You can click on the image to zoom in on it. I am working on re-thinking the whole concept of fretting, and putting in snapping points that correspond directly to the spectrum. Most current attempts to find a set of fixed notes appear to be rooted in the use of instruments with discrete notes, and trying to wrangle MIDI into doing Microtonality correctly (it won't! it never will! the protocol deeply assumes that you play 'notes' and bend with 14bits of resolution at most, usually not more than +/- a whole tone, want to bend them all together or put them all on separate channels. the MIDI spec for alternate scales still assumes there is some small number of notes within an octave. The whole mentality is broke; it should be more like a channel being a string that continuously takes on arbitrary pitch values. if all MIDI did was take on a frequency orientation, this would all be simple and work correctly. But 30 years later, it hasn't changed. OSC is still not well supported, and is vague like XML, not guaranteeing that you can just plug in an instrument and something reasonable will happen.).

In Pythagoras, I have decided to tune the rows to the fourths of the spectrum rather than of 12tet. You can intuitively see the ratios and just play at red-green line intersections, or follow the lines around if you want to calculate what it actually is for some reason.

I don't want to obsess on the math, but here is a demonstration of why this works: The ratio 3/2 is the bright diagonal line. The next angle counter-clockwise (straight up!) is 4/3, then 5/4, then 6/5, and 7/6. To follow the line up is to multiply by it, to follow the line down is to divide by it. As an example, if you go up, diagonal up right, down, diagonal down left then you navigate: root, fourth, octave, fifth root. In numbers: 1, 4/3, 4/3 * 3/2 = 2, 2 * 3/4 = 3/2, 3/2 * 2/3 = 1.

So if you want to have a chord with some whole number ratio, just make sure that lines match up at the intersections with the red line. I have an overwhelming preference for intervals made purely by navigating the fourths, as these are all the sounds I settled on after months of fretless play before I came up with the idea of using markers to explain them away.

http://www.youtube.com/watch?v=2X9AqiUVF_U

An older video showing the microtonal markers closer up:

http://www.youtube.com/watch?v=FOPL6d41Ofg&feature=related

Notice that I have not made any mention of any number of equally spaced smallest intervals. I don't see the usefulness of anything except 12 or perhaps 24. Though you will see a tantalizingly close to equal pattern when you play in Pythagoras a median second interval (quarter-flat whole-tone); I haven't counted exactly, but it looks like the 53 tet that I have read about. But in any case, it's a uselessly large number that might as well have you playing completely fretless.

Also note that in the discrete fourier transforms (turning signals into frequencies electronically), the lowest detectable frequency is the fundamental and only integer multiples of it can be represented cleanly (ie: frequency 1 is lowest detectable, 3 is a fifth and an octave up...there isn't a clear bin available for 2^(17/12) times the fundamental frequency to get the pitch at 17 'frets' up. ). This integer limitation is microtonality, and it is completely natural in the sense that it is tied into computing, math, and physics rather directly. It is also the practical sort of microtonality created by whole ratios.

Anyways, this is where Pythagoras is on progress. It hasn't surpassed Mugician yet, but the plan is to have the best instrument for playing fretless and learning microtonality in a free-form manner. It is not open source like Mugician is now.

Impressive work.

ReplyDeleteBy the way, Erv Wilson would have loved your work. His work on generalized keyboards might interest you. There's a beautiful system of mappings between generalized keyboards and the 'Scale Tree' (Stern-Brocot tree). Erv Wilson spent much time writing about it and you can see some of his papers at...

http://www.anaphoria.com/wilsonkeyboard.html