believe the piano is 'detuned' or even 'out of tune'. It is not 'detuned' by an expert tuner - it is tempered. Any diatonic (8 note octave) or chromatic (13 note octave) scale, will either be deliberately and skillfully tempered, or it will contain dissonant or out of tune intervals, or it will be arbitrarily or 'randomly' tempered.