An illustration of the non-existence of God

Somehow this blog has got sidelined for a while, which is sad. Being unemployed is getting me down and I am tending to not do any of the things that I should.

However, that changes today, with this post. I was slightly hesitant about writing about this topic, because this knowledge caused a bit of a musical crisis for me earlier this year and I wouldn’t want to wish that on anyone. However, under the assumption that what you don’t know hurts you more than what you do know, the time has come to talk about temperaments.

The universe is broken. Irremediably. And in order to explain why I have to use Maths. Don’t worry, it won’t last long.

You can split any sound into components at different frequencies, called ‘overtones’. Many sounds, especially musical ones, are ‘harmonic’, which means that their overtones are close to what is called the ‘harmonic series’. What makes a sound distinctive is the magnitude of these overtones.

Pythagoras, of triangle fame, is supposed to have noted that if you took a string and shortened it to half the original length, the resulting plucked note sounded an octave above the original. If you shortened the string to a third of its length, the note resulting was between the octaves and sounded pleasing and sweet. We call this a perfect fifth. If you shorten the string to a quarter of the length then you get the octave above, if you shorten it to a fifth then you get a major third. This is the harmonic series and those lengths are the wavelengths of the harmonics of the string divided by two. So the harmonic series is an intrinsic part of nature.

It turns out that all of the intervals we use in western music can be represented by whole ratios based on the harmonic series and it is these whole ratios which result in intervals sounding clear and pleasant. If a note is out of tune, it means that the interval is not a whole ratio and as such a ‘beat’ is set up, which can be quite unpleasant. If you’ve ever tried tuning a mandolin (or a Charango) then you will know all about beats.

So far, so good. Unfortunately it was soon found that there was a problem. If we start with C and multiply the frequency by a ratio of 3:2, we get to G, a perfect fifth above C. If we do the same to G then we get to D. If we continue on this path then we get to A, E, B, F#, C#, G#, D#, A#, F and C. So the ‘circle of fifths’ leads us back where we started (albeit seven octaves higher). Or does it?

In mathematical notation, we can describe this as the fundamental frequency multiplied by the number (3/2)^12, which means that we multiplied by 3/2 twelve times. This brings us back to C. This C that we have obtained should be related to our original note by a whole ratio, specifically by a power of two. Our original factor of (3/2)^12 comes out to approximately 129.75, which try as we might, isn’t exactly divisible by two. It is very close to the factor 128, which would be exactly seven octaves above our original note, but the fact that it isn’t exactly 128 means that the note at the end of the circle of fifths is not C, but is slightly sharp.

If we bring our new note down seven octaves, then the ratio between it and the original C is around 1.014:1. That may not seem like very much, but it is significant. That ratio is called the ‘Pythagorean Comma’.

So when you go round the circle of fifths, you do not get back where you started from. It turns out that if you go round a circle of perfect fourths (ratio 4:3), you also end up with a comma. In fact none of the pleasing intervals which are used in modern music will lead you back in the way that you expect, it is a fact of mathematics. Even the major third, going from C to E to G# back to C will leave you short by a tiny interval known as the ‘Diesis’.

Now, this is an interesting piece of mathematical trickery, but what does it mean for music? Well, it is important when we are constructing a scale (and believe me, there are hundreds of ways of doing so). One of the earliest scales known is the Pythagorean scale. This uses the harmonically pure circle of fifths; in this scale, the relationship between all the notes in the circle of fifths is harmonically pure except for the last one, which is unpleasantly flat by a Pythagorean comma (often known as the ‘wolf interval’, not to be confused with the ‘wolf note’, which is annoyingly something completely different). The wolf interval was placed between two notes that were rarely used in that particular scale, meaning that most of the time it was not an issue. However, most of the other intervals, such as the major third, are dissonant and unpleasant, meaning that music based on the Pythagorean scale tends to have fifths as the most common interval. It is thought that some music written before the 15th century in Western Europe was written for the Pythagorean scale.

If we want play in one particular key then we have another option. Instead of using the circle of fifths, we can construct our diatonic major scale using whole ratios of our tonic (i.e. a fifth is 3:2, our major third is 5:4 and so on). Using this system, at first everything is pretty much OK. Our fifth sounds nice, our third sounds nice, our sixth sounds nice. However, if we then start to use intervals not involving our tonic, we run into a problem. Say we are in C major. Our major second, D, is a ratio of 9:8 to the tonic, our sixth, A, is 5:3. The ratio between these two notes should be a perfect fifth, or 3:2. However, we actually get 40:27, meaning that this interval is not pure (the ratio between the two is 81:80, otherwise known as the ‘Syntonic Comma’). It turns out that even in one diatonic scale, it is impossible to have every interval harmonically pure. However, using whole ratios to construct a diatonic scale does result in being able to play in one key with thirds that sound nice, which is more than the Pythagorean scale gives you. Unfortunately if you try playing in a different key using the same scale then it sounds atrocious. This is called “Just Tuning” and was used some time in the 17th century. A variation of Just is still used for Cajun Accordions.

However, most composers want music to be in more than one key; they want to modulate. In addition, they want to use interesting intervals and notes not in the diatonic scale. Just Tuning is not suitable for chromatic music like that. So a new system was invented called “Quarter Comma Meantone”. Like the Pythagorean scale, it was based on the circle of fifths. It took the Pythagorean comma (which we have already met), divided it by four and flattened (“tempered”) every fifth by that amount. The fifths were therefore not pure, but were close. Obviously there was still a wolf interval, where the circle of fifths fails to meet, this time even worse than the Pythagorean one, because it was three times wider than the Pythagorean comma. However, for a handful of keys, the intervals were relatively pure, although for other keys they were impossibly unpleasant. The interesting thing about this temperament was that every key had different intervals, so every key had a very different character. Meantone lasted a long time, but was eventually supplanted by what we use now, which is ‘Equal Temperament’.

In Equal Temperament (‘ET’), the gap between each note of the chromatic scale is the same – the octave is split into 12. In terms of the circle of fifths, the Pythagorean Comma is split into 12 and each fifth is tempered by that amount. This seems at first glance an entirely straightforward and logical way of proceeding, and this is certainly the way that it is commonly presented. With this system every interval in every key is constant, you can modulate freely and there are no complications. However, there are disadvantages as well. Specifically, in ET no interval other than the octave is harmonically pure. Some come very close (the fifth, for example), but the major third is extremely sharp. This is audible to the listener, but for two reasons it is seldom commented upon. The first is that intonation problems are often masked by the instrument or the player, either through human error or through techniques such as vibrato on a fiddle or the musette on a melodeon. The second is because so much of the music that we hear is in ET, we are conditioned to accept it and any other temperament sounds ‘wrong’ to our ears.

So why do I say that it gave me a musical crisis and why the title of the post? Well, for all of my life I have played fixed pitch instruments, specifically the piano and now the melodeon. The closest to a variable pitch instrument that I have ever come to is the Clarinet and even then altering the tuning of notes in an utterly controlled way is far from easy. When I learned about temperaments not that long ago, it made me realise that the instruments that I play are helping to perpetuate ET – they are part of the problem. And indeed, it is difficult to see how they could not do so. Modern music to a certain extent demands ET and modern keyboard instruments definitely do. So I realised that a whole side of music was lost to me. Now that I know about the temperament problem I can hear it – there are a handful of chord changes on the piano where the sharp third sticks out like a sore thumb. More problems with training your ears… I know that it is just one of the limitations of the instrument, but the question is whether it is one that I can live with.

The title of this post is a reference to a conversation I had with a couple of physicist friends of mine. I posited that a universe where the circle of fifths did not close could not have a creator, as any creator would not have such a fundamental part of human existence – music – so fundamentally broken. However, it was argued that it was impossible to have a rational universe where this was not the case, as for it to be otherwise, arithmetic would break down. So it appears that my watertight argument for the non-existence of God has a leak. But to be honest, it probably has a greater leak than that, for the existence of the Pythagorean comma is not really a problem with music, it is instead an opportunity for musicians. Using temperament to introduce consonance and dissonance into a work gives a whole new slant on music making. Microtonal music is often associated with dissonance, but in reality it is as much to do with consonance, making intervals beautifully pure in ways that ET cannot manage. It just saddens me that as a piano and melodeon player, it is a part of music that in all likelihood is barred from me for life.

p.s. In a previous post, I mentioned that the thirds of the chords on the bass end of a melodeon are often tuned slightly flat. It is easy to see why this is. The melodeon is usually tuned to ET, but when this is done, the chords on the left hand sound rather unpleasant. Flattening the third brings it closer to a just chord, and the musette in the right hand takes care of any discrepancy between ends.

p.p.s. I mentioned Cajun tuning earlier, but actually many different temperaments are used for accordions and melodeons over the world. I have even heard of anglo concertinas being tuned to quarter comma, which would be a fascinating experiment.

p.p.p.s. You didn’t think that I would waffle on for 2000 words without a video at the end did you? There are many videos on the web which deal with just and ET major thirds, this is just one. And if you are interested in seeing a melodeon tuned to Just then click here for an example made by Olav Bergflodt and played by Stein Ove Lognseth. If you are really keen on this sort of thing then click here…

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