Re: Digital vs. Binary Computers

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

Date: Sun, 25 Apr 2004 09:14:53 +0100

T.M. Sommers wrote:
> Annie wrote:
> > Beth wrote:
> > > Oh, and then there's - C, C#, D, D#, E, F, F#, G, G#, A, A#,
B
> >
> > Ummm...technically, there's no D-sharp, G-sharp or A-sharp,
> > Bethie-poo. They're E-flat, A-flat and B-flat, respectively...
> > although music people would certainly understand what you meant.
>
> If you are talking about notes, D#, G#, and A# certainly exist,
> and depending on the termperament being used, are not necessarily
> the same as Eb, Ab, and Bb, respectively. If you are talking
> keys, then you are correct for the major keys, but d#, g#, and a#
> exist.

Thank you; Saved me a reply there...

You know, in Germany, they sometimes call "A#" by "H"...I kid you
not...don't go automatically assuming even the notation is
internationally consistent...it mostly is but not totally...if I'd
have listed "H", you'd have thought me completely bonkers...unless you
knew about this "deviation" from the usual conventions we're more used
to in our cultures...

And, indeed, T.M.'s on the money about "temperament" here...there's
more than one way to "tune up" your instruments and this _makes a
difference_...that's why there is "sharp" as well as
"flat"...otherwise, why not simply say "there's only sharps" and kick
out "flats" completely because they are merely "synonymous"?

Well, you _could_ get away with this attitude in the "temperament"
that your average electronic keyboard is automatically tuned to...in
fact, this is why I just put the "sharps" because, in such a
temperament, I _do_ tend to simply consider them "synonymous" for the
sake of convenience (and to explain to Frank without complication,
leaving out the more "obscure" bits helps make it clear :)...that is,
D#'s frequency _is_ Eb's frequency _when_ you have equal
temperament...but if you don't have equal temperament, then this isn't
necessarily the case...

I collected some information about this before (off the 'Net...pardon
lack of references but I just didn't write them down in any way
because I wasn't interested and just wanted the raw facts ;), so let
me copy some over to give an appreciation:

------------------ 8< ------------------

Pythagorean Tuning

Western music is generally considered to have started from Pythagoras,
the ancient Greek. Pythagoras devised a system based on mathematical
principles. He defined the scale around the ratios of the fifth, being
in the ratio 3:2 exactly, and the fourth being 4:3 exactly. The
difference between these two was then 9:8, which he defined as the
tone, or whole step. He then divided the octave so that there were the
seven notes, just as we have today, but to get the mathematics to add
up he was left with two semi-tones which he defined as 256:243.

Thus the Pythagorean scale has the following intervals:

Cumulative Intervals: 1 9:8 81:64 4:3 3:2 27:16
243:128 2
Note: C D E F G A
B C
Incremental Intervals: 9:8 9:8 256:243 9:8 9:8
9:8 256:243
Cents: 204 204 90 204 204
204 90

It is interesting to note that Pythagoras did not recognise the major
third, which is distictly sharp at 81:64 compared with the ideal of
5:4.

The chromatic Pythagorean scale is formed by inserting semitones equal
to 114 cents in such a way as to keep all perfect fifths true, except
for the interval G#-Eb, which needs to be adjusted so that the
intervals add up mathematically. This difference is known as the
"comma of Didymus".

Ptolemaic Tuning

Ptolemaic Tuning is generally referred to as "just intonation". To
create perfect major third intervals, this system alters one of the
fifths, D-A. (And thus the major sixths are also perfect, except for
F-D.) This makes the triad D-F-A quite unusable, although the others
are perfectly in tune. The Just Intonation scale employs two different
sized tones in the ratios 9:8 and 10:9, and thus it can hardly be
considered satisfactory even for purely melodic music.

Cumulative Intervals: 1 9:8 5:4 4:3 3:2 5:3
15:8 2
Note: C D E F G A
B C
Incremental Intervals: 9:8 10:9 16:15 9:8 10:9
9:8 16:15
Cents: 204 182 112 204 182
204 112

The above comments hold true for scales and intervals based solely on
the "white notes". Adding the "black notes" to give a full chromatic
Just Intonation scale creates more perfect fifths, but major thirds
and sixths which are not true. In fact the result is three different
sizes of semi-tones. (16:15, 135:128 and 256:243.) It is thought that
this system, although considerably debated, was not used much.

Mean-tone Temperament

In this scheme the major thirds are made exact. This results in the
fifths becoming slightly flattened but in such a way that the error of
the Ptolemaic system is spread out over four fifths. This reduces the
dissonance and makes the fifths more acceptable. The whole tones are
also all equal in size, being half the major third. Melodically the
Mean-tone scale is more acceptable than the Ptolemaic scale.

Cumulative Intervals: 1 5:4
2
Note: C D E F G A
B C
Cents: 193 193 117 193 193
193 117

The chromatic Mean-tone scale has semi-tones of two very different
sizes: wide 117 cent semi-tones in the diatonic scale, with 76 cent
semi-tones balancing the whole-tone interval of 193 cets. Mean-tone
termperament was designed for keyboard instruments and it was an
acceptable compromise as long as the "black notes" beyond Eb or G#
were not used. The G#:Eb fifth was so bad as to be unusable - it was
often given the name "the wolf".

Equal Temperament

In equal temperament the octave is divided into 12 equal semi-tones
each of 100 cents. The intervals are then built from the semi-tones.
For example a fifth is 7 semi-tones and a third 5. The Equal
temperament scale is universally used today in Western music.

Cumulative Intervals: 1
2
Note: C D E F G A
B C
Cents: 200 200 100 200 200
200 100

The chromatic Equal temperament scale with all semi-tones = 100 cents
means that perfectly tuned intervals have been totally eliminated.
However, the mistuning on fifths is only 2 cents and on thirds 14
cents which the ear does not appear to mind. The big advantage is that
all keys are equally usable.

------------------ >8 ------------------

What is most interesting to note is that this is, mathematically, an
"unsolvable problem"...that is, if you insist on the set of notes as
we have them and that you want "pure harmonies" between them all, this
is actually an impossible task...you'll note that all of these
"temperaments" are "adjusting" things to suit one ratio, often at the
detriment of some other harmony...

Hence, there's many different "temperaments" (if you this is a lot,
then you should see another webpage I copied that lists tens and tens
more than this...there was a bit of a "frenzy" in inventing new ones
all the time at one point...also, there are "temperaments" with some
pretty major "biases" in them to make them unuseable but for
particular applications: e.g. play in D major, say, and everything is
fine...try to "transpose" to anything else and it turns into a
complete mess everywhere...an "extreme" kind of bias...makes that
particular piece in "D major" absolutely pure and sweet but you can't
really take it anywhere beyond that ;)...and, really, _all of them_
are equally as good as the next one...there is no "solution" to the
problem as it stands...it's a case of "biasing" it in particular ways
that are useful but not so horrible to listen to that you're whincing
all the way through a tune...

What your ear is listening to in frequency terms is
_exponential_...what everyone's trying to do is fit them in with some
"linear" scale of "notes"...and then what is expected is that the
"ratios" will all be nice perfect, "pure" harmonies
throughout...sorry, you can't have _all_ of those requirements at the
same time...I mean, Pythgoras? Ptolemy? And, yeah, old Wolfgang
Amadeus and Johannes Sebastian had a go at making up their own
"scales" too, just for the "fun" of it all...if these people - the
best mathematicians and musicians you could name drop - couldn't work
it out then, really, it _can't_ be worked out...

It's not a question and, therefore, there is no "answer"...it's about
"balancing" out your requirements...

And, as the stuff above hints at, what the "modern" scale we're used
to - that's now so common that, yeah, it's surprising how many
musicians, not just people, have no idea of "temperament" because,
like, everything is scaled to "equal" temperament and is never
questioned if it could be anything else - is simply trying to achieve
is a kind of "they are all slightly 'out of tune' but the 'mean error'
overall is reduced"...that is, you play a little tune in one scale
and, yeah, how conveniently you can just "transpose" it up without any
concerns for "temperament" at all into some other scale...this is the
"balance" that "equal temperament" goes for...a _practical_ one,
basically...you can tune up all your instruments - or, with electronic
keyboards, pre-program them - to a certain set of frequencies and,
there we have it, no need to give a crap about any "temperament"
thereafter...Western ears are totally accustomed to the "error" on all
the harmonies (yup, they actually aren't "pure" in all cases but this
is where "culture" comes into it...if the ear hears something often
enough, it gets used to it :)...

Hence, in "equal temperament", it might not be totally correct or
"convention" to do so, but I can presume "A# is the same as Bb...they
are synonyms" and get away with it...yeah, not strictly technically
correct, but, hey, we're already working with a "flawed"
temperament...and, in terms of frequency, I'm making no error
whatsoever because that's how equal temperament "cheats"...

And, top of the list, I listed only sharps because that's _easier_...a
delibrate simplification...I was just listing out all 12 to
demonstrate that, yes, there is 12...providing all the detail
necessary to explain musical theory, its relation to actual
frequencies and acoustics, as well as "temperament" just to list out
12 notes? Again, some of you people have got to learn to stop being so
"anally retentive" about everything...yes, I'm _ABSTRACTING_...but
short of providing novels after novels to teach someone who knows
nothing every single piece of musical theory (heck, I don't know it
all...and, from the responses, seems only T.M. really has some
reasonable understanding of what's involved...the rest of you are
replying _thinking_ you know it...I _know_ I don't understand it all,
to be honest...but what I do know, I attempt to report as Faithfully
and sincerely as I can ;)...we're going to _HAVE TO_ "abstract" this
because it's a very non-trivial subject...

I'm _STILL_ abstracting and simplifying, by the way, in that longer
response I gave to Frank (like, I've copied _some_ of the information
I have about "temperaments" in this post and I've got lots more stuff
like that besides...never even said "Just Intonation" in the response
to Frank...I could have because I have the information at my disposal
but, again, one has to understand the need to sometimes _ABSTRACT_ and
_SIMPLIFY_ things to make them clear...this can often, yes, make them
deceptive or technically "slightly off"...but it gets the basic idea
across...if someone is interested and does "further research" - heck,
not like they are _only_ going to take one post from me and consider
that as the sole "resource" on the subject that exists on Earth (and,
if they do, one could say "they deserve what they get" ;) - they'll
learn which bits were "abstracted" and which weren't...but at least
the "basic appreciation" given to them allows them to understand
enough to _begin_ making that quest into finding out what's
what...and, in truth, Frank probably doesn't care...yup, as simple as
that, Frank don't give a crap...so, yup, just "chatting" with Frank,
is all...NOT an attempt at some "New Communist Manifesto"...please,
people, calm down with the pedanticism, thank you very much ;)...but
my aim was "basic appreciation of what's involved" so that Frank's
"B#" question was at least roughly addressed...not the final word on
musical theory forever more...there's probably some "technically
objectionable" simplifications in it...but, please, so long as it's
not grossly in error then appreciate it's only meant to be providing
an appreciation of the _spirit_ of what's going on...it's not meant to
be a "how to compose music like Mozart could in 9 easy steps"
instruction manual...heck, only Mozart himself could even begin to
Hope to write such a "manual" and then it's quite possible that
there's simply stuff he could never express in words or teach to
others directly - "intuition" and "having a good ear" is certainly a
big part of music - that even he couldn't manage it...

Beth :)