Re: Digital vs. Binary Computers

From: Beth (BethStone21_at_hotmail.NOSPICEDHAM.com)
Date: 04/25/04 

Date: Sun, 25 Apr 2004 06:08:57 +0100

Frankie say:
> Beth wrote:
> > Yeah; Duodecimal (base 12)
>
> Didn't they used to use the Duodecimal System in Libraries? :)

Don't know; Ask a librarian...but that will be an avenue I'll research
for my "Duodecimal restoration campaign" ;)...

> > Oh, and then there's - C, C#, D, D#, E, F, F#, G, G#, A, A#, B
>
> What have they done with B#? Seriously, musical notation strikes me
as
> weird - were they afraid they'd run out of letters, or what?

The ones without sharps or flats happen to be the "C major
scale"...it's quite possible - actually, I think it was the case -
that they didn't originally exist but were added later...

Right, let's see how easy this is to describe (ignore if you like, I
just want the practice at trying to explain it as easily as possible
:)...right, for simplicity, our "piano" only creates pure sine waves
at the respective pitches...now, as is the "concert pitch" convention
(that is, this is _by definition_ to make sure all instruments tune to
the same pitch to play together but is otherwise not "special" in any
way :), if I hit the "A" note above "middle C", our "sine wave piano"
will produce a 440Hz sine wave...

Now, what makes notes sound "harminous" together is actually rather
simple in an abstract sense...if we were to half the pitch - 220Hz -
or double the pitch - 880Hz - and play this alongside our 440Hz "A"
note then the notes would fuse harmoniously together...they'd sound
very pleasing to the ear...and, in fact, the "A" note below our 440Hz
"A" _is_ at 220Hz...and the "A" note above it is at 880Hz...all the
"A" notes are multiples of this basic frequency (pitch? Frequency? For
the purposes of this discussion so far, consider it identical
:)...they are what's termed an "octave" apart (why it's "_oct_ave" as
in eight will be explained later...for now, just treat it as a "name"
for this "half / double" relationship between musical notes :)...

Great, so we play 55Hz, 110Hz, 220Hz, 440Hz, 880Hz and so forth and
it's all very "harmonious" sounding...but, ummm, kind of a "limited
vocabulary", isn't it? Not too many tunes you're going to make just
whacking "A" notes all day long, however "harmonious" these notes may
be together...we need some more "notes" in between our "A" notes...

Right, how about something like a "one and a half" relationship? That
is, we play 660Hz to go along with our 220Hz and 440Hz and 880Hz...and
then all the multiples of 660Hz - like 330Hz, 1320Hz, etc. - are
included as well, just like we did with "A"...yeah, that works...a
different kind of harmony that's not quite so "perfect" sounding as
the "octave" half / double thing but still very harmonious...

If we put this into an oscilloscope so we can _see_ what we're
hearing, then play our 440Hz with an 880Hz (the "phase" - most easily
describe as "where we start on the sine wave" - is going to be exactly
the same for both...BUT, this does not actually matter, in the sense
that the brain will still find them "harmonious"...it only effects the
"quality" of the sound - constantly changing the "phase" gives a
"spaceman"-like special effect onto the sound - but not the "harmony"
between notes :)...well, as these two are multiples of each other,
then the resultant combined wave (being sound waves, playing two notes
together, one doesn't in any way "stop" the other - just like two
ripples on a pond seemingly "pass through" one another - so that our
resultant sound is merely a case of _adding_ one wave to the
other...so, yes, play a 440Hz sine wave and a 440Hz cosine wave
together - both the same amplitude (or what we usually call "sound
loudness", the same "volume") - when added together they exactly
cancel each other out and you'd get silence ;)...is itself perfectly
"fitting" with the notes...choosing a "horrible" harmony quite
delibrately, say 437Hz and 861Hz, we can see what isn't
"harmonious"...these two played together don't "fit" and their phases
don't match...resulting in a very odd combined wave where we have
"beats"...that is, it's "stopping and starting"...though entering the
realm of psychology a little, things don't naturally resonate like
this - all harmonies from the same naturally vibrating object, like a
string or banging a steel ***, would all resonate at frequencies
that "fit" because they are all coming from the same pitched
vibration - and the ear picks this up and, well, just doesn't like the
sound of it too much...it's pure psychology and evolutionary _why_
humans don't much like such "unfitting" frequencies and like "fitting"
ones BUT, regardless of why people like music - which I won't pretend
to explain - this is the underlying "good" and "bad" things in the
actual oscilloscope ;)...

Carrying on with this basic idea...you can, in fact, more or less go
around all the notes with "1.5" - because it has "wrap-around" - and
come up with the entire musical scale out of it...well, not
quite...you're trying to fit a "linear" kind of thing onto an
exponential scale (because it doubles all the time to go up an
"octave", the "gaps" between things keep growing apart)...you'll come
back around and find you're fractionally "out" from where you
started...

This problem is, in fact, unsolveable...sound is actually exponential
in nature (not just the pitch like this but amplitude too...but that's
another topic ;)...but, for playing convenience, we want the same
notes - repeating octaves - all the way along...rather than, for
instance, there being more notes at the higher end of the scale than
at the lower end...remembering, also, that we need for all instruments
to follow the same system or they can't play with each other (those
pitches have to "fit" or, even with a minor variation, it'll sound
horrible...that's the sole origin of our "A = 440Hz" thing at the
beginning...this is "concert pitch"...we just "define" a particular
note to be at a certain pitch and work from it to tune everything
else...there is actually nothing "special" about it other than being
an "agreed" standard :)...

Which is another point to mention, if you started with, say, 400Hz
rather than 440Hz but did exactly all the same things as before -
200Hz, 800Hz, 600Hz, 300Hz, etc. - then it would still be "harmonious"
and would sound the same - if you were playing a particularly famous
tune then everyone would still totally recognise it - just that the
overall pitch is a little higher or lower...it's all about the
relationships between them, rather than their absolute frequency or
anything...

Right, so, we have all of our musical notes - C, C#, D, D#, E, F, F#,
G, G#, A, A#, B - and then I decide that the first note of my song is
going to be "C"...I have, in fact, _doomed_ my choices
immediately...and, you'll actually tend to find that much music
_begins and ends on the same note_ (this is how with those really over
the top endings that some songs have - where the singer holds out
really long notes - even if it's the very first time you've heard the
song...you exactly _know_ which is the final note and only that note
has the proper "finished" sound to it...if you check it out with your
oscilloscope or look at the *** music - though clever composers _do_
sometimes put things together differently - this final note just so
happens to be the same as the first note :)...if I hold down my "C"
note, then if I was to play a "C#" with that, it's a horrible,
horrible sound...oops, not all notes go together from our scale...each
note _does_ somewhere in that scale have other notes which "harmonise"
well with our "base note"...but not all of them...if you've got a
piano or something, press C and C# and prepare to whince ;)...

Hence, as I'm holding down "C", the other notes from the overall full
scale of notes here that I can in some shape get away with playing
alongside my "C" (there _is_ a touch of "cultural bias" in how much
"tolerance" to "ill-fitting" harmonies a person has...Indian, Chinese
and Malaysian traditional music have their own scales - still obeying
ideas like the "octave" because these are _human_, not cultural -
which tend to have far more notes in them...their cultural ears are
more "tolerant" than Western ears to the Western scales...but it _is_
all based on the same ideas...as they say, "music is the universal
language" because _everyone_ finds an "octave" harmony "nice" - 440Hz
with 880Hz - as it's a "pure harmony"...the "fifth" harmony - 440Hz
with 660Hz - is still good with most people but it's not so "pure"
sounding...hence, you kind of get the "cultural" point,
Hopefully...just how far you can push this and what kinds of harmonies
are "usual" all depends on what the ear is culturally attuned to
hearing...that's why some of that "traditional" music over towards the
East can sometimes "sound odd" to Western ears...never totally
unpleasent - same theory underlying it all - but they've made slightly
different choices...normally a choice of "more notes, worry less about
pure harmonies" which means Western ears sometimes feel that the
"harmonies" go a little too far to the "dischord" end of things...but
listen to it long enough and you'll get used to the fact that such
music just has "more tolerance" for less harmonious sounds :)...

The notes I can "get away with" - by Western "standards" - to play
along with the "C", in some form? C, D, E, F, G, A, B roughly
speaking...yup, all the notes without sharps or flats...this is the "C
major scale"...interestingly, once you select a "scale" like this out
of the full choice of notes, the mind kind of "remembers" as it is
listening...hence, the whole piece of music tends to all be in this
same "scale" (_very_ clever composers _do_ actually change scales
half-way through a piece of music and then change back later
on...this, though, is NOT an easy thing to get away with unless you
know _exactly_ what you are doing...basically, Mozart and John Lennon
get away with doing this...but most pop music has the sense not to
even try and stays in the one "scale"...it's a _really_ cool thing if
you can manage to pull it off but doing that really ain't too easy to
do...in a sense, you have to "fool" the listening ear a little as
you're going on - also, you couldn't change to any old "scale" but one
that's pretty close - so you've just got to understand music
inside-out to really know what you're doing...great composers even
delibrately use the disharmonious notes I'd mentioned earlier with
subtle skill - "accidentals" (notes not part of the "scale" you're
currently in) - but, again, if you don't know exactly what you're
doing, you'll never get away with it and everyone will whince at all
the "bum notes" you've dropped into the music...okay, "bum note" is
not quite a "technical term" but when you hear someone playing a tune
and they hit that wrong note and you just can't help but whince at how
horrible that noise just was...yeah, they've hit a "bum
note"...whacked a note outside the "scale" they are currently
playing...and it just sounds horrible ;)...

Although, if you hit, say, "D#" first then the notes that work with it
are - D#, F, G, G#, A#, C, D - the "D# major scale"...you'll note that
the "gaps" between notes are the same "distance" as they were for the
other major scale...you could even play the same tune as before but
just "transpose" - moving all the notes up or down the same amount -
it all up to this scale and, yeah, would sound exactly the same...if
you were playing some famous Mozart tune then, yeah, they'd all
recognise it (but the obsessive classical music buffs will get really,
really annoyed that you're playing it in the wrong scale...yeah, they
get _really_ fussy about that kind of thing...must be _exactly_ as
Mozart composed it...now you might begin to "get" why classical music
often has "in D Major" or "in A Minor" often placed after the
name...actually, there _is_ more to it than just obsession with
playing it at the right pitch but it's a story in itself to explain
;)...it just sounds "a little higher" or "a little lower" in pitch...

Singers, of course, can _exploit_ this when their voice is basically
only good at a certain pitch...that is, for Barry White or Johnny Cash
(unfortunately, neither of you are still with us anymore ;), you'd
probably want to be transposing everything _down_ to their
voices...although, in the case of those two, that's an awful long way
down to the bottom there ;)...you might sometimes see, when someone's
playing live with a guitar and there's someone else singing along with
them, that they stick a "capo" thingy across the notes...this just
raises the whole pitch of the guitar by a certain amount - by
effectively shortening the neck of the guitar and, correspondingly,
the length of the strings - and they can play away as they would have
played without it but it's all raised in pitch...it's a kind of way of
"cheating" with a guitar to "transpose without thinking"...just stick
on the "capo" at the right place for the singer's voice and play away
as you would have played, anyway...just further down the neck of the
guitar...although, you can only go _up_ with this method because you
can't stretch the length of the neck and put it "further down" than
the end of the neck of the guitar...anyway, if you see one of these
things on a guitar, then now you know what it's there for...raising
the pitch a little in an easy way...usually to suit the singer's
voice...well, only the real good singers have a massive "range" and
can sing low or high without problems...most people have an actually
rather limited "range" and it's easier to just move all the
instruments to match than try to force the person to sing deeper or
higher than they usually do...a woman won't be singing any Barry White
hits (at least, not at the original pitch ;) and not too many men
could be as "squeaky" as Kate Bush at the start of "Wuthering
Heights"...the BeeGees could probably manage it but not too many
others...best not to try if you're not confident, though, as going too
high with a male voice? It just cracks and can't make any sound at
all...NOT a particularly clever thing to happen in the middle of a
live performance...don't try the "falsetto", lads, unless you're like
Prince or the BeeGees and are "masters of the sqeaky voice"...mind
you, fair play, at least men can _attempt_ it to a degree...if it's
too low, then, with women, it's just a case of "sorry, can't sing that
at all...raise the pitch a little...actually, no, a lot" ;)...

Hence, musical notation is, indeed, weird...but there's good reasons
for all of it...if you want to play without sounding too dumb, Frank,
then you can do what a lot of those awful, awful cheap pop musicians
do and follow the "always play the white notes" or "always play the
black notes" (you lose a few potential white notes that are
"permissible" doing that, mind you ;)...where "white notes" means "the
notes without any sharps or flats"...those notes _are_ the so-called
"C major scale"...and, if you want to learn the other "major scales",
it's actually kind of easy...pick a number between 0-11, then from
each white note, count up that many notes...do that for all your
original "white notes" and, hey presto, a new "major scale"...so, "C
major -> D# major" is just "move up four notes" (a "fourth" ;)...

[ For the musically minded, I'm delibrately ignoring "flats" and
"minor scales" because they _are_ effectively synonyms, anyway..."G# =
Ab"..and, yeah, now you see why I pick "sharps only"...you can _type_
the sharp sign at a keyboard...you can only kind of "approxmiate" the
"looks a bit like a b" flat sign ;) ]

As for why the notes ain't "symmetrical" looking - e.g. why no "E#" or
"B#"? (Actually, not strictly true..."E#" is just "F" and "B#" is just
"C", traditionally...but you know what we mean ;) - that's because
they aren't arbitrarily made-up...it's the old "trying to fit a linear
set of notes onto an exponential scale" thingy there...that is, it
_looks_ odd but when you know the theory behind all the "frequencies"
stuff underlying it then you get the reason why...the notes are all
"factors / multiples" of some kind with each other (some more
"fitting" than others ;)...this is just the pattern that emerges when
you try to get all your harmonics nice and equal...well, when you're a
Westerner following Western cultural biases...there are other ways you
can do things - hence, it _is_ significant that "12" or "duodecimal"
was "selected"..._that_ part _is_ an "arbitrary" decision for the
so-called "Western scale" - as some traditional Indian, Chinese and
Malaysian (amongst others) show by adopting different "scales"...the
Indian one has almost twice as many to choose from, for example...

In fact - and I was initially amazed to find this out - the Germans
actually often write "H" for "A#"...nope, all the rest are the
same...don't quite know what that's all about...but they sometimes do
that over there...which makes me wonder if Mozart and Beethoven really
_did_ use "H" notes in their music...that would be funny: "Yes,
Beethoven used H notes" / "What? You're talking nonsense! There are no
H notes! It only goes up to G!! 'Everyone' knows that!"...but, ah,
that's where you are wrong...though mostly universal, there is the odd
"deviation" in musical notation here and there...so, everyone will
laugh, thinking you have no idea what you're talking about, but,
secretly, you can have the last laugh because, you know, he most
probably _did_ use "H notes", as stupid as that might sound...

Anyway, not a case of "worried about running out of letters" or
anything...though I'm not sure of the exact history, it's probably
more like they "discovered" the C major scale and were playing that
all the time - "just the white notes" - and came up with "C, D, E, F,
G, A, B" for that...then someone comes along and goes "oh, you know
there's, like, other notes and stuff, don't you?" / "ummm, okay, where
are they?" / "Neatly in between some of the notes you've already got"
/ "Cool...we can just squeeze them in...so, there's 'C', 'C and a
bit', 'D', 'D and a bit'...yeah, like we're 'sharping' it up...yup,
let's call them 'sharps'"...makes you wonder why the opposite, though,
is "flat" rather than "blunt", eh? ;)...

The real interesting thing here is the "unsolvable problem"...it's not
actually possible to completely 100% "fit" a linear thing like the set
of notes onto the exponential range of frequencies (if you start with
440Hz and do the "one and a half" thing wrapping-around until you get
all the notes, then you'll find you'll be "slightly out" by a fraction
when you return back to "A" again ;)...in the olden days, they
actually did stuff like decide "right, we're going to be in D major"
and then just tune things in to "bias" that all the notes work
perfectly with "D" - the "base note" of the chosen scale - and, yeah,
that's the "other reason" why classical music often has "in D
major"...if trying to reproduce the music exactly as it was written by
Beethoven, then you also follow their "tuning up" methods and the way
that their pitches were just "biased" to whatever scale they were
playing in...

The "modern" scale we see today is actually a case of trying to "equal
it all out"...a case of "we'll tolerate a small 'error' on each of the
harmonies"...mind you, calling it "modern" is a touch incorrect in
that Pythagoras was the guy who, being very mathematically minded so
_would_, indeed, think "must distribute the 'error' and get a 'line of
best fit' for the problem", came up with the idea in ancient Greece
(not sure if this was anything but "theory", mind you..._could_ they
actually measure frequencies accurately enough then to be able to make
use of his idea? Would have to ask a music historian...there _are_
"tricks" employed to get really quite accurate stuff from listening
alone...after all, a piano tuner just carries around a bunch of
"tuning forks"...bangs them the side of the piano and tunes the notes
to that...how does the piano tuner do it? Well, 99% is about having a
"good ear"...but what he's specifically listening for are the
"beats"...the ear is _very_ sensitive and, as noted, if harmonies
don't exactly "fit", then the ear can hear it...when the two exactly
match up, they are perfectly "in phase" with each other...when they
don't quite match up, there's an audible "beat" to it...you know, one
note plays steady but the other is out of tune so it keeps "missing"
that "perfect phase" and has a regular "beat" in missing the phase all
the time (hard to describe but Hopefully you get the idea :)...when
you can't hear "beats" in the sound, they must be nicely in phase with
each other...so, you have a bunch of tuning forks for particular notes
and just tune directly to them...then you can tune the other notes in
relation to each other...and can "double check" you did it right when
you reach the pitch of your next tuning fork...if it's _still_ "in
tune" with the next tuning fork, you got the pitches tuned to each
other right beforehand...if not, well, the piano tuner pops back down
the piano and makes "adjustments"...this, mind you, makes it sound
deceptively easy...as it were, "that's the theory" but it ain't so
easy in practice...and a lot of piano tuners are blind, in fact,
because of the old "when you lose a sense, your other senses become
more acute" (not really true, mind you...more a case of "when you lose
one sense, you _pay more attention_ to your other senses", so to
speak...and that focus _does_ make them better because your eyes,
ears, nose, skin, etc. are only "half the battle" when it comes to
your senses...a big chunk of it is to do with the brain's
processing...so, blind people really do hear better...but it's not
because their ears physically "improve", like they were "gaining super
powers"...well, not unless they happen to be the superhero
"Daredevil"...who might be the only disabled superhero in comics...oh,
no, wait...that "Professor X" in the X-men films is stuck in a
wheelchair...nice touch, that, really...he is "the world's greatest
mind" but his body? Ah, not so good...can't even walk...mind you, I
like the X-men stuff because, underlying the "superhero" stuff, it's
clearly all about "prejudice" and "discrimination"...people persecuted
and killed off because of their genetics? Yeah, no accident that the
bad guy, Magneto, is seen outside a Nazi concentration camp at the
start of the film, watching his parents murdered that, yeah, you can
kind of understand why he went totally insane and on some "kill all
humans" philosophy...the dark irony being that he, of course, is being
a total Nazi himself thinking that way...thinking from "revenge"
only...then there's other little "touches" in the X-men that show that
though it might be a "comic", the writers were actually putting a lot
more into it than your typical "Batman and Robin" crap...that
Wolverine is, basically, a purpose-built killing machine but, yet, the
"professor" has no prejudices so he still tries to help him and make
him fit in...that Nightcrawler - the blue guy who teleports -
basically looks completely like your stereotypical "devil" with tail
and pointy ears...but is actually one of the kindest people in the
team and a very devout Roman Catholic...and perhaps the only superhero
team where their "international" nature is actually _part of the
point_...as I say, it's all really about "prejudice" and
"discrimination" under the surface but is wrapped up in a "metaphor"
that rather than "black skin" or "being Jewish", the thing they are
persecuted for is having "weird powers"...actually, very _frightening_
weird powers at that...in a sense, the "in joke" is that they are
turning the "superhero" thing on its head...and actually facing a
human reality that if there were people with "powers" like this,
they'd be persecuted and discriminated against because,
understandably, everyone would be scared crapless by some guy who can
just wander into your mind whenever he feels like it or a "devil"
shaped person just teleporting wherever he likes...anyway, I know what
comes next because my brother used to collect X-men comics when he was
a kid and I'd read them when I was bored - but, as I say, under the
superficial surface of "magic powers" and "big fights", there's
actually a very strong and compelling story about people persecuted
for being different...hey, I Love "Cartoon Network" and don't care if
something is "kid's stuff" so long as it's fun...but, in this case, it
actually _isn't_...maybe other comic books were about that but this
one never was...the "prejudice" and the "politics" were always at the
front because, for instance, the "professor" and Magneto - Good Guy
and Bad Guy - actually were friends and do _talk_ to each other about
their "philosophies" without actually fighting over it...yeah, a comic
book with "magic powers" but, umm, the Good Guys are trying
_diplomacy_ rather than a big fight?? Yeah, it still fits in with all
the typical "superhero" fighting crap but there's another level that's
very much not simply "kid's stuff" to it too - and "what comes next"
is "Dark Phoenix", where, again, we see things turned on its head in a
non-typical way for "superhero crap"...Jean Grey...well, let's not
spoil the plot for those that don't know...it was actually done a
whole lot better in the comic book because it happened over a really,
really long stretch of time and was an unbelievably "gradual" thing (a
"subplot" taking _two years_ from start to finish! Chris Claremont,
the writer, must be one of the most patient writers I've ever
seen...he was the one that also turned it from "typical superhero
crap" into a kind of "soap opera about persecution" and, well, kind of
_completely ignored_ what people usually think about "comics = for
kids"...he broke the conventions by having stories with *gulp* people
talking about "philosophy" and "relationships" for an entire issue
rather than just having a "big fight" about everything all the
time...just wrote the story he wanted to write and didn't care what
people might think of it...in fact, the movies kind of "traitor" that
a bit by going back to "typical superhero crap" too much for the
popular audience and don't focus on the messier subjects...but you can
still see some of it poking through...a disabled guy running the team,
a killer trying to "reform", a guy that looks like a "devil",
etc....shame they left out "Colossus" but I guess the Cold War thing
of having a "Commie Russian" on the team might not work the same way
it used to...and I guess they think that his half-girl, half-demon
sister - kind of a "Good Guy" and "Bad Guy" at the same time because
her powers actually come from hell - was too confusing and would be
objected to by the "Bible Belt"...as per bloody usual, may I say...and
how Kitty - who walks through walls - is actually scared and
distrustful of "devil-looking" Kurt at first but they end up
bestfriends...really, the entire thing is one long story about
"prejudice" and "discrimination" that's really cleverly wrapped up in
a "superhero" metaphor...such as calling them all "mutants", which
sounds really distasteful and twisted as a name...that's kind of the
running irony of the thing: They have all these great "superhero
powers" like kids often imagine of flying around and they save
people...but, at the same time, they are "lepers", they are _hated_,
they remain hidden or those they try to protect would very likely try
to kill them off in their fear of "the mutants"...to be honest, that's
the thing which kind of sucks about the films...they chickened out of
telling the _proper_ story...it's actually, I'd say, comparable to
Orwell using farm animals to make his political points...or Aesop
using animals for his tid-bits of wisdom...exactly the same literary
device but because this comes from a "comic" rather than a "book",
it's discriminated against...yeah, exactly the topic it tries to
address: "prejudice"...hence, mock it and you're actually proving the
point)...but, well, Hollywood movies for you...you know how they
mangle famous books the same way...cutting out seven important
chapters just to make it fit a few hours viewing...

Anyway, how the hell did I get onto that subject?

> > All we need to do is have a look to see if the Arabs have another
two
> > squiggly symbols we can use for ten and eleven...well, it would be
> > nice to "keep it in the family" and make the two new digits
"arabic
> > numerals" too...
>
> Or maybe a couple of Hebrew symbols for "diversity" :)

Oh, you could have great fun with something like that...you know,
coming up with your own symbols and some "system" of writing them
out...but, yeah, make a leap too far and you're bound to "leave
everyone else behind"...like you're scribbling all these weird symbols
all over the place and everyone's, like, "what on Earth is going on
there?" / "I'm doing some maths" / "Yeah, really?
Okay...whatever"...and you can hear them mutter "nutter" under their
breath as they walk away...

> > *shakes pom-poms*
>
> ... I didn't have any comment on that... just a nice image, so I
left it
> in...

:\

> >> The slide rule is definately a decimal digital computer
>
> Actually, I think I'd class a slide rule as an "analogue" device
(unlike
> an abacus). At least the old wooden ones... maybe the newfangled
plastic
> ones are digital :)

Well, you're now answering someone else, not me, here, actually...and
I'd agree with you, Frank, clearly "analogue"...hey, you even spelt
the word properly too!! ;)

> >> simply chop down a tree and build a log table. ;-)
>
> How big a crowd are you expecting? Do we want a four place table or
a
> five place table? :)
>
> >>But since "bits" are "BInary digiTs",
>
> So in a three-way system, whether we call it "trinary" or
"tertiary", it
> would be a... never mind...

"Trits", _obviously_...or, perhaps, "terts", if it's "ternary"...why,
what you thinking, oh filthy-minded Frank? ;)

Beth :)