### Our Medium Is Handwriting

In Standard Musical Notation, one has so-called “sharps” and “flats”. This is one of the things that makes it hard to read music. It’s as if the notation were designed for pianos in particular: the sharps and flats show up as black keys on the piano.

From “C” to “C”, on the white keys only, is the C-Major Scale. Since only white keys are involved, there are no superscripts on the names-of-notes (i.e., no sharps, no flats); the C-Major Scale is C,D,E,F,G,A,B,C. So we need only to know which piano key is “C”… consult the picture; it’s the white key to the left of the cluster-of-two black keys… to discover that the pattern A_BC_D_EF_G_A emerges if we start instead at the “A” note.

In particular, the “one half-step” gaps… a single “jump” of the so-called “chromatic scale” obtained by playing a single guitar-string at each fret successively from 0 to 12 (or by playing each successive piano key from left to right without regard to color-of-key; I’ve marked such a scale above the keyboard in my drawing) is, annoyingly, called a “half-step”… the one half-step gaps, I say, occur from B to C and from E to F. This is one of the first pieces of “music theory” I ever learned (and you should go ahead and learn it too). All the other “one-letter gaps” are so-called “whole step”s. Anyhow I digress.

Because what I’m getting at here is that one can now play any minor scale… on the guitar, say… by starting on any note (some letter-superscript combination; E-flat, say): just use the gap-lengths I’ve indicated just above my hand in the photo (after computing ’em on the drawing of the piano).

(Hmm. I appear to have just set an exercise. Here’s its solution, worked out live at the typer. [Good thing there’s a picture of a piano handy.] E-flat, F, G-flat, A-flat, B-flat, B, D-flat, E-flat. This seems to contradict my prejudice that “B-flat” and “B” shouldn’t appear in the same scale. Oh well. I’m reasonably sure this is so for Major scales.)

So I calculated out a bunch of ways to play A-minor scales in “boxes” four frets wide. There’s at least one mistake in the shot above (one of the circles has, like the Emperor told Wolfgang, too many notes)… sufficient proof that this diagram is, not only the result of certain calculations, but the work itself. I’m a long way from “just knowing” what note I’m holding down when I’m fretting away at the neck of the guitar, so I had to keep reminding myself [as I drew this] of the “B-to-C and E-to-F” business I was pointing at a little while ago); then the “calculation” turned to “how should I draw these bent-up ‘circles’ around the scales [clearly]?”

The upper-right scale has a “cheat”: its last note is outside the four-frets-to-a-scale condition (way outside; two frets worth). The frets get real close together here though (if you can reach ’em at all, if they’re there at all) so it’s not necessarily much of a cheat. The upper-left scale has the same kind of cheat… it’s the same “box”, twelve frets lower, after all… but this time it’s only a one-fret cheat since we get the “open” strings (the 0th fret) for free.

Then finally (for now) I cleaned it up a little. And looked at a few ways to make an A-minor chord.

I’ve done similar stuff for major scales. I anticipate at least a little more work along these lines. Certain things seem to be falling into place.

#### 1 Comment

1. evidently the done thing is to cheat
and call the B note in E-flat-minor
by the name of C-flat.
http://en.wikipedia.org/wiki/E-flat_minor

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Vlorbik On Math Ed ('07—'09)
(a good place to start!)