brian capleton


the great circle of fifths 2




The Great Circle is not always shown with all the internal connecting lines, as it is here. When it is, it represents all musical intervals in the Western scale system, except that here, we haven’t included double sharps and double flats. The only way the octave is represented in the circle, is by implication.

Travelling around the circumference, or across the lines, we could pass through any sequence of notes, and the function of the circle remains the same, whichever inversion of each interval we are using. So, for example, starting at the 12 o'clock position, and going clockwise, we go from C to G. It doesn't matter whether we consider this to be a rising perfect fifth, or its inversion, a falling perfect fourth. The circle encapsulates the musical relationship between the two notes in a way that is independent of where they are in the compass.

Contrary to popular belief, the Great Circle as a whole, without any additional information, isn't a picture or representation of a set of 12 notes and their relationships to each other, in any way that relates to what we might actually hear. There is nothing in the circle as it stands to say what a perfect fifth actually is, other than as five steps in the alphabet.

The actuality is that a perfect fifth is a musical interval of a definite kind. What makes it what it is, is the relative tuning between the two notes. This isn't just a matter of musical pitch, because musical pitch itself is only one characteristic of a musical note. Musical notes have timbre as well, and actually, though many people may not realise it, the two things are connected. What gives a note its timbre includes what gives it the quality that we perceive as musical pitch.

In general, musical notes are not simple, indivisible things, but are actually complex recipes based on sound ingredients at many different frequencies, each with their own musical pitch, if we were to hear them separately. Modern acoustics calls these ingredients partials, and when richly-toned notes such as those from musical strings or pipes, seem to us to have a definite musical pitch, then it is because the frequencies or wavelengths of these partials fall into a set called the harmonic series, or at least something approximating to it.

It is this same pattern of partials that also gives rise to the fact that the timbre of a musical interval itself, changes according to the relative tuning between the two notes. The tuning of an interval is not just a matter of relative musical pitches. It's also a matter of timbre.

So in the very badly tuned "perfect" fifth (the wolf interval in the other Great Circle article) here:

It's not just a question of bad pitch intonation. What really shows up in the timbre of the interval is this:

And the sound ingredients or partials this sound is made from are:



And this:

This "beating", as it is called, arises from the fact that both notes have sound recipes in a harmonic series, but are not themselves tuned in the same way as the perfect fifth that occurs between the 2nd and 3rd partials in the harmonic series. The discrepency between one and the other causes a misalignment between the partials in the harmonic series of one note and the partials in the harmonic series of the other.

Basically, to tune the perfect fifth so that it sounds optimally good, would require tuning it the same as the perfect fifth between the 2nd and 3rd partials in the harmonic series. These two partials, or harmonics, have frequencies in the ratio 2:3. A consequence that follows from this ratio between these two harmonics is also that two pipe lengths or (otherwise identical) string lengths (at the same tension) in musical instruments, in the same ratio, also produce a perfect fifth between their two notes.

The whole business of ratios being associated with musical intervals, whether as pipe lengths, string lengths, soundwave lengths, or frequencies, comes from the ratios inherent in the harmonic series. Long before our modern knowledge of the harmonic series, the ratios were known about through a mixture of practical experience with strings and pipes, and the construction of theory to go with the practice.

The Great Circle of Fifths is part of that theory. What it is, is a mathematical tool, in much the same way that an equation or a matrix are mathematical tools. Without additional information, the circle as it stands is rather like saying

a + b = c

That's all very well, and it may be true, in some situation or another, but without quantities to fill in the placeholders, it's actually meaningless.

It's the same with the Great Circle of fifths. All the segments around the circumference, and all the internal lines, are actually placeholders. They need to be filled in, with ratios, in order for the circle to be fully meaningful. But just as in the simple equation

a + b = c

you can't just put any old numbers in. What you put in, has to be true. And just as in the simple equation, if you have some of the values, you can work out the other ones.

If you look at it like this, you can see that the Great Circle is not only a network, but is like a multidimensional equation.

Just as you can’t say that

2 + 2 = 5

so you also can’t say that all the perfect fifths round the circumference have the ratio 3:2. This is because the rules of the circle are that whatever ratios you put in, the lines connecting notes must all be arithmetically correct ratios, and any complete circulation around the circumference back to the starting note, must either be a ratio that is a multiple of 2, or it must be 1. This is because the ratio of an octave is 2:1, or a unison is 1:1.

The ratios that the harmonic series contains can't just be injected into the Circle, without breaking the rules. So always, at least some of them have to be compromised.

Remember we were talking about how the partial ingredients that facilitate the quality we hear as musical pitch, can also affect the timbre of an interval by introducing beating? Well this means that this compromise amounts to deliberately introducing beating into the timbre of some intervals, in a controlled way, rather than ending up with something sounding like this:

What piano tuners do, for example, is to put only 1/12 of the "mistuning" in this interval in every perfect fifth, which means the rules of the Circle are still obeyed. If you tune 11 of the perfect fifths in the ratio 3:2, as Pythagoras would have liked, the rules of the Circle mean that the remaining fifth would sound like the example above.

Controlled "mistunings" away from the ratios of the harmonic series are what tempered intervals are, and a complete system for all the intervals, applied to the Circle, so that its rules are obeyed, is a temperament. The Circle is then a temperament circle. The modern, Western tuning system is Equal Temperament, in which all the fifths around the circumference are tempered in the way I just mentioned, which results in the ratios for all the internal semitone lines being the same.

The result of Equal Temperament is that the enharmonics all become equivalent names for the same note, even though correct musical grammar continues to distinguish between them as though the intervals are all harmonic rather than tempered. That's why on the keyboard there is only one key, for example, for both A-sharp and B-flat.

The first reason I've avoided the real nitty gritty of the math in this explanation is because it doesn't really enlighten anyone with what really goes on in piano tuning practice by expert tuners, and makes it look as though it can all be understood with arithmetic, which it can't.

The second reason, which is more important, is that going into the detail of how musical intervals fit together as ratios, is useful for understanding the general behavioral principles of musical intervals in tuning them, but can give the false impression that temperament theory, or this science of harmonics, as it was once called, is a scientifically accurate description of how musical interval tuning works in practice.

It's not really, and this is reflected in the background context of the science. The science of harmonics was once an integral part of astronomy in Europe, because it was believed that the universe was constructed, and moved, according to musical principles. That's why the science of harmonics, which was just called music at Oxford in the Middle Ages, was one of the four subjects of the quadrivium, together with arithmetic, geometry, and astronomy.

This model of the universe (the Ptolemaic model) was based on the idea that the Earth is at the centre of the solar system, and the planets and stars move around the Earth on perfect spheres and circles. The underlying idea was that God, being Perfect, used the principle of perfection in the construction of the universe.

You can, to some extent, appear to explain the motions of the heavens in terms of harmonic ratios, which is why the old system persisted until observations got better to the point where it was no longer tenable.

For a long time the effort was made to get more into the nitty-gritty of ratios and circles in the attempt to fit the theory to the observations. But eventually, the whole theory had to be thrown out and replaced with Newtonian mechanics. The planets don't really move in perfect circles, they move in non-circular ellipses. And they don't orbit the Earth. They orbit the Sun.

In a not dissimilar way, temperament theory, or the science of harmonics, although it still has its uses, has been superseded by modern acoustics. Strings and pipes seldom produce a perfect harmonic series, and most musical intervals aren't things you can mathematically encapsulate in a single arithmetical ratio. You can't really, accurately describe what goes on in the tuning of musical intervals, just by describing them in that way.

The theory of temperament or the science of harmonics goes to great lengths to get into every nitty-gritty of musical intervals as described by the theory, but it's all theory, or pure mathematics, without what scientists call observation. If we want a deep and detailed understanding then it often bears little relation to how nature actually behaves, or what we experience in tuning musical intervals. In that, it's rather like continuing to insist that the heavens move in perfect spheres and circles.

© Brian Capleton 2016
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