Temperament 2

(cont. from previous post)


When I was first experimenting with pitch as a teenager (I was completely off the rails with my wild experimentation back then) I analysed a number of songs by Deftones and other metal bands and I found that often they would use a scale which went tone, semitone, tone, semitone, tone, semitone...  At the time I thought they just didn't know any better, but this is actually a very valuable idea.


One of the features of this scale is that it takes two octaves to come back to the root note.  Every note in the second octave is out of key with every note in the home octave.  You would think this would cause problems, especially when the only way this pattern can manifest is when you go up into the second octave, and considering the fact that a bass guitar is an octave below a normal guitar.  But it doesn't mess everything up, it creates interesting effects.  There is very much you can do by disrespecting the octave.  It's a short cut to incredibly complex modulation.


What this proves is that you can abandon one of the most basic assumptions of western music.  Notes which are an octave apart do not have to modelled as the same note.  It's just a harmony like any other.  A very clean and pure harmony, but just a harmony.


Let's say we're tuning in just temperament starting on D, and we're moving up from the home octave to the second octave.  How should we tune A in the second octave?  It's a fourth above a fifth above a fifth.  This gives us a note 2 cents out from equal temperament.  But what if it's five minor thirds and a major third up?  This time, the note is 22 cents out.  It is a different note.


There's four solutions to this:
0. Use the most fundamental intervals you can to construct each note.  Octave > fifth > fourth > major third etc.  This is the historical solution.
1. Build a keyboard laid out in such a way as to offer all possible combinations.  You would need 12 times the number of combinations of 12 intervals for every octave.  That's a huge number of buttons.
2. Build an instrument that retunes itself based on the root of whatever you're currently playing.  Programming this would be an exciting challenge.  Googling reveals several attempts have been made.
3. Optimise your scale for a certain root note and a certain favoured interval.


I want to explore 2 and 3.
For 3, the root note will be D = 294 Hz.  If we optimise for the fifth or the fourth instead of the octave, we end up with a scale that is very close to equal temperament.  I hope that by optimising for either the thirds, the tritone or the seconds, we may be able to build new musics with different harmonies that are emotionally affecting in unexpectedly nuanced ways.  There's a whole realm out there that has been very ill explored.


The conclusion of DIftW is a violin duel with the devil.  The devil plays all this fancy filigree stuff, but the simple woodsman just plays a single harmony of unexpected nuance and purity.  The secret chord that David played that pleased the lord.  I want to find new chords, and I have found a gap in which they may exist.


(part 3 to follow)

Original comments from posting at woodideas.blogspot.com:

2 comments:
madKharman said...
I'm on board - I estimate option 3 should take a days' programming using virtual control voltages and some mildly complex logarithmic equations. Option 2 is, of course, grail territory, but not out of the question.
12 Feb 2010 14:23:00
This I Did Not Do said...
For option 2 we would have to do two things:
2.1 Design an algorithm to find the root note based on which keys were being pressed and the active motion of the performance (not impossible, but tricky... some parameters in the algorithm could be altered with knobs during performance.)
2.2 Retune the keys as we go, which is just algebra.

I'm sitting down with pen and paper now playing with stuff. I can't find enough information on the harmonic series from the music end of things (the first 30 terms are not enough) so I'm going to calculate it all from scratch. There's wonderful shapes in here.
12 Feb 2010 16:27:00