2D Synthesizer

I went out to see a friend of mine play with his band and It was the first band I had seen play in many years.  My wife was sick and so I went by myself and on the way home I spent the long ride back alone thinking about music theory and instruments.  I had played trumpet for many years so I already had what I thought was a fair handle on music theory.  My thought experiment was to consider how I would approach defining a scale if I were to reinvent music.  My ignorance became very evident when I started to study up though on my seemingly intuitive approach.

I approached the subject by first accepting the concept of the octave;  That given some fundamental frequency, N multiples of it have a natural order to them.  Between these octaves though, I sought to divide the frequency fractionally into some number of notes.  This is to say that the note halfway between the octaves would be 1.5  time the fundamental and the note a quarter of the way up would be 1.25 as much.  This seemed natural and is how I had assumed that Tets were defined in music but I was excited to find that I was completely wrong!  Here is an excerpt with the punchline, but I strongly recommend reading up on Equal Temperament, it is mind blowing if you take the time to really understand the math and internalize its implications.  It never ceases to amaze me how all of our sense all seem to measure on proportionality on logarithmic scales.

In an equal temperament, the distance between each step of the scale is the same interval. Because the perceived identity of an interval depends on its ratio, this scale in even steps is a geometric sequence of multiplications. (An arithmetic sequence of intervals would not sound evenly-spaced, and would not permit transposition to different keys.) Specifically, the smallest interval in an equal-tempered scale is the ratio:

where the ratio r divides the ratio p (typically the octave, which is 2/1) into n equal parts

Now continuing with the idea while driving and under the false impression scales were based on arithmetic sequences, I envisioned an instrument where the keys were arranged in a grid.  The top left note would correspond to the fundamental frequency for the instrument, all other notes would be derived from this note.  Each note in the column below the fundamental note would be a multiple of the fundamental frequency.  Every other note would be a multiple of the frequency in the first column.  Below is a snippet of a screenshot from an excel simulator that sets up the keys dynamically based on a settings panel and that plays the frequency of the note when you click on the cell (green highlights appear when the frequency matches any chromatic note).  In it you can see it is setup to be a 12×12 matrix with 0.5 multiplier increments and a base frequency of 64hz.

With this key arrangement you can see that all the keys in a given row or in a given column are all be based on the same fundamental frequency.  This, it seemed to me, would allow a musician to play any note in a row or column and always produce notes that would be consonant with one another.  This is not entirely accurate as a matter of fact, but selecting a multiplier progression of 0.5x as I did supports this goal.

So that was the idea and on it I built an excel simulator and then a working prototype.  The prototype is based on a switch matrix connected to a microprocessor with frequency-output capabilities tethered to my pc soundcard.  As far as switching goes, I was surprised to find that there doesn’t appear to be a more elegant way to prevent keyboard ghosting than to construct a diode matrix, so that is what I did.  Here is a video of the prototype in action:

I researched a MIDI interface, but unfortunately MIDI does not support microtonal or frequency-specific communication.  Notes are defined by number within the normal octet scale only.  I considered pitch shifting calculations on each note, but pitchshifting in MIDI is not specific to each note but to the entire channel, so you could only really support one voice.  This would be a sloppy workaround anyway though.  End of the story is that a PC interface needs to be analog, I am ok with that… has some home-brew charm to it!

My observations in playing around with the prototype is that the more keys the better!  Building a 256 key diode matrix though would be a serious pain in the rear, let alone soldering all those keys.  Before I got too far into this, I had a whole other idea which was my Water Keyboard… so for now this is collecting dust on my shelf and I am not sure that I will do anything else with it.

As an aside, I did look into how this grid approach could be applied to our 12-tet scale, and playing around with the simulator it is actually a lot of fun.  I think it illustrates quite simply how the major scales are defined within the chromatic and allows someone who has not studied their circle of 5ths (like me!) to switch between keys with ease: