today’s morning ramble: music theory

January 26, 2011 in Cool Tricks, Music, Rambles

how to study guitar as if
you were being taught math
by me.

part zero.
(in which i pretend we’re here
together in a room with a guitar.)

okay. gimme the guitar for a second
while i get it tuned. okay. over
to you. grab a pick if you want.

(a lot of *your* work will be about,
essentially, training your body
[mostly the hands]. but it’s
inconvenient to talk about that
stuff in this format so let’s
postpone that discussion
and go for the “high theory” stuff.)

right. we count the strings from the highest
to the lowest (pitch, not position in space:
the highest pitch [i.e., the highest *note*]
comes from the thinnest string… the one
closest to the floor as we normally hold
the guitar).

go ahead and play the high string, good and loud.
now try the low string. those tones
(yet another name for “note” or “pitch”)
are both called “E”. in fact…
here’s something for your *left* hand to do:
hold this fourth string here down
right here at the second fret.
push it good and hard; don’t worry
about the pain. okay. see how well
you can make it ring out.
compare it to the other E’s.
(singing: “do, re, mi,… , do”;
the E’s land on the doh’s.)

here comes some mathy stuff.
notation: let the symbol
[a,b]
denote “the a-th string, fretted
at the b-th fret”.
open strings (so-called; unfretted)
are said to be “fretted at zero”
(or “at the nut”).
thus, the three notes we’ve considered
are [1,0], [6,0], and [4,2]:
the high-E, the low-E, and,
well, what the heck, the middle-E.

you can play ’em all at once if you
drop the pick and put your thumb
on the bass and one finger each
on the 4 and 1 strings: *pluck*
that sucker! fabulous!

okay.
let me have the guitar one more time.
when you add in a couple more notes
like this (
[[0,0,1,2,2,0]]…
i’ve introduced another notation…
this denotes “first string open
(i.e. “at zero”), second string
open, third string fretted-at-1,
fourth-at-2, fifth-at-2, sixth
string open”: a “standard E” chord)
you get what’s called an
E-major *chord*. (strums;
reluctantly hands guitar
back to student.)

there’s yet *another* E at [1,12].
(try it.) also at [6,12].
in fact… yet another notation…
[6,12]~[4,2]: the notes sound
the same (anyhow, they do when
the guitar’s in tune).

the important fact for us right now
is that *twelve* frets-up-the-neck
(from [6,0] to [6,12], for example…
it’ll be convenient to work on
the bass strings for a while now
i think…) is *one* (so-called) octave.

and not *just* for right now.
“12 frets equals one octave”
lies behind *all* of the (little bit
of) music theory i know.

part one.
(in which “12” is confused with “0”.)

twelve frets up the neck is one octave.

DOH to DOH. (i’ve changed the name of
the “tonic note” for the major scale
here… not so much in honor of homer
simpson as because “do” looks to my eye
like a common english word and when
i see it i hear “due”.)

not by coincidence, consider a piano.
C_*_D_*_E_F_*_G_*_A_*_B_C
there are “black keys” at the *’s;
the pattern repeats several times.
anyhow, *twelve* different notes
on the piano before the whole thing
repeats itself an octave higher.

all thirteen notes sounded in succession
form the so-called “harmonic scale”.

at this point i’ll introduce *two* notations.
0,1,2,3,4,5,6,7,8,9,10,11,0′
i’ll call (for now; i just made this up)
“absolute” notation for the harmonic scale
(C, C-sharp, D, D-sharp, E, F, … , C…);
meanwhile i’ll also call
0,1,1,1,1,1,1,1,1,1,1,1,1
the “relative” notation for the same scale.
(so-called “first differences”; the *gaps*
between successive “absolute” positions…)

the Major Scale i can now define as
0,2,4,5,7,9,11,0′
in “absolute” notation and by
0,2,2,1,2,2,2,1
in “relative” notation.
the easiest way to rediscover these
(when we inevitably forget ’em) is
“white keys on the piano, starting at C”;
anyhow, that’s essentially what *i’ve* done,
any number of times. of course
as long as we can *sing* a major scale
we’ll always be able to fiddle it out
by plucking around on the guitar
(or what have you).

played backward, we have
0,11,9,7,5,4,2,0 (absolute)
and
0,-1,-2,-2,-2,-1,-2,-2 (relative).

now. *sing* it.

the Minor Scale is white-keys-starting-at-A.
thus:
“oh, two, one, two, two, one, two, two”
or
“oh, two, three, five, seven, eight, ten, oh!”.

and, so, *backward*, we have
“oh, ten, eight, seven, five, three, two, oh”
or
“oh, oot, oot, no, oot, oot, no, oot”.

“hold it right there!”, i hear you cry
(you need a vivid imagination to teach
ideosyncratic notations to imaginary
audiences). “what’s this OOT business?”

well, you didn’t think i was gonna sing
“oh, minus-two, minus-two, minus-one,…”,
did ya? too many syllables!

so. last new notations on the day.
i’ve made up my own names for
negative-one through negative-eleven.
to wit. (here comes the *harmonic*
scale backwards.)

oh, no, oot, eeth, roe, vie,
kiss, nev, tay, yah, net, neal, oh.

just like in miss di baggio’s class
in ’68. when, with the world in flames,
we settled down one week to develop
“base eight arithmetic” in my 6th grade class
and made up new names… and new handwritten
squiggles… for the objects ordinarily denoted
by the digits 0 through 7.

one does this *all the time* in mathematics.
“here are some phenomena we intend to investigate.
fiddle, fiddle, fiddle. hmm. thingum again!
something interesting seems to be happening.
how can i *describe* it? well, i’ll need to
have a name for… this… um… well, what
the heck *is* going on?”

this is why the good lord invented variables,
for example. (why the good lord waited until
the late middle ages to clue *humanity* in
on the concept is a topic for another lecture.)

here are the opening bars to
“sunshine of your love”
in relative-vlorbik notation:

oh-oh, oot, two!
vie, no-no, vie-three, eeth.

twelve-oh, oot, two!
vie, no-no, vie-three, eeth.

sev’nteen-oh, oot, two,
vie, no-no, vie-three, eeth…

sev’n-oh, oot, two,
vie, no-no vie-three, eeth!

five, oh-oh!
oh, three, two, oot, eeth, three…
two, oh-oh!
oh, three, two, oot, eeth, three…

in the sun, shine of your lah,
ah, ah, ah, ahh,
ah, ah, ah, ahhh…

part two.
(all knowledge is found in 60’s paperbacks.)

recall that the major scale is
C, D, E, F, G, A, B, C (piano notation) or
0, 2, 4, 5, 7, 9, 11, 0′ (absolute vlorbik).

all i know for sure starting out is that
the 0′ tone has *twice the frequency*
as the 0 (and the string is fretted
at 12 to achieve this… which is
*half the length* of the string).

from the harvard brief dictionary
of music (willi apel & ralph t. daniel,
washington square press [8th printing,
november 1966]), i learn that, by
assigning a “frequency” of one
(1) to the C note, the frequencies
for the major scale are then:

C 1
D 9/8
E 5/4
F 4/3
G 3/2
A 5/3
B 15/8
C 2
.

there’s no “C” string on the guitar though,
so it’s much more convenient to use
absolute-vlorbik notation. in practice,
i actually tend to *think* “C, D, E, F…”
even when i start on E or A
(for example; chosen because there are
E and A strings and one can…
and should… be plunking along on
one or another string at least from
time to time in this discussion).
but i consider this habit to be *damage*
(because i’m *lying* to myself
about the names of the notes as i play ’em);
part of the reason for introducing
my new notations.

so let me change this display to

0 1
2 9/8
4 5/4
5 4/3
7 3/2
9 5/3
11 15/8
12 2
.

the left-hand numbers can now be taken as
*fret* numbers: holding down
a string at the *seventh* fret,
for example, produces a tone
with 3/2 the frequency of the
open string.

it can hardly be a coincidence
that the 7th fret is 2/3 of the
way between the bridge and the nut
(the right- and left- hand ends
of the string). i measured this
the other day on my guitar (and
urge you to do the same on yours).
of course i used *tools at hand*
(“get the right tool for the job”
is for the incredibly rich)…
in this case a length of paper
towels rolled from the bridge
to the nut and then folded in thirds.

likewise, the *fifth* fret is 3/4
of the way down the string (recall
that “down” here means “in the
direction of lower pitches”…
from the bridge toward the nut).

now, this translates to *one*-fourth
of the way *up* the neck… a more
pleasant number to think about.

no math major could now resist developing the
transformation x |—> (1 – 1/x) and displaying

0 0
2 1/8
4 1/5
5 1/4
7 1/3
9 2/5
11 7/15
12 1/2
.

these are (conjectured; i’ve only actually
*measured* the 5 & 7 frets) the distances
nut-to-fret for the notes of the major scale.

finally… we’ve had more than enough
for today, i think… when *i* start
burning out, it’s a sure sign that
everybody *else* has checked out long since…
let me remark that in *another* version of the
whole theory… that of the famous
“well-tempered clavier”… one obtains
the frequences for the harmonic scale
(0, 1, 2, 3, … ,11, 12; a so-called
“arithmetic sequence” with constant
*differences* between successive terms)
by taking
1, x, x^2, x^3, … ,x^11, x^12:
a “geometric sequence” with constant
*ratios* term-to-term.
since we need the last frequency to
be twice the first, this results in
. now, this
turns out to be a (so-called) *irrational*
number. so the values in the table (following)
are “estimates” of the theoretical values
for the frequencies. nonetheless, they
should be considered *highly accurate*:
*no* “real world” (measured) quantities
have the perfect precision of so-called
“real numbers”, after all.

on this model, we have
0 1
2 1.123
4 1.260
5 1.335
7 1.498
9 1.682
11 1.888
12 2
.
the decimal values for the rational-number
scale, by contrast are
0 1
2 1.125
4 1.250
5 1.333
7 1.500
9 1.667
11 1.875
12 2
.
darn close. especially at the 5th & 7th.

it appears to be some sort of miracle
that natural-number-powers-of-2^(1/12)
should, right when it matters, be
so close to shifted-reciprocals-of-fractions-
-with-very-small-denominators.

but there it is.

questions? comments?

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