Archive for the ‘Music’ Category


at least one stranger “liked” last week’s
hell hound on my trail, so here’s another
guitar-neck diagram in the same format.

this one’s in “open G” tuning: DGDGBD
(“dog dug bed”). in key-neutral notation
one has 0-5-12-17-21-24 (“half steps” above
the lowest note). (the thing to memorize
here is probably the gaps-between-strings:
5, 7; 5, 4, 3 [“twelve & twelve”; there are
(of course) 12 half-steps… guitar frets
or piano keys for example… in an “octave”];
in the same notation, the “open D” tuning
from the previous slide would be “7,5;4,3,5”.)

adding “7” (and reducing mod-12 again) gives us
7-0-7-0-4-7; the point of this admittedly rather
weird-looking move is that the “tonic note”
for the open chord (“G” in the dog-dug-bed
notation) is placed at the “zero” for the system.
(“open D” is 0-7-0-4-7-0 in this format; the
tonic note falls on the high and low strings.)

my source informs me that open-G tuning is also
known as “spanish” tuning; a little experience
shows me that it’s even more convenient for
noodling-about-on-three-strings. that is all.



here’s a diagram showing some of the notes for
a guitar tuned in “open D” tuning. i posted it
if f-book not long ago but of course one soon
loses all track of whatever is posted there.

three “inside strings” (the 3, 4, & 5; marked
here with a purple “{” [a “set-bracket” to me;
i’m given to understand that typographers call
it a “brace”])…

three “inside strings”, i say, can be played
as shown here to produce “the same” major chord
(the 0, 4, and 7 from a 12-note scale) in
three different “inversions” (namely, 7-0-4, 0-4-7,
and 4-7-0).

meanwhile, quasi-random bangings of the “outside”
strings—i like to use my thumb for these some-
times—particularly at frets 5, 7, & 12—will
blend in nicely and one needn’t worry much about
“muting” strings-not-played.

gaff my wheel (2009)

Here we go again
It’s another phony friend
Pretending that they’re oh so glad
They’ve found ya
Until you can escape
It’s emotional rape
You haven’t got a chance
When they’re all around ya
And they always seem to know
Right where to find me
And start right in
To own me or define me
I will somehow make ’em see
That that can never be
Get Away From Me With Your Lies

GAFMWYL Mister Salesman
GAFMWYL You flirt, you tease
GAFMWYL You politician
Get away from me you dread disease
Pretty please

Get away from me
With your phony sympathy
I can see what you want
In your eyes
All you want from me
Is that I should agree
That I’m the kind of guy
You should despise
But you really oughta pick
A better victim
Some sucker who won’t even know
You’ve picked him
I will somehow make you see
That your victim can’t be me
So get away from me
With your lies
Posted by r. r. vlorbik at 2:18 PM
(last year’s model.)

the way we live now

get away from me with your phony sympathy
i can see what you want in your eyes…
get away from me with your lies
(2009; now with much smarmy spam).




Photo on 11-28-15 at 10.22 AM

Photo on 11-28-15 at 10.23 AM #2

Photo on 11-28-15 at 10.20 AM

Photo on 11-28-15 at 10.20 AM #2

Photo on 11-28-15 at 10.21 AM

Photo on 11-28-15 at 10.23 AM

Photo on 11-28-15 at 10.34 AM

(seven chords seven ways, part mercury)

ten for maria.

1 – There isn’t enough user-generated content or “making your own math artifacts.”

equations, most likely, first.

but wait. zero-th.
by-hand copies of the *symbols*
for the material at hand.
“the student learns essentially
nothing until the student’s
pencil makes marks on the page”
is a pretty good first approximation
a lot of the time… or anyhow,
i’m far from the only teacher
given to *saying* stuff like this.

i’ve got plenty to say, too, *about*
this but i’m hoping for a list of ten
in under 2^12 characters (for a little
longer; i’ve begun to despair already
at least a little though if you want
to know the truth).
“unions” should look different from “u” ‘s
as an example more or less at random.

*our medium is handwriting.*

out-loud discussion of and…

…written sentences *about*
those equations. written
at leisure without the
instructor (or fellow student).

similar or exact versions of such equations,
repeated, or, much better of course,
improvised, in a “public” setting
with small or, slightly better i
suppose, large *groups* of fellow
students. oral presentation of
the sentences themselves is not
only okay here but much to be
preferred (the board should not
be littered with sentences).
the “correctness” of the sentences
should nonetheless be at issue
throughout the presentation.
said “correctness” is to refer
explicitly to “code”…
utilizing (hey! ed jargon!)
the symbols from our step zero.

it does not escape my attention
that the “artifacts” created by
the student presentations i here
imagine are scribbles of chalk
on a board, soon erased. so be it.

leaving some out…

yick, computer code.

student-designed exercises,
exam templates, lesson plans…

songs and other verse, games,
comics and other graphics,
something to astonish even me.




Sue VanHattumMarch 7, 2010 at 7:08 PM
Maria, I loved your list.

Owen wrote:
>”the student learns essentially
nothing until the student’s
pencil makes marks on the page”

Maybe for higher math, but not at all for young kids. The mathematical issues they’re working on don’t usually require pen(cil) and paper.

My son is thinking so much about what I’d call place value these days. “60 and 60 is 120, right?” “Yep.” No writing – at home, anyway. Lots of mathematical thought.

I’ve long been puzzled by your emphasis on “the code”. Maybe someday I’ll get it… :^)


AnonymousMarch 8, 2010 at 6:57 PM
My son is thinking so much about what I’d call place value these days. “60 and 60 is 120, right?” “Yep.” No writing – at home, anyway. Lots of mathematical thought.

I’ve long been puzzled by your emphasis on “the code”. Maybe someday I’ll get it… :^)
—sue v.

maybe today!

the “places” of “place value” are
places *in* certain symbol strings!
it sure doesn’t matter that you
*speak* of such strings without
having actual *written* code
in front of your actual eyes…
that’s not what i’m always
going on about at all…

presumably gets its interest
from 6+6=12,
together with, right,
the “place value” concept…
*as it manifests in base ten*.

now of course you and your kid
don’t have to have spoken of
bases-other-than-ten for
the essential *role* of “ten”
in discussions of place value
to have become quite clear
all around.

“what’s so special about ten?”
i can now imagine asking
some kid of the same age
if i were lucky enough to
know any…
and i’d sure enough expect
(maybe with a *little*
stack-the-deck prompting
from me) pretty soon to
start hearing about the
role of *zero* (in, again,
certain symbol-strings).

and when our conversations
*without* written work begin
to break down… and if we
still *care*… why then,
we’ll break out some *pencils*
and take a look:
“what do you mean, *precisely*?”.

we’ve been talking about code all along.

calculating with numbers
is the very *model* of
is just flat-out wrong.
and this is our greatest strength.

in principle, anything worth
talking about passionately
in a math class should have
the *same* character:
there *is* a right answer
if we could only find it.

in order to have this happen,
we have to agree on things.
we *can’t* agree… and be
*sure* we agree… and be *right*…
without certain so-called “rigorous
definitions”: marks on paper
(generally; otherwise
*verbatim verbal formulas*
memorized syllable-for-syllable
[mostly… i don’t seek a
“rigorous” definition of “rigor”…
“one is *this* many”
and its ilk (so-called “ostensive
definitions”) are all the rigor
we can *get* sometimes]).

generally the “rigor” one speaks of
is… i think… pretty *close* to the
thing i spoke of (with reference to
elementary arithmetic) a moment ago.
and this comes from “code”.

again. our power in mathematics
comes to an amazing extent from
being-able-in-principle to emulate
some doesn’t-know-anything-*but*-code

now i’m as much of a luddite as the next
guy, if the next guy figures the wrong turn
was somewhere around “domesticated animals”.
but one *glaring* benefit of computers
in math ed is that students will work
for *hours* on getting code letter-perfect
(if they know no human being can see
their failures happening), that wouldn’t put
in five *minutes* of homework on paper
without getting so frantic about each
“move” that they fall apart before even
getting started. it’s that “interactivity”.
this used to break my heart but it’s true.

if schools were for clarity,
command-line programming
would begin in about first grade.
it’s much *easier* than almost
any other thing you can do
with a computer (which is why
it emerged much *earlier*
than the hugely-user-unfriendly
[from a “code” point of view]
*graphical* interfaces that
erased it from the national
consciousness in around 1984).

(somebody mention “logo”.)

math *is* hard.
but it’s much easier than anything else.
because we’ve got *all* the certainty.
(programming on this model
is of course a subset of math).


or flat the high four

five-string exercise for guitar

(lose the sixth string.)
tune the fifth up
a half-tone higher
(than its usual “A”,
to B-flat [or A-sharp;
B-flat to us today]):
X B♭ D G B E. so far.

now we’re gonna “barre”
four strings more or less
throughout the rest. in
across the universe
notation, playing
[[X,0,1,1,1,1]]— i.e.,
pushing down the first four
strings at the first fret—
in this tuning gives, as it
were, a “shifted-ordinary”
tuning: the familiar
A D G B E (learned by every
beginner) is “shifted up” to
B♭ E♭ A♭ C F.

the good news is, forget the
“accidentals”: the exercise
is to play on the high four
with the “A-string” (now in
some sense “really” a B♭)
droning away (or silenced…
but otherwise untouched by
the left hand) the whole time.

so new-school “D”:

this is (of course… ) just
“old school” D
“raised by one fret”.
but, alas, putting that second
“barre” down? (the first finger
is barring four frets “throughout”;
to get the “D” here, i have to
*also* barre three strings with
my third finger [and finally put
my pinkie in the middle of the bar,
one fret up.) that’s pretty brutal.

more good news: one already plays
[[X,X,3,2,1,1]] (regular tuning):
this is Beginner F (typically one’s
*first* “barre” chord… note that
only two frets must be barred).
well, new tuning *improves* on the
sound of that cord since we can now
loudly *play* that fifth string:
[[X,0,3,2,1,1]] (“new” tuning) gives
us an extra bass note, as it were
“for free”.

the real payoff, though…
or anyway my reason for having developed
this whole line of investigation…
the “A” chord and its variants
(as i think of ’em while playing
actual B-flats and *their* variants)
take the cool-sound-making form
where you’re just mashing down the
major-chord in the middle and can
drop in the boogie-woogie treble
with very small movements of one’s
left hand.

this A-form trick (e.g.) also works
in ordinary tuning at the fifth
and seventh frets and that where i
first figured it out.
the “raise the bass” trick lets me
play in the same style but lower
on the neck.

“G” form is another useful near-freebie.

songwriting 201

Photo on 8-19-14 at 11.55 AM

our medium is handwriting. but
*don’t* trust even this little
bit of score; i can’t keep a beat
*even in real life*… not like a
drummer, say… with my rhythmic
body and *darn well* haven’t learned
how to put the *numbers* in.
syllable counts and beat counts just
blur up fast when i try to get a handle
on suchlike matters so i always give up
right away.

but scribbling out this display here
*was* helpful in learning to play the
damn thing out note-by-note. (i’ve been
wandering around the house singing this
for a few weeks now).


math-&-music presentation announcement
at hyperbolic guitars (NCTM; 4/26 in philly).

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):
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
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,
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!

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
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!

let me have the guitar one more time.
when you add in a couple more notes
like this (
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.
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.
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
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
in “absolute” notation and by
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)
0,-1,-2,-2,-2,-1,-2,-2 (relative).

now. *sing* it.

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

and, so, *backward*, we have
“oh, ten, eight, seven, five, three, two, oh”
“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
x = \root{12}\of2. 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-

but there it is.

questions? comments?

Photo 182