### across the universe

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…
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
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!

1. Standard tuning has evolved to provide a good compromise between simple fingering for many chords and the ability to play common scales with minimal left hand movement. The separation of the first (e’) and second (b) string, as well as the separation between the third (g), fourth (d), fifth (A), and sixth (E) strings by a five-semitone interval (a perfect fourth) allows notes of the chromatic scale to be played with each of the four fingers of the left hand controlling one of the first four frets (index finger on fret 1, little finger on fret 4, etc.). It also yields a symmetry and intelligibility to fingering patterns.

you could look it up (w’edia).

2. so here for something to do
that somebody might get some use out of
(or maybe just for the exercise):

A = [[0,2,2,2,0,(0)]
C = [[0,1,0,2,3,X]]
D = [[2,3,2,0,(0),(2)]]
E = [[0,0,1,2,2,0]]
F = [[1,1,2,3,X,X]]
G = [[3,0,0,0,2,3]]

standard “first position” major chords.

the new (0) notation is for “optional” notes
… not to be considered part of the “standard”
version of the chord (where they’d be
replaced with X’s). the bass note
of the chord-without-optionals is
the “tonic” note that give the chord
its name.

minor chords:

Am = [[0,1,2,2,0,(0)]]
Dm = [[1,3,2,0,(0),X]]
Em = [[0,0,0,2,2,0]].

it should not escape our attention
that Em is the *easiest* of the lot:
only two fingers! and those close together!
so start here if you’re just starting
if you want my advice (of the moment;
“my advice” is very much a moving
target).

and, hey! look! Am is real *similar*!
you keep the (middle) two fingers
forming the Em and move ’em
(together!) up one string; then
the first finger [2nd fret] for
the completed chord: strummmm.

then back and forth. a steady rhythm
is what we want here; changing
chord-to-chord with the left tends
to disrupt the beat we’re keeping
with the right. go for something like

Am, 2, 3, 4, Em, 2, 3, 4,
Am, 2, 3, 4, Em, 2, 3, 4, …
(endlessly… or so it should seem
to your neighbors long before you’re done…)
with the numbers denoting
“strum again without changing chords”.

down, down, down, down….
and
down, up, down, up

both make fine strumming patterns here.

somebody congratulate me for not making up
some *notation* for these. if things get messy,
though i will, you bet. DDDD and DUDU are
just *too simple* to need special notations
(unless… oops i did it again… they’re going
to be *repeated* or something).

i think i learned from a teacher
that i studied with for a few weeks
(but jacking around was more fun
than practicing so i soon quit; i did
it again for maybe a little longer
with another)… but maybe it was
out of a book… that dylan’s
“tombstone blues” can be set
to an arrangement of just Em’s & Am’s.

i banged around on it quite a lot
for my first year and have never
given it up altogether yet.

“the sweet pretty things are in bed now of course…”

3. seventh chords:

A7 = [[0,2,0,2,0,(0)]]
A7 = [[3,2,2,2,0,(0)]]
B7 = [[2,0,2,1,2,X]]
C7 = [[0,1,3,2,3,X]]
D7 = [[2,1,2,0,(0),X]]
E7 = [[0,0,1,0,2,0]]
E7 = [[0,3,1,2,2,0]]
.

various combinations of E (E7), A (A7), and B7
make up the key-of-E-blues, a centerpiece
of my guitar life for *most* of my guitar life.

i first played the “boogie-woogie”…
my motivating example of “twelve bar blues”…
in G. in fact, this’ll’ve probably been the
first bass-line i ever doped out “by ear”.

[6,3],[5,2],[4,0],[4,2],[4,3],[4,2],[4,0],[5,2]
[6,3],[5,2],[4,0],[4,2],[4,3],[4,2],[4,0],[5,2]
[5,3],[4,2],[3,0],[3,2],[3,3],[3,2],[3,0],[4,2]
[6,3],[5,2],[4,0],[4,2],[4,3],[4,2],[4,0],[5,2]…

(the first 8… you don’t have to look too carefully
to see that the 2nd line repeats the 1st
and that the 3rd repeats the same
fretting pattern, one string higher…
my point if i were making a point.
anyhow, it typically wraps up with
a flourish so i’ll leave you to work it out.
bang on the D7 chord if all else fails
[and end in the G of course]).

to actually *play* the SOB, form
a G chord (fingers middle,
annular, and pinkie are to be
preferred… though i wouldn’t’ve
believed you if you told me when
i started and you should probably
be no quicker to believe me now…)
when you’re moving fingers around
to the bass notes indicated, keep
the high-G note [0,3] covered
so you can pick (or strum) the
high strings without loss of the vibe.

similarly with C for the third line.
wrap up, like i said, with D7.

CHORDS TO TWELVE BAR BLUES

G (8 beats)
G (8 beats)
C (8 beats)
G (8 beats)
D7 (4 beats) C (4 beats)
G (8 beats)

there’s all this excitement
built into the end where
the new chord comes in
(and goes away quickly).
do some tricky solo.

anyhow… and lunchtime *is* creeping up here quick…
the E blues is then EE,EE,AA,EE,B7-A,EE.

anyhow, in theory. actually i use A7 more often
than A itself here. probably. and lots of other tweaks. ideally the same old blues comes out new every time after all.

what no minor-7ths?
nope. like i said: lunch.

• vlorbik

i can play [[X, 5, 5, 7, 8, 7]]… a “D-form G”…
and its ilk ([[X, 7, 7, 9, 10, 9]]… D-form-A…, e.g.).
i’d’ve never believed it when i started getting good
a couple years ago (“barres” with *two different
fingers*!—who do you think i am, plastic man?).

by “i can play” it, i mean it sometimes doesn’t
sound as bad as all that. especially on the electric.
(note here that both my guitars are strung with
only 5 strings… this means, e.g., i don’t have
to worry about the “X” and can just bang on
the whole chord “strum”-style). a lot of the time
the high string gets missed; mostly it’s okay this way.

• ## (Partial) Contents Page

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