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)