Tuesday 23 March 2010

Intonation 6: Constructing a Tonic/Supertonic scale

Western music gave up just intonation because it wanted to modulate between keys.  One reason you might want to do this is laziness.  You can make a whole album of very similar songs if you make small alterations to the key and rhythm: fast blues in G, slow blues in D, shuffle in E etc.  I am chiefly interested in exploring dronal music (using only one chord), but I also recognise that there's all sorts of awesome thing you can do with it modulation.  So let's see if we can construct a justly-tuned scale that allows for some limited modulation.

As an example I'm going to pick one very simple modulation: that between the tonic and the supertonic.  The supertonic is simply the chord one tone above the tonic.  So we might be moving from C to D and back again.  This is a useful technique often used in post rock.  It sounds very epic.

I believe that the wholetone (major second) is the difference between two fifths stacked on top of each other and an octave.  This is 3/2 * 3/2 * 1/2/1 = 9/8.  Harry Partch gives both 9/8 and 10/9 for his major second(s), but this is a product of his methodology: he first found all ratio combinations of small numbers, and then looked for identities for them.  It is possible that there is a good argument for using 10/9 or both, but I don't know it.  The ratio used in traditional Pythagorean tuning is also 9/8 so I think it's pretty safe ground making that assertion, but I'd still like to be very explicit that this is totally subjective.

Let's say I want to make ordinary major chords using the tonic, the fifth and the third.  The fifth and the major third correspond to the third and fifth partials respectively, and are universally agreed to be 3/2 (the fifth) and 5/4 (the major third).

If we make a note one tone above each of these, we get a six tone scale as follows:
1/1 Tonic
9/8 Supertonic
3/2 Major Third
45/32 Super Major Third (this is somewhere between the perfect fourth and the lower just tritone)
3/2 Fifth
27/16 Super Fifth (this note is less than the limit of human hearing out from the Major Sixth)

So it should be easy to construct more complex scales that allow for a wider range of modulation, such as a justly intoned blues.

No comments:

Post a Comment