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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13 thoughts on “An illustration of the non-existence of God

  1. Pingback: An Inside and Out Update – Sibylla Brand G/C/F | Music and Melodeons

  2. Oh my God, you are SUCH a dork! And I’m a dork, too, for loving this post! You’ve managed to go deeper into the temperament issue than I’ve seen before, and I pay attention to such things. The only quibble I have is with your framing question is that you are presuming upon God’s motives. That is, you are assuming that the Almighty actually desires some sort of symmetry (probably not the right geometric term) and the aesthetic it implies. The Almighty comprises infinite possibilities (just go with me on this for a minute) and those possibilities include, for example, the 40:27 ratio. And just as pot smokers find Doritos utterly fascinating, it is encompassed within the infinite capacities of the Almighty to find 40:27 VERY FASCINATING. Well done. Remember, the blog is a form of procrastination … that should encourage you to post frequently.

  3. I think I would construct my argument thusly. We are, apparently, built in the image of God and so logically, by looking at ourselves, we can deduce the characteristics of God. Traditionally all of the good characteristics of human beings, such as kindness, love, compassion, justice, honour and so on, are attributed to God and the fact that humans seem to be as intrinsically nasty as they are intrinsically nice is overlooked or ascribed to Satan/Free Will. However, the desire for elegance is a defining human characteristic. Scientists and artists alike seek it, the moment that comes when you look at something and it is just *right* and cannot be improved. The temperament issue is not elegant. At all. It is impossible to have even a major diatonic scale with all the intervals harmonically pure. Even if you have an equal temperament with loads of notes (there is a particular one with over 50) then it doesn’t help, since a simple chord progression will inadvertently modulate you into a slightly different key. It is a chaotic and a nightmarish scenario.

    The key I think is to exploit it. An old teacher of mine taught me that if you make a mistake and can’t cover it up, then make it a feature. Temperaments should be a feature of music, not something which (as it is at the moment) is suppressed and ignored. I had no idea about any of this until I’d been playing music for 14 years.

  4. Ah well, do Theists have a consistent position anyway, on whether their God is subject to the laws of physics, maths and logic, or whether he/she invented them? A fascinating article regarding temperament, whatever!

  5. In my description of a new type of 31 button Melodeon see the sequence :
    (c’/d’) (e’/f’) (g’/a’) (b’/c”) (d”/e”)
    Make the intervals between the push and pull of the “whole tone” button a just minor tone 10/9 and the interval between the push and pull of the “semitone” buttons a Just “semitone” 16/15.
    Make the interval between the pull note on one button and the push note on the next button a just major tone 9/8. This will give both alternative Just scales, and you will be using the pull Ds when you need to play a D which harmonises with D minor and a push D when you need to play a D to harmonise with G major. All 3 in a row buttons will play a perfect major or minor chord.
    over the whole instrument leave out the “chromatic” button and half buttons so that a (g#/x) becomes a (g#/a) and add a (b/c#) button to the A scale sequence.
    Of course a 3 voice instrument will have to be tuned L M H or a very dry L M M, or M M H.
    Brian Hayden.

  6. Have you come across the Fluid Piano? It’s a piano with sliders that pull each note up to a semitone flat or sharp, thus allowing you to set the instrument to any temperament you want, and even incorporate small pitch bends into your playing.

    Then, there are some people operating under the name of ‘Blues box’ who have devised a pitch bend system for accordions, which uses, as far as I understand, little chambers that resonate at a pitch a whole tone flatter than the natural pitch of each reed, and which can be coupled to the reed by pressing harder on the keys, thus pulling the note flat.
    http://www.bluesbox.biz/

    It strikes me that if the degree of coupling could be controlled by one slider per note, like the fluid piano, rather than the push-the-key-harder device, then you could have a fully temperament-adjustable accordion (albeit one with a lot of little sliders sticking out of it). This would also make life easier for accordionists who want to play Arabic music, who presently have to get an accordion specifically retuned to the scale with quarter-tones that it requires.

    Sadly, I have not the engineering skills to either design or build such a thing, and I’m sure one would need to negotiate around a lot of patents, but I put it out there as a thing that someone, someday, ought to be able to build.

    • This I find very interesting. I was aware of the fluid piano, but hadn’t thought of it’s application to the box. In fact, it might be more practical for the box than the piano, due to the playing style. That is very interesting. If I had the money then I would approach Tom Tonon (he of bluesbox fame) and see what he thinks. Should be possible I would have thought.

      You might be interested in my latest blog post David :)

      • Indeed, that does look fun. Though, as a non-player of bisonoric systems, I may not be the best person to appreciate how it would feel to play. Still, shouldn’t be too difficult to make one – I assume you could just take the reeds out of any old 3-row melodeon and slot a new set in in the new arrangement?

  7. Pingback: Harmonetta Bass | Music and Melodeons

  8. Pingback: 19 Tone Equal Temperament | Music and Melodeons

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