The Attraction of the Almost Perfect

Some more musical theory, because the last one went down so well 🙂

Musical harmony is all about notes played together. When two notes with frequencies in particular ratios are played together, the results are pleasing. So play two notes, one double the frequency of the other, and you get notes an octave apart. Notes an octave apart sound so familiar we treat them as the same note (they might both be an A, for example). Three times the frequency and you get notes that are an octave and a bit apart. If we take the high note and half its frequency, then it brings it down by an octave, so we can see the ‘bit’ on its own. The ‘bit’ is called a ‘perfect fifth’.

(Bear with me, this is going somewhere)

Writing this in numbers we can say that 2x frequency is an octave 3/2x (i.e. 1.5x) is a perfect fifth. The next interval is the gap between that fifth and the next octave up, which is 4/3x (i.e. 1.333…x) the frequency. And in fact, the intervals build up in this way. So the “major third” (the interval between C and E) is 5/4, and the minor third 6/5.

Now we can do some maths. For example, to find the difference between one interval and another, we use division. We did this above, we took the 3x frequency, and divided it by an octave (2x), to get the fifth (3/2 x). So the difference between a fourth and a fifth is 3/2 divided by 4/3, which is 9/8, this is called a “whole tone” or just a “tone”. Division allows us to subtract intervals, and multiplication allows us to add them. So to add a fourth to a fifth we do 3/2 x 4/3. The threes cancel, giving us 4/2, or 2/1. So add a fourth onto a fifth, and you get an octave. Fun eh?

(No, really, there’ll be a moral eventually)

So here the problem creeps in. Any pianist will know that if you put 4 minor thirds stacked on top of one another (a so called “diminished 7 chord”), you’ll span a whole octave. So 6/5 (the minor third) to the power 4 gives 2 (the octave). Only it doesn’t. Well, not exactly. It gives 2.0736. Slightly out. Similarly 3 major thirds stacked (an “augmented chord”) gives you an octave, but 5/4 to the power 3 actually gives 1.953125. Again, slightly out.

This has been a constant problem for music and musicians. If you pick one interval and make it sound perfect (the major third, for example), the other intervals will be slightly out (the octave will be too small). Pick a different one (the minor third), and all the other intervals will change (the octave is now too large). There is mathematically no way to add multiple copies of one beautiful interval and arrive exactly at a larger interval. I’ve read some things that claim that harmonies are a problem when you have more than one musical key (you can play in C major or G major, for example). Well, that’s true. But it misses the point. Even in one key, you can’t make your intervals add up.

(And, onto the tenuous point)

So the solution we’ve settled on, as a musical culture, is to compromise every interval. To try to distribute out the errors, so that they are as small as possible. Mathematically this is called the ‘equal temperament tuning’ (ET) and in theory this does keep pure octaves. But even that isn’t what is really used. In a regular piano tuning, for example, the octaves are slightly stretched out, because strict ET on a piano sounds flat for the high notes and sharp for the low notes (because our ear is drawn to the other compromised harmonies that don’t quite sound right). The result is a mixed bag, where everything makes sense compared to its neighbours, but nothing is quite accurate. And when you play the result as a whole, there are no pure ratios sounding. And this means you loose the full force of waves reinforcing one another. You lose the power and purity of the harmonies. The quality of natural harmony, the real world of vibrations and frequencies, is lost against the need to have scales, predictable intervals, key signatures, and regularity.

And that, dear patient readers, is what the frameworks of religious dogma feel like to me: stretching out the harmonies and natural beauty of reality in countless little compromises to have it fit in a neat, structured, controllable, but ultimately discordant scheme. You can play any melody you like, but nothing quite gels together into a convincing whole.

(*bow* thank-you, thank-you very much, I’m here all week, try the veal)

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8 Comments

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8 responses to “The Attraction of the Almost Perfect

  1. Boz

    I have no idea about music. but i like maths.

    so,

    you said: “So 6/5 (the minor third) to the power 4 gives 2 (the octave). Only it doesn’t. Well, not exactly. It gives 2.0736. Slightly out.”

    can you change 6/5 to 6.2844/5.2844? that means that (6.2844/5.2844)^4 = 2

    ?

  2. Ian

    Yes, that is basically what happens in Equal Temperament tuning. You make the intervals slightly out, so that when stacked they give the octave. But if you think about wobbling strings, it isn’t possible to have 6.2844 wobbles in a string – only integers (because the ends are fixed). So you do lose those beautiful overtones. And that is the bind – either have impure intervals, or stacked intervals that don’t add up.

  3. A few thoughts:

    * It sounds like “leap year” for harmonies.

    * When you made division equal to subtraction and multiplication equal to addition, I thought you were going to build a revolutionary new logarithmic morality to match your theory of a Meta-God. 😀

    * Your last paragraph was fantastic ! But the veal sucked. But don’t tell the cook, I haven’t got my desert yet.

    * Dude, you needed a diagram or two to illustrate your dance throught the mathematics of music theory — jeez, be more kind. I wonder how many readers gave up half-way down due to a run-on-mathematical-sentence induced headache.

    * Seriously, good fun! I did not know that about music and hope to learn more as I have just started studying the blues as a future project in order to jam with a friend. So much to learn. But this issue of mathematical symmetry and human hearing has puzzled me. I have always expected there to be a disconnect. You confirmed it.

    * Indeed most religions often do what you accuse them of, but it seems some forms of Buddhism say, “Stop trying to twist everything neatly into your artificial ideal categories — watch what is and learn to rest in that.” They seem to be speaking to your point. OM !

    * A Funny FINAL thought just came to me: In American English we say, “MathematicS” but we also say “Math” (no “S”). Though you Brits sound funny saying “MathS”, you are at least consistent. But is this not the same silly compromise you accuse “religions” of? For you said:

    “…fit in a neat, structured, controllable, but ultimately discordant scheme.”

    For though you Brits know damn well that “Math” (without an S) just sounds a hell of a lot better. You resist changing because this insight was made by your wayward offsprings — Americans, who saw the silliness of consistency and embraced hedonistic pragmatism.

    Brilliant ! Your essay just help confirmed my long-held jingoistic prejudice against Brits. Thank you. But wait, wasn’t this post about all religions and not about all Brits? I get so easily confused.

  4. John Clavin

    As a musician and sound designer I always have a table of the equal tempered tones (based on the 12th root of 2) and the perfect harmony ratios, near by for reference.
    Ian, you wrote an amazing article once about equal temperament that I can’t even begin to describe. I had never seen anything like it. Is that article around anywhere?

    I like to use the phrase “the continuously evolving enlightenment of the human race.” If we keep getting more and more connected, like we are, there will be no need for religion. No dogma, everyone will live in the “now”, and we will have dynamically tuned musical instruments that always play perfect harmonies. But then we might become a species like The Borg.

  5. BTW, couldn’t your whole point have been illustrated by just discussing tuning the octaves of “C”?
    Didn’t you just introduce a lot of other complexity? Doesn’t even just 4 or 5 octaves of C (4 0r 5 notes) require tempering?

  6. Boz

    thanks for the point to equal temperament tuning. the wikipedia article is very interesting!

  7. Ian

    @John – I think that’s the article linked to at the start of the post – the one where I calculated the number of notes in an Even Temperament tuning that come withing some percent of pure-harmonies?

    @Sabio – Well, I think the point was the musical theory, rather than the religion. (Hence the lounge comic gag). But octaves don’t need tempering on their own, no. There’s no absolute musical scale, so everything is relative to some interval, and the octave is normally the baseline for everything else to fit it. It would be unusual to see a tuning that didn’t keep the octave. My comment about pianos is more complex again – real octaves don’t quite sound right because of the way the high-tension tuning distorts the equally tempered harmonics, so in a piano, because of the rest of the equal temperament tuning, the octaves sound odd and need to be extended (by a tiny fraction). Not all piano tuners do this, but the best do.

  8. Well, if octaves aren’t adjusted, then I must misunderstand the tuning issue. Oh well, I tried. This is tough stuff for me. (Visuals may have helped)

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