across the universe

January 28, 2011 in Music, Rambles

so let’s take this puppy for a spin.

[[0,0,1,2,2,0]] =[[E]]

the “standard E-chord” learned
by every beginner i know about.

(i learned it at lesson one:
“the house of the rising sun”;
Am, C, D, F, Am, C, E
Am, C, D, F, Am, E, Am.
[thanks, darrell!])

i threw the notation out on the fly
in my last post; it works like this:
the six numbers between the double-
-square-brackets [[…]] are the *frets*
(0 through, well, about 15 with any luck
[but anyway at least 12])
at which one’ll hold down the six
*strings* [[1st, 2nd, … ,6th]].

ringing the neck

now the F chord darrell showed me
was the usual [[1,1,2,3,X,X]]…
i should have said (and *would* have,
if it’d’ve *occurred* to me) that
when a string is “muted” (i.e., not
played at all), i’ll use an “X”
in place of the 0–15 i already
told you about… my first “barre” chord.

(one typically holds down both of
the first and second string with
one finger; it takes a while to
get this at all close to right.
it’s quite the done thing to
*fake* it somehow while waiting
for the skill to grow.)

but the *real* barre-chord-F… the one
i’m using for exhibit A of “ringing the neck”,
is the *six*-string barre (harder still, of course):
[[1,1,2,3,3,1]].
got that? the first finger goes across
*all six strings* at the first fret;
then add in the other three notes
with the other three fingers.

okay now consider
[[1,1,2,3,3,1]] = F and
[[0,0,1,2,2,0]] = E.

it jumps right out at you, right?
each note of the “F” is *one more*
than the note-of-same-string in the “E”.
so once you’ve learned to barre six strings…
and can arrange your middle-finger-through-pinkie
in the “E-form” on strings 3, 4, and 5…
you can play (the “E-form version” of)
*any* major chord you like.

as follows.
[[0,0,1,2,2,0]] = E.
[[1,1,2,3,3,1]] = F.
[[2,2,3,4,4,2]] = F-sharp = G-flat
[[3,3,4,5,5,3]] = G.
[[4,4,5,6,6,4]] = G-sharp = A-flat
[[5,5,6,7,7,5]] = A
…
[11,11,12,13,13,11]]= D-sharp = E=flat.

and this begins to look like a pretty good notation.
*so* good, in fact that other guitarists will already
have invented many very similar others. back in the
typewriter era, they’ll even have been commonplace
unless i miss my guess. so i’m not claiming any credit
for thinking it up.

just for blogging about it in a math-ed blog.

keeping the damn thing in tune

a real obstacle for many of us; some take to it
without any apparent effort at all.

*another* of yesterday’s notations:
[a,b] for “a-th string, at the b-th fret”.
get your low-e (6th string) in tune.
(by any means necessary. i myself
typically bang on it once and invoke
the sacred formula “okay, good enough”.)

then play [6,5], good and loud.
now play [5,0] and tune the 5th string
until it resembles the [6,5] note.
closer is better.

same thing for the rest (almost):

[5,0] :~ [6,5]
[4,0] :~ [5,5]
[3,0] :~ [4,5]…

and a clear *pattern* seems to’ve emerged…
but then

[2,0] :~ [3,4] (!)
[1,0] :~ [2,5].

so the second string is funny: there’s
only a “jump” of *four* (so-called) half-tones
(i tend to just call ’em “steps” even if
the books call ’em half-steps… four *frets*,
if you will [though the very point we’re
now discussing is how to change notes
string-to-string and not just fret-to-fret]…
it appears that i’m forced again to invoke
keys-on-the-piano-without-regard-to-“color”
and have done with it…) from the third string
to the second, whereas any other two adjacent
strings differ by *five*, um, notes-of-the-
-harmonic-scale (yet another way to say halftones).

here it comes again, for ease in cut-and-pasting.

[6,0] :~ [what you will].

the “:~” can be pronounced
“by definition, sounds like”.

(the colon identifies this symbol
as an “assignment operator”, turning
the “sounds like” symbol, “~”, into
a loading-in-a-new-value-of-a-variable
symbol. i use this convention all the
time [and urge others to do the same].
thus “r := d/t” for “rate, by definition,
equals distance-divided-by-time”, e.g.)

anyhow, then

[5,0] :~ [6,5]
[4,0] :~ [5,5]
[3,0] :~ [4,5]
[2,0] :~ [3,4]
[1,0] :~ [2,5].

now, i know precisely nothing about the history
behind this standard tuning, so i can only
speculate about “why” things are done this way.
here goes.

[1,0] on this tuning is 24 steps…

a step, dammit, denotes a one-fret gap in tone…

[1,0], i say, is 24 steps higher than [6,0]: two octaves.
both are “E” notes. so maybe somebody thought it was
a good idea to *have* the highest- and the lowest-
pitched strings two octaves apart.

and if we take this for granted…
along with “there should be exactly six strings”…
and “the tonal ‘distance’ string-to-string”
should be *the same* at every instance…
we arrive at a mathematical impossibility:
5 (gaps string-to-string)
doesn’t go into
24 (gaps [6,0]-E to [1,0]-E).

but it’s *close*, so we just kludge it by making
one of the gaps a little smaller: four gaps of 5
and one of 4. okay. twenty-four. satisfied?

(it wouldn’t do to have only 5 strings, as i imagine
this playing out, with gaps-of-six in each of the
[four] gaps-between-strings, because “six” is
*too big a gap*. as it stands now, there’s
this perfectly good major scale going, say,
[6,5],[6,7],
[5,4],[5,5],[5,7],
[4,4],[4,6],[4,7],
[3,4],[3,6],[3,7],
[2,5],[2,7],
[1,4],[1,5],
that one can play with *one left-hand finger per fret*,
the whole way across the neck; the gaps-of-six tuning
would forbid this.)

(exercise: recall that a major scale has gaps
0,2,2,1,2,2,2,1; check my work. [to do it
*without* a guitar you’ll need to make reference
to the four-gaps-of-5-&-one-of-6 pattern of course;
it’d be best to just *play* the damn thing though.])

having said all that, i’m very nearly ready to display
a certain relationship between some chords… the idea
that started me off on the drawing-and-typing binge
you see me in today. but before i change the subject…
the subject is *tuning*, in case you’ve forgotten…
i’ll remark that i don’t actually rely on these
“definitions” exclusively. there’s a trick with “harmonic
frequencies” i like (two notes on one string);
also quite often i’ll tune by octaves.

for example, to tune the B string [2,0],
i’ll play [5,2] (A-string, two frets up…
another, lower, B). in fact, what the heck,
making up notations is fun and easy,
[2,0] ~ [5,2]{+12}.

also, maybe even more commonly, i’ll tune *down*
via, e.g. [6,0] ~ [5,7]{-12}. anyhow, it ain’t
always done according to the rulebook, kids.
not by me it isn’t, and probably not by
your favorite players either. you’ll find
a way. if you stick with it long enough.

jai guru deva

[[0,0,1,2,2,0]] = [[E]]
[[0,2,2,2,0,X]] = [[A]]

so the A-form can be derived from the E-form.
just move everything up one string, right?
but be sure to adjust for the weird gap-of-four
between the third and second.

move it up *another* string: voila, the D-form.

[[0,2,2,2,0,X]] = [[A]]
[[2,3,2,0,X,X]] = [[D]].

likewise, starting at the G-form
[[3,0,0,0,2,3]] = G
[[0,1,0,2,3,X]] = C
we get the five-steps higher “C” chord
just by going-up-one-string
(with the one-fret “slide up”
at the 2nd string).

i’ve been *vaguely* aware of how all this works
for most of my life. but have only now (a couple
of weeks or so) been able to make it explicit.

anyhow, now just slide the C a couple frets up
[[0,1,0,2,3,X]] = C
[[2,3,2,4,5,X]] = D
and get a variant on the D chord.

(no funny stuff here as far as the theory goes…
just add two to each note… but the *chord* is
hard because you’ve gotta barre three strings
and stretch out four frets; *i* can play it
but i wouldn’t advise a beginner to try.
maybe with an electric action it’s do-able.)

okay. burning out. lunch!

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