Music and recursive descent

I've been trying to understand the concept of "geometric algebra" for a while. This discipline was developed by Clifford, hence the alternate moniker Clifford algebra.

I can demonstrate an algebra of a geometry, with the following example: musical algebra and geometry. Here the algebra (A) is the notation, but also any "playing" by those musos who don't "learn" any theory or how to read & play simultaneously (perhaps instead being content with being shown the chords and the harmonic structures - the progressions, "breaks" and when to change, i.e the meter or 'time metric', given a signature in the notation).

Start the exercise with the guitar. Now a guitar is an extension of an instrument, with strings, there are many examples of stringed instruments. Bowing and plucking, or strumming are different ways to play them - think Jimmy Page, Jimi Hendrix, even how you can run a pick along a wound string and make it "bow" like a cello of sorts - but you need both hands. A piano "frees" up this requirement - you can in principle play a tune one-handed, but it's harder to cover a decent range of notes that way.

So on a piano the "bowing, or plucking or strumming" is arranged for you, you just press the button, the noise depends on how hard and how long you press it.

Back at the fretboard, there's a string action to contend with. That's the distance you have to move a string to fret it, so you can sound the fretted string without "buzzing" or attenuation - proper fretting is analogous to fingering the violin properly, so the note when you bow the fingered string is 'accurate'. The action can be raised or lowered to make the fretting "action" easier. This won't affect the picking or plucking end of the "action".

So if music on a guitar is the integral of all the "action", including fretting, sounding the strings in a number of "techniques" that include left-handed ones like hammering on and off (much more effective with pickups and electric current) sliding, etc. Right-handed techniques are strumming types, picking (individual or Spanish-style, bluegrass, etc), and 'beating' the strings; harmonics can be played by touching strings at points without fretting - a tuning up technique.

So the fretboard, with a single string is where to start. Let's tune it (down) to D, and make it the largest gauge or 1st, on a 6-string. D is a symmetrical note on a piano keyboard (the extended or parallelized version of a guitar's fretboard). You can see that D has two "black" keys either side, and these have two white keys each side of them; descending from D, we have D-flat, C, B, going up or ascending, we have E-flat, E, F. The intervals D,E,F. and B,C,D map to the same number sequence: {1,2}, if you exclude the black keys.

Think about it like this: assume you can exclude all the black notes (keys) by making a cover of some kind (a sheet with a curled over edge, that completely hides all the keys except the white ones, so all the keys look identical). Now you need to find groups of 4 adjacent notes (here, adjacent doesn't necessarily mean one key apart), that correspond to major and miinor part-scales, or the first or second 4th, or both, for these modes, maj & min in the diatonic scales left on the modified piano.

If you know about the theory and harmony you can narrow the search to just two sets of 12 notes, as subsets of 4 notes within the 2x12 = 24. The lowest tuned string on a guitar scales by twice the 'tune' when you fret at the 12th. D is sounded an octave higher because there are 8 notes that are 'in" a scale, a subset of the 12 - 1 = 11 left between an open D and D at the 12th fret.

Why is the 5th fret (the 6th note if you count the first D) a perfect 4th above D? The 4th is a (5-1)th interval if you exclude the open D. Because of the 8-note octave there are 8 elements to "play" with, like the corners of a cube you can change the colors of.

The same domain as some quaternion group a subgroup of Q[sub]8[/sub].