## Archive for January, 2011

### Blogging 104. Exam 1.

The first exam was last week. I’ve posted grades into the appropriate intranet doo-hickey; students can look up their own grades, administrators can tell I’m actually doing the part of the job that matters most (to the machine), and I have some insurance against the almost-unthinkable fate of losing my gradebook. I haven’t been posting Quiz or Homework grades since these are subject to tweaks. The “daytime” classes have Lecture and Recitation meetings, with Q’s & HW’s in the recitations; in my “evening” class, I’m both Lecturer and Recitation Coach (both “sage on the stage” and “guide on the side”… holy moley, I’m beside myself). Anyhow, this situation, as you can imagine, improves the communication between Lecturer and Recitation Coach immeasurably and it’s, well, better for morale if I hold back on posting the “recitation” grades.

Alas, there’s been a marked tendency on the part of a sizable fraction of the class to treat Tuesdays (when I administer Q’s and pick up HW’s) as the “recitation” section and Thursdays as the “lecture”… and then skip the lectures. Attendance is a lot better on Tuesdays, in other words. For this and other reasons, I didn’t do much “post mortem” work (“going over the Exam”) when last I met the steady-attenders on Thursday. Here’s a rundown now.

Very pleasant for me overall. We use the same Exams as the Day Class versions of 104, so until the exact day, I didn’t know exactly what to expect. It’s a real good test: do-able in an hour by appropriately-prepared students. The temptation to use tricky questions has been suppressed to my relief. This is somewhat in defiance of a claim in the syllabus to the effect that any HW problem is as likely to appear on an Exam as any other. So be it. A custom more honored in the breach than the observance, sez I. I got a pretty typical distribution… too small (n = 15) to be a good fit to “the curve”, but with the much-to-be-expected “lots in the middle and a few at each end” property just the same. One perfect score; two failing scores. Mean $\mu = 75.4$; Median $Q_2 = 78.5$.

Solutions to “two linear equations in two variables”; check. The class-as-a-whole has, anyway, learned the basic moves for both the “addition method” and the “substitution method”, and proved it on the first page. One variable in one solution had a value of 0; this causes more confusion than one would like; the “no solution” and the “infinitely many solutions” cases are typically more confusing still… so I don’t say we’ve mastered the methods. But everybody’s at least prepared to have a conversation about how this stuff works (if there were world enough and time).

On the more abstract interpret-the-graphs version of the “systems of two linear equations” situation, there is, predictably, more confusion. The same page revealed more of the zero-versus-nothing bug… a common difficulty for learners of math (ever since the introduction of “zero”… and yet, confusing as this seems to be for the laity, “zero” is one of the best ideas of all time…). A classic instance of the classic general complaint of teachers about students, “they don’t want to think”. But we’ve got, anyway, kind of a grip… somewhat tenuous I have to admit… on translating from data presented graphically to equations and inequalities. (Going the other way one has the Graphing Calculator so this is the hard way.)

Speaking of inequalities: interval notation was the source of the worst difficulties here. Our examiner very tastefully avoided absolute value inequalities altogether. I don’t say that this topic is more trouble than its worth; I do say I’ve seldom seen it done right. Our text has rather a high-concept presentation that I liked… but I sure didn’t feel like we’d taken enough time to’ve improved the overall skill level in this area by much if at all. I note with pleasure that we skip the section on graphing-systems-of-inequalities. This topic was very badly handled at Crosstown Community College (in a mostly-very-different course, also called 104): in particular, there was a flat-out mistake, for years, on the key to the Final Exam (concerning the “shading” of a certain “boundary point”); nobody seems to have been scandalized by this but me: this is one of those areas (like “set-builder notation”) where the instructors quite often aren’t qualified to present the standard treatment of the material. The industry is aware of the situation and appears to like it that way.

The slope of y=3x+7 is 3. Not “3x” (dammit). If the variable were part of the slope, we couldn’t get the slope by “plugging (four) numbers into (all four variables of) the well-known formula for the slope”. (Now, could we? Think, doggone it! Think!) But… of course… the question of “how variables work” is one of the trickiest of all: this takes practice. Of course the classic area for confusion-as-to-the-nature-of-variables is “word problems” and we’re still seeing quite a bit of it here. It pleases me much more than it should that we had a “mixture” problem since I laid a lot of stress on those; my class would have done even worse on some of the other problem “types” (precisely, on my model, because of the nature-of-variables issues; problems-by-type is essentially a way of “routing around” students’ astonishingly-stubborn refusal to discuss what variables mean and how).

Intercepts are points, not numbers (when we’re being careful). It’s common enough when talking to say, for example, that “3” is “the intercept” for “5x + 3”, when we mean “(0,3) is the y-intercept for [the graph of] the equation y = 5x+3″. The fact that we require more precise language in certain contexts than in others should create no confusion.

But “should” has nothing to do with much of anything, and it turns out that this “be more formal in work to be handed in than when you’re banging away talking and calculating” thing has been a major problem for a sizable fraction of any class I’ve taught at this level of the game. They don’t want to do it and think I’m just being mean. “Well, I meant…” [such-and-such], they’ll tell me, refusing to believe that it’s my duty to grade what they wrote and not what they meant. This misses rather a big part of the whole point of bothering with “mathematical precision” at all: code can be perfect.. and “perfect” is a lot better than “almost perfect”. It should be helpful to think of computer interfaces here: one wrong mousetweak can botch the whole environment. But somehow it never is. Helpful, that is. “Should”. Feh.

Hey. Madeline just woke up. See you later.

### 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!

### today’s morning ramble: music theory

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,
[mostly the hands]. but it’s
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

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
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

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-
-with-very-small-denominators.

but there it is.

### time considered as a sequence of many-named values (modulo twelve)

E__F__*__G__*__A__*__B__C__*__D__*__E

B__C__*__D__*__E__F__*__G__*__A__*__B

G__*__A__*__ B__C__*__D__*__E__F__*__G

D__*__E__F__*__G__*__A__*__B__C__*__D

A__*__B__C__*__D__*__E__F__*__G__*__A

E__F__*__G__*__A__*__B__C__*__D__*__E

so here’s a picture of the neck of a guitar.
i’ve been drawing it a lot; here it is in cold type.

it looks all wrong already and will no doubt be
munged up altogether by the miracle of the
god-damn internet by the time anyone sees it.

but in the meantime, i can cut-and-paste to produce
displays like
E__f__*__g__*__a__*__b__c__*__d__*__e

b__C__*__d__*__e__f__*__g__*__a__*__b

G__*__a__*__ b__c__*__d__*__e__f__*__g

d__*__E__f__*__g__*__a__*__b__c__*__d

a__*__b__C__*__d__*__e__f__*__g__*__a

e__f__*__g__*__a__*__b__c__*__d__*__e

where i’ve boldfaced the notes of a
C chord. on paper i mostly draw circles
around scales or chords or individual notes;
i also use a certain amount of underlining.
bold face comes into play, too.

there are some examples in my last post but one
(with mistakes intact… i know of three so far). anyhow,
this looks very much like the music-theory-for-guitar
tool of choice. chord diagrams are common of course
but they generally leave off the names of the notes.
for me, right now, these are pretty close to being
the whole point. sure, i’m mostly working out
*numerical* relationships (like the gaps in a
minor scale: 0, 2, 1, 2, 2, 1, 2, 2 [or the absolute
numbers 0, 2, 3, 5, 7, 8, 10, 12])… but their
instantiations in specific examples long known
to me by kinesthetics are there to make
the results vivid and memorable.

so. the high three notes on the chord in the diagram
are GCE; the middle three are EGC; the high notes
are CEG. these are so-called “inversions” of the
C-major chord. by putting the “tonic” note–C–
at zero and counting so-called “half-steps”
(notes on the piano without regard to
color-of-key… one-fret differences on
the guitar; steps from the 12-tone “chromatic
scale”), one arrives at
0,4,7 for the root-in-the-bass inversion;
-5, 0, 4 for the root-in-the-middle;
and -8, -5, 0 for the root-in-the-treble.

the “negative number” approach is probably
pretty unusual but comes very natural to *me*.
it would be just as true… but maybe less useful…
to say that these patterns are described by
047, 059, 038. (one has moved the “0” to
the bass-note-whatever-it-is; one drawback
of this notation is that we can’t “read off”
the name of the chord as easily).

also one is of course working “modulo 12”:
twelve half-steps above any given note
is the next-higher note having the same name
(and twice the so–called “frequency”:
the string vibrates twice as fast).
in “mod 12” arithmetic, 12 is replaced
with 0; other numbers with their
remainder-on-division-by-twelve.
it follows that, for example
-8 = 4 (mod 12) and
-5 = 7 (mod 12), so
the weird-looking -8, -5, 0 inversion
turns out to be nothing but
4, 7, 0… the E, G, C by
their ordinary chromatic-scale names
(with C at 0).

e__F__*__G__*__A__*__B__c__*__D__*__E

B__c__*__D__*__E__F__*__g__*__A__*__B

g__*__A__*__ B__C__*__D__*__e__F__*__G

D__*__e__F__*__G__*__A__*__B__c__*__D

A__*__B__c__*__D__*__E__F__*__G__*__A

E__F__*__G__*__A__*__B__C__*__D__*__E

in raising the CEG from the (ordinary, first-position)
C-chord of the first display by 12 half-tones,
we of course get another 047-inversion
C-major chord. for the C and E this is
accomplished by going one string higher
and seven frets higher… but for the G
by going one string higher and *eight*
frets higher… this is due to the quirk
in tuning known to all guitarists: the
gaps between strings are 5,5,5,4,5
(in the standard tuning; of course
there are many others… a few of
them known even to me).

guitar players will recognize the resulting
“shape” as an F-chord slid up seven frets.
i’ve been looking into the relationships
among the long-familiar “shapes”…
that, essentially, *follow* from the
5,5,5,4,5 pattern. and having a blast
doing it.

but. i exaggerate for rhetorical effect.
the *real* result of all this scribbling
and calculating is that i’m not thinking
about *money* or *social life* or any of the
million other things i can never seem
to get any satisfaction out of (but
can sure as hell get incredible
amounts of *frustration* out of).
if this isn’t “having a blast”,
well, it’s mostly the closest
i can expect (and lucky to get
even this).

i “had fun” typing all this up, too: same effect.
until i got to the part about my actual life,
i was solving little abstract “problems”
like “how can i say this clearly” or
“how many notes apart are C and E”
or what have you and using the results
to make *improvements* in the manuscript
for this blog post. the feeling that
one is making *something* better
in this crazy life, even something
as meaningless as a blog post,
is the feeling of *something going right*,
and damned hard to get. but…
better than a poke in the eye with
a sharp stick… when we feel “good”,
we’re mostly really feeling *nothing*.

this is, i think, how “obsession” works.
why are so many math-heads nutjobs?
because so many nutjobs become
math-heads… it’s the only thing that
musicians too.

a very lucky few in each field of endeavor
are able to find other people that’ll care
about what they’re doing. most of us
have to be content with the doing itself
and count ourselves lucky to get it.
god knows it beats TV.

probably i’ll become a better guitar player
from all this. and, sure, i’ve had some
satisfaction *playing* for other people;
there is *some* investment-paid-off quality
to be had from all the hours and hours
of this-is-no-good, not-this, not-yet,
work-work-work, beavering away.

but that’s guitar. the real motorcycle
one is working on is a motorcycle
called “teaching-and-learning”….
where i can sometimes convince
*myself* i’ve gained some momentary
“insight” into some part of The Art…
but i emphatically do *not* feel myself
becoming a better teacher. (of others;
*maybe* of myself though heck knows
i’m still a very slow learner like always
and anyhow, this is better described
as being-a-better-*student*).

so it’s very frustrating. meanwhile, the students
most in need of my help are the very ones that
avoid me the most effectively etcetera etcetera.

nothing will bring you peace but the triumph of principles.
stated more clearly: nothing will bring you peace.

### remarks on recent work

there’s info on assignments here;
i’m hoping everyone knows this already.

i only graded three homework problems.
three points possible apiece, plus one
for appearing-to-have-done-most-of-the-rest:
ten points possible. the 5-day-a-week classes
have a scale of “grade *two* problems at
*two* points apiece, plus one for completeness”.
so, while any slight mistake (using my scale) is
this is a heck of a lot better than 20%.

when i noticed something without even trying…
an inappropriate long-digit decimal approximation,
for example… on an *ungraded* problem, i went
ahead an remarked on it. but there’ll be quite a
bit of interesting work going *unremarked* here
of course. ideally, every student would talk over
the homework with at least one other student…

a few papers had “no slope” for “zero slope”;
don’t. i let these go by… but, unfortunately,
some writers use “no slope” for *undefined* slope.
so it’s best not to use this language at all.

using graph *paper* for graphs seems to correlate
(positively) with “good grades on HW1”.

intercepts are points, not numbers…
the x-intercept might be (3,0) for example
(not 3). not an enormous big deal…
but it pays to try to be as precise
as we know how.

there was a system-of-equations having
*no solution* on the quiz. i gave full
credit for “parallel lines” in one case…
but we’re looking for “no solution” here.

at least one student panicked pretty badly
on this problem… and *erased* what looks
to’ve been pretty good progress toward the
answer. when you get an equation that
*can’t be solved*… remember that this
doesn’t necessarily mean you’ve done
anything wrong! (and *whatever* you
do… don’t “blank out” on a problem!).

when there *is* a solution for a system,
we’ll prefer *ordered pair* solutions
(for “abstract” problems like the ones at
hand… for “word” problems, it is of course
more appropriate to give “word” answers
[typically including units]).

fractions are *more algebraic* than decimals
and much to be preferred. the calculator is
pretty good at making the conversions, too.
so-called “mixed numbers” like $3{1\over8}$ are
much harder to work with than (so-called)
“improper fractions” like 25/8. students
of algebra should make the effort to get
used to this situation. again, the calculator

### 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.

### smallest repunit prime base 8

gas station without pumps on high school computer science.

### #(A)\times#(B) := #(A\cross B)

Devlin on multiplication. Again.

Joshua Fisher’s take on the bilinear functions approach at Republic of Mathematics; the key Number Warrior thread of May ’09.

Math doodles by Vi Hart. Wow.

Oh. George Hart‘s kid. That explains much.

### Blogging 104. Week One.

(Homework and other administrative stuff is here.)

Blogging 104. Week One.

There were an unusual number of walkouts on the first night… and a much smaller class the second night. We’ll see how things shake out when the points-for-a-grade start going in the log next week.

For all I know, One-Oh-Anything students at Big State might habitually “shop around” on the first night of these Night-for-Day classes (most of the sections meet a Lecturer in a big lecture hall and a TA in smaller “breakout” sessions… I do, in effect, the lecture and the recitation [with less time to do ’em in; the big classes have extra sessions for exams (all the sections get the same exam), whereas I’ve gotta give up class time…].).

And, of course, for all I know, I’m just the world’s worst lecturer-slash-recitation-coach and they’re just running away fast for their own good. You’ll forgive me if I find this option hard to believe. Things’re going pretty well by my lights. Several students responded early on to questions I tossed out to the general room. In some cases (when I needed to refer again to the result), I asked their name (and then immediately used their name in referring to the result). In principle, this is part of my getting-to-know-you process but in practice I tend to forget the names and it’s really about letting ’em know they can and should speak up when they have something to say (and that it’s good idea to know who the other students are and what they can do).

Then the much-smaller Day Two class was all over me with the questions. Actually, it was almost entirely the same three people during the “lecture” bit… but I spoke with several more on the “problem solving” bit.

And the problem solving itself? Well. Too soon to tell. No scarier than one should expect, I suppose. One is mostly doing recap for most of the students here or they wouldn’t stand a chance at this pace. This week we “did” four textbook Sections, “covering” slope-of-a-line, forms of (two-variable) linear equations (general, slope-intercept, and point-slope), and systems of (two) such linear equations, via (a.) algebra (“substitution method” only; the “elimination method” is in Section 5 [but naturally I looked ahead and did one this way]) and (b.) the Graphing Calculator.

The second half (five weeks) of Math 102 at Crosstown Community College where I’ve presented the same material countless times to students mostly even more doomed than, let’s say, the bottom quartile of the 104 students here. I may be misunderestimating something or somebody somehow of course but make no mistake. There is a great deal of doom in these courses in their nature. Math departments pay the rent by “weeding out” students in “required” courses… in programs where Algebra plays no other part. It feels like telling tales out of school putting it thus bluntly. But actually, outside of school it’s pretty well understood. It’s just taboo in school because teachers are incredibly touchy about their grading practices (and administrators are worse).

So. My team on the first day… those who stayed to the end and handed in a Problem Of The Day… did “Find an equation for the line through (-1, 3) and (2, -6)” [or somesuch pair of points]; half of ’em got it perfect. Of course, one should routinely check such work (and not hand in until you know it’s right), but this is still a pretty good sign. What’s more, there were no blind-fumbling-with-formuli papers at all (handed in). All but two or three appeared to have a pretty good handle on the nature of the procedure.

And last night? The shrunken class made it easy and comfortable to talk over most of the papers with their authors and it felt pretty right. I plotted (-7, 11) against some co-ordinate axes and sketched the horizontal and vertical lines through (-7, 11); also the line through (-7, 11) and the origin; find the equations. This used to give 102-103 students fits back at Crosstown. Also a straight-up “solve the system”.

And, like I say, so far so good. On the actual quiz Tuesday? The “special case” stuff… systems with no solution or infinitely many solutions, for example… will probably throw more than a handful for a loop (despite my having… emphatically… “reviewed” it just before the quiz [as I intend to do]). Beyond that, I’m unwilling to predict. Don’t want to jinx it.

Oh, and I’ll draw a map of the room just before they take the quiz and study the names. If I time it right, I might be able to take my “quiz number one” and recite ’em all off (with the “map” hidden from my eyes). Yay, small classes.

• ## (Partial) Contents Page

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