Intonation 4

(cont. from earlier post)

We've made a lot of progress on the just intonation front. Tim has lent me a sound module with a "Just" preset that plays in a form of just intonation. I don't know which flavour of just intonation they used, but it disrespects the octave and is centred on middle C. It rings and sings and sounds a bit mediaeval and it's exactly how I expected it to be even though I'd never heard it and all I had was maths and faith.

I had noticed there was a pattern in the arrangement of partials in the harmonic series. To recap, I had seen that:
1.1 All overtones have overtones of themselves.
1.2 There is some kind of complex exponential relationship surrounding the position of overtones of each particular order in the series.
1.3 Some partials, like the 15th partial (commonly identified as the major seventh), can be seen to be made up of two earlier partials, in this case the fifth and major third identified as partials 3 and 5 respectively.

Mark was talking about how the odd numbered partials were the interesting ones. I had seen this written on the internet but it sounded like guff: the early even numbered partials are very useful. Mark also mentioned that he had got "carried away" with prime numbers. Then it all fell into place and I saw how it worked.

Each overtone in a prime numbered position in the harmonic series (we have been calling them "prime partials") has a strong identity. We get the following identities:
2 - Octave
3 - Fifth
5 - Major Third
7 - unassigned (this is the forbidden note on a natural trumpet)
11 - unassigned
13 - unassigned
17 - Minor Second (semitone)
19 - Minor Third
etc.
If you get too far into the harmonic series it gets diluted to the point where it may as well be dissonance, so we'll arbitrarily stop there.

If you multiply these partials together you get an interval which is musically the sum of the elements you put into it. So 3rd partial x 5th partial = 15th partial, and fifth + major third = seventh. This is true for all intervals. You can find the elements that make up an interval by finding the prime factors. How awesome is that?

Next we tried generating ratios to plug into software synths Mark was programming as we went. We weren't happy with the ratios given on wikipedia so we decided to find better ones ourselves. We realised that there wasn't a good prime seventh interval that corresponded with modern tuning, so we used the 15th partial instead. This shows that modern tuning approximates a mixture of prime and second-degree partials. There is thus nothing particularly archetypal about prime partials, but they are elemental in a way.

We were interested in the 7th partial (which is the 7th note playable on a natural trumpet using only embouchure, that is the shape of your lips and how hard you blow). We quickly found that it occupied a place in the scale somewhere around the minor seventh in modern tuning. (It's a coincidence that the 7th partial makes a "seventh").

At this point Mark made a machine with three keys. It produced a constant tone until you pressed the keys, at which point the tone changed. The lowest key brought it up a major third. The middle key did a fifth and the third key the interval of the 7th partial. If you held down combinations of the keys, it altered the pitch by both intervals. If you wanted a major seventh you held down the major third and fifth keys. In this way we produced an eight note scale spread over several octaves. If you had one in both hands you could begin to make open-handed harmonies. Very exciting. Afterwards, when we played a normal keyboard to check our tuning and identifications it sounded disgusting. The normal intervals used in modern tuning sounded dissonant and ugly. The entire history of recorded music needs to be thrown away and it all recorded again. You would hear it too once you had heard how it is supposed to be.

One other thing we did was to understand more thoroughly how to make ratios. I used the following method to generate some simple ratios, scrawling out a table on a whiteboard. I took the numbers of prime partials and used them as the numerators. For the denominators, I used 2 raised to the power of the number of octaves above the fundamental the partial sounds in (this is always the power of 2 below the value of the partial). It is necessary to divide by this amount to bring all of the notes into the same octave together. This gives, in ascending order:

tonic 1/2^0 = 1/1
minor second 17/2^4 = 17/16
minor third 19/2^4 = 19/16
major third 5/2^2 = 5/4
? 11/2^3 = 11/8
fifth 3/2^1 = 3/2
? 13/2^3 = 13/8
? 7/2^2 = 7/4
octave 2/2^0 = 2/1
etc.

Mark and I noticed that all of the denominators were powers of 2 before we understood what it meant. We turned the relationship around to try and find the meaning given above, and found the relationship (sounding octave) = log2 (denominator).

We were struggling with some intervals, particularly the fourth. The fourth is one of the most important intervals, especially in the 20th century when the guitar dominated music. I understand the fourth as the anti-fifth (a fourth and a fifth make an octave exactly, even in just temperament), but what does that mean in terms of ratios? We couldn't work it out.

I noticed that in traditional Pythagorean tuning many of the denominators were powers of 3. I saw that this meant they were related to the third partial (the fifth) rather than the second partial (the octave). I modified my spreadsheet to respect the pentave as the basic unit of harmony rather than the octave and it fell into place. By pentave I mean that every seven semitones is considered to be the same note as the fundamental. There's a much stronger argument for using twelve semitones as we do (the octave), but why not find overtones of the third partial as well as overtones of the fundamental and the second partial? Here is another set of intervals which are just as valid, and when we trialled it, they sounded sweetly compatible.

I made an important error. I assumed we would have to use powers of 3 for the relationship because it was the third partial instead of the second that we were operating from. It is actually powers of 3/2 that you must use. This is because we have defined the fifth in the home octave and not in the next octave where it's partial belongs. If you use 3, you are actually using an interval of an octave plus a fifth as your basic unit of repetition. I could see by trial and error that I was out by a factor of 2 but it took a long while to understand why.

Using the pentave rather than the octave, all of the partials get new enharmonic identities.
2 - Fourth
3 - Fourth
5 - Pentave
7 - unassigned
11 - unassigned
13 - unassigned
17 - Major Second (wholetone)
19 - Minor Third
etc.
I shouted, "I've found the fourth!" In the first pentave (so already a fifth up) the interval left to the first partial is a fourth. If you use octaves the first partial is a whole octave away. Fifth + Fourth = Octave.
fourth 2/(3/2)^1 = 4/3
Which is exactly where it is in Pythagorean tradition. We made another key on our machine for this interval and it sounded great. It was 2am and we stopped there.

One final thing, when we were making the machine we needed a process to drop the pitch back down when you release the key. We made a mistake here at first, using the ratio of the note in the octave below in the belief that this would negate the effect of the ratio increase. We made this mistake because in common musical practice, if you go up an interval you can go back down the same interval by moving down the keyboard. What we actually needed was the inverse of the ratio we had gone up by.

This may seem trivial, but a week or two ago I posted about my big idea that the Dorian mode is the genuine minor mode and that western practice got it wrong when it picked the Aeolian. Well, the way we mistakenly tried to come back down the scale was rotationally symmetrical in exactly the way I had meant when I tried to justify why Dorian was the bee's knees. Here is the objective proof that the Dorian mode is the real minor. Before, all I had on that was aesthetics. Now I have maths.

(cont. in part 5)

Original comments from posting at woodideas.blogspot.com:
3 comments:
Anonymous said...
Have you look into the work of Harry Partch? He was really into microtonal scales and stuff for similar reasons to yours.
18 Feb 2010 16:51:00
This I Did Not Do said...
I've never heard of him before. Reading the wikipedia article, he seems very similar in character to me, and his ideas seem a very good match to what we're exploring at the minute. I'm sufficiently excited to have ordered his book Genesis of a Music off amazon. Thanks for that.

I'm listening to some of his music and I'm disappointed. He's really not using this to its potential. I have a vision of something quite different.

Also, I am turned off by the idea of microtonality. I don't want to piddle around in tiny insignificant details. I want to fix our broken system and draw something purer from it.
18 Feb 2010 18:11:00
mark said...
I now understand the revelations that you had while I was hacking away trying to get the synth working... nice write-up.
19 Feb 2010 09:12:00