Mike Whitaker wrote:
>
> > cb 278.43 277.2
> > d# 274.69 277.2
>
> You mean C# and Db, surely?
Of course. Sorry. I had some font convertion problems. The flat and
sharp signs actually were * and * (that's how they're mapped in
MetTimes) and I messed up the search-and-replace routines.
Here's the correct table:
Pythagorean Equal
c 260.74 261.6
c# 278.43 277.2
db 274.69 277.2
d 293.33 293.7
d# 313.24 311.1
eb 309.03 311.1
e 330 329.6
f 347.65 349.2
f# 371.25 370
gb 366.25 370
g 391.11 392
g# 417.66 415.3
ab 412.03 415.3
a 440 440
a# 469.86 466.2
bb 463.54 466.2
# 495 493.9
Phil Taylor wrote:
>
> That's not what I understand as a Pythagorean scale.
Yes and no. Your description of the Pythagorean system is the correct
one, but I chose to leave out a couple of intermediate steps to simplify
things a bit. If you go through the cycle of fifths (dividing with two
every now and then to stay in the same octave) you end up with the
ratios I gave for semitones.
>
> >These two temperements have two things in common, they are simple to
> >define mathematically and they are pretty useless musically.
>
> It is indeed a pretty useless scale for any music which wanders very
> far away round the circle of fifths.
>
> We wouldn't get very far without the equal-temperament scale though
> would we? The equally-tempered scale distributes the comma of Pythagoras
> around all twelve intervals so all intervals are very slightly wrong.
> It's the only way you can tune an instrument with fixed tunings and
> have it sound reasonably OK in all keys.
This only applies to keyboard instruments. All other insruments are to
some degree intonated on the spot - whether the player is concious about
it or not. A for keyboards - well, any decent piano tuner will tell you
that he or she does not use strict equal temperament. You have to adjust
the intonation slightly to get a good result.
John Henckel wrote:
>
> Is "well-tempered" and "equal-tempered" the same thing?
By coincidence, I aked that question at the smt maillist (where all the
high-browers in music meet) a week or so ago. The answer seemed to be a
rather clear "No". There's nothing to suggest that Bach used or tried to
use equal temperament, and he didn't seem to have any problems writing
in all major and minor keys.
Laura Conrad wrote:
>
> John> One time I watched a professional piano tuner and was
> John> surprised to see that he didn't use any electronic pitch
> John> measuring device.
>
> Yes, but he still tuned the piano to an equal tempered scale. Piano
> tuning is an art because piano strings are stiff, so the harmonics of
> the string are not the same as the mathematical overtones. Also, the
> tone sounds better if the 2 or three strings that are struck for one
> note aren't exactly in tune.
There are a number of factors a piano tuner have to take into account,
the overtones are never completely pure, unison strings should be
slightly detuned for a fuller sound, bas strings has to be tuned
slightly sharp...
One of the important factors is that he has to somehow compensate for
the shortcomings of the equal temperament system.
Bruce Olson wrote:
>
> Can anyone
> tell me where to find out what Pythagoras said in a reliable
> translation?
Unfortunately not. It's important to remember that Pythagoras lived
quite a bit earlier than the other famous Greek philosophers and most of
our information about him is based on myths. Platon seems to be one of
our main sources, and he was born more than sixty years after Pythagoras died.
Simon Wascher wrote:
>
> To my understanding, there are two groups of tuning systems which both
> are forming the basis of western music:
> 1) tempered intonation scales
> Including everything from pythagorean to equal tempered...
>
> 2) just intonation scales...
Right except that Pythagorean isn't a tempered scale. It firmly belongs
to the just intonation group.
Phil Taylor wrote:
>
> I stand corrected. However, if the system used involved distributing
> the accumulated error from twelve perfect fifths among all the notes,
> the result will surely be an equally-tempered scale, even though it's
> mathematical basis is different?
Not at all. The art of temperament is all about evening out the
"deficiency" of the original fifth based system. There are literally
thousands of ways to do this, all producing quite different results.
John Walsh wrote:
>
> In fact, the even tempered scale hasn't completely taken over.
I'll go further than that. Equal temperament is only used by electronic
instruments. That's one of the main reason (perhaps *the* main reason)
why a musical piece played by a computer sounds "unnatural" and "synthetic".
Frank Nordberg
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