## Archive for the ‘Graphics’ 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.

the blister and still-forming scab is a trophy

from a recent encounter with some hot pigfat.

(the pigfat won.) i knocked off the fat with

the other hand within a fraction of a second

and very soon had the wound in cold water.

& so was able to pop the blister many hours

later quite painlessly… the juice squirted

out a hole too small to see and the burnt skin

went right on lying on top of the still-healing

layer of much-less-badly-burned skin.

our text is #2681: t.~izawa’s _my_ABC_book_ (1971).

digitization has put some weird distorto-mojo on

some of the lettering (as seen on my screen).

also you can barely see my penciled-in notes

and drawings and diagrams. they’re clear enough

in the actual object (or, of course, i’d ink

’em in). anyhow, for the record, i’ve drawn

12 notes of a piano keyboard at “X” along with

“EGBDF” and “FACE”; on the first two pages i’ve

copied out the whole alphabet (and indicated

the pagination… thus providing sort of a

“table of contents”) and copied out each of

A, B, C, and D as a cartoon. the rest is left

as an exercise.

the left-hand photo shows

a nine-point plane: an “ordinary

two-dimensional plane” over the

field with three elements (and its

label is, therefore, ).

such a plane is ordinarily co-ordinatized as

(0,2) (1,2) (2,2)

(0,1) (1,1) (2,1)

(0,0) (1,0) (2,0):

the set of (x,y) such that

x & y are both elements of

the set {0, 1, 2}.

one could convey the same information more

concisely as

02 12 22

01 11 21

00 10 20.

it’s useful for our purpose here, however,

to consider our plane as belonging to a

*three*-dimensional space… (x, y, z)-

-space, let’s say… and as having a

*non-zero* “third” (*i.e.*, “z”)-co-ordinate.

thus, in the photo, our plane is represented by

021 121 221

011 111 211

001 101 201.

the colors come into play in displaying the

solution-sets for various (linear) equations.

the reader can easily verify that the Green

equation—x=2— is “true” for the points of

the vertical line at the right… *i.e.*, for

{ (2,0), (2,1), (2,2) } (old-school), *i.e.* for

{ 201, 211, 221} (“our” version).

likewise “y=x” (the Red equation) gives us

{(0,0), (1,1) (2,2)}… *i.e.*,the “Red line”

{001, 111, 221}.

now for some high-theory. by Algebra I, one has

a well-developed theory of Lines (in the co-ordinate

Plane). the usual approach there is to use the

(so-called) Slopes. the (allegedly intuitive) notion

of “rise over run” allows one to calculate—for any

*nonvertical* line—a number called the Slope (of that

line). vertical lines are said to have “undefined”

slopes. one might also say that they have an “infinite”

slope… though this invites confusion and is usually

best left unmentioned.

y = Mx + B

x = K

are then our “generic” *equations of a line*.

any particular choice of numbers M & B will

correspond to the a set of solutions lying

along a (nonvertical) line having the slope

of M (an passing through (0,B)… the so-

-called “y-intercept” of the line); each vertical

line (likewise) is represented by some particular

choice of K.

now. having different “forms” for vertical and for

nonvertical lines can be devilishly inconvenient,

so, also in algebra I, one sometimes instead uses

the “general form” for an equation of a line in the plane:

Ax + By = K

(with A & B not both zero).

likewise (but typically *not* in algebra-i)

Ax + By + Cz = K

(with A, B, & C not all zero)

is the equation of a *plane* in *three*-dimensions.

thus far i’ve been as precise as i know how.

but now i’m going to start waving my hands around

and making leaps-of-faith all over the place.

in the second photo, four new “points” have been

added into our framework (namely

{010, 120, 110, 100}—the “line at infinity”).

the upshot… without the details… of this move is

that one now has an algebraic theory of “Lines” in a

“Plane” containing precisely our 13 “Points”. more-

over, this theory is “structurally” very similar to

“ordinary” linear theory. in particular, we dealing

with solutions to

Ax + By + Cz = 0—

the “K=0” case of the “general form” for (the 3D case

of the “ordinary” theory).

the Green equation—which must now be written without

its “constant term” (x = 2 is “the K=2 case” of x = K)—

becomes x – 2z = 0;

similarly, rather than (the three-point “line”

of ) “y = 1” (concentric black-and-

-white circles), the (“homogeneous”—for us, right now,

this can be taken as meaning “having no constant term”)

equation is “y – z = 0” (and, again, we pick up a “new”

point at 100).

when the smoke clears… which won’t be here and now…

we’ll have a *very nice* geometry. just as in “ordinary”

space, two distinct points determine a unique line.

but… *unlike* “ordinary” space, it’s also true that

(*any*) two lines determine a unique point.

the same “trick”—homogeneous co-ordinates in finite

fields—converts any plane having p^2 as its number

of points to a *projective* plane having

p^2 + p^1 + p^0

as its number of points. thus there are PP’s having

7, 13, 31 = 5^2 + 5 +1, 57 = 7^2 + 7 + 1, ….

as their number-of-points. there are also some others.

but the margin is too small.

that 7-space has seven-way symmetry is obvious

(rotate the drawing by 2\pi/7—about 51.43\degrees—).

but the *three*-way symmetry isn’t so obvious.

here it is in its 7-color wonder.

we’ve chosen to fix the “Mud” point at the top.

we then chose the “secondaries” line

{Green, Purple, Orange} and permuted;

the “primaries” permute accordingly;

voila.

*************************************************

a 2-way symmetry can be displayed by “swapping”

each primary with its “opposite”:

(RG)(BO)(YP).

i call this one “reflection in the Mud”.

(one may also… of course… reflect

in any of the other colors;

for each (there are three)

“line” through a given color,

interchange the positions for

the other two colors.

the lines-on-blue are

{bgy, bmo, bpr}, so

“reflection in the Blue” has

(GY)(MO)(PR)

as its permutation-notation.)

or, the fano plane presented symmetrically.

each of the three triangle-edges

found along any of the “long lines”

(joining vertex-to-vertex

on the biggest 7-point “star”)

is a “line” of rainbow-space.

check it out. the “points” are

Mud Red Blue Green Purple Yellow Orange

the lines are MRG, RBP, BGY, GPO, PYM, YOR, OMB.

here’s a version from last year.

this new one’s much cooler.