Re: [ARSCLIST] Temperament--was: Are we at the end of the road musically??

[Don Cox <doncox@xxxxxxxxxxxxxx>]

Hello Steven

On 09/10/06, Steven Barr wrote:
> ----- Original Message ----- 
> From: "Don Cox" <doncox@xxxxxxxxxxxxxx>
>> Certainly. You can start from any note, such as A=440Hz, and then
>> apply a constant ratio of the 12th root of 2 from note to note.
>> 
>> It doesn't give the exact ratios for any chord, but is considered
>> near enough for practical purposes on pianos, which go out of tune so
>> quickly that the ratios are never likely to be exact.
>> 
> 1) What is the numerical value of this "twelfth root of two?" I don't
> think I have either a sufficiently advanced calculator (or program),
> or sufficient calculation skills, to figure that out...

According to my calculator, 1.0594631

 
> 2) Just a thought...if, as you say, "It doesn't give the exact ratios
> for any chord"...is it not, then, a practical approximation...or, as
> we used to say back in my USAF days, "Close enough for government
> work"...?

I guess it depends on how good your ear is. The thirds come out sharp
and the fifths flat. The Bach family had a special tuning system for
keyboard instruments, to give more accurate results. (Hence the
"well-tempered" klavier - the music is written to demonstrate the
advantages of this tuning system.)

See http://www.larips.com/ 

The violinist Joachim, friend of Brahms, the Schumanns, Mendelssohn and
others, adjusted his intonation as he played to fit the harmony. Any
singer can do the same, if there is no keyboard accompaniment.

Yes, it is close enough for government work. 

> 
> 3) Does "from note to note" allow for the fact that some notes in
> the major scale differ by full tones, while others differ by
> half-tones (i.e. there is no C-flat!)...or were you speaking of
> the series in a twelve-tone "scale?"

The terms "tone" and "semitone" are a bit misleading. In standard
western keyboard music, each "octave" (what Partch used to call a "2/1")
is divided into 12 equal steps. For some reason these are called
"semitones" while double steps are called "tones". 

Each of the 12 steps is 1.059 times the one below, in equal temperament.

Any seven of the steps can be used to make a 7-note scale (or key), but
some sets of seven are commonly used and others are rare. 
> 
> I'm not challenging you...I'm just trying to get as full an
> understanding as possible of how...and why...western-world
> music "works!"

Mainly by a mixture of trial and error and tradition. Much of it was
already in place before the frequencies of sounds could be measured,
although wavelengths in strings and pipes were understood in the West.

The Chinese got themselves confused by doing their research on metal or
stone bars rather than strings - a bar has more complex modes of
vibration. But they did evolve an equal temperament system in the 16C
AD. The Europeans probably got the theory from China.

See Needham, "Science and Civilisation in China", Vol. IV, part 1, pages
126-228 for a fascinating account (illustrated) of the history of
music in China. 

Big subject, I am sure there are people here who know far more than I
do.

Regards
-- 
Don Cox
doncox@xxxxxxxxxxxxxx