From: roger@crux.Princeton.EDU (Roger Lustig)
Subject: Re: Define "Well Tempered"
Originator: news@nimaster
Reply-To: roger@astro.princeton.edu (Roger Lustig)
Organization: Princeton University
Date: Tue, 26 Jan 1993 23:40:10 GMT

In article <1993Jan26.164237.19034@digi.lonestar.org> gpalo@digi.lonestar.org (Gerry Palo) writes:
>In article <1993Jan22.155421.21393@netcom.com> hatunen@netcom.com (David W Ha
>tunen) writes:
>>In a scientifically proper musical scale, related harmonics are
>>related by integer relationships. This was discovered by the
>>ancient Greeks. Thus, the frequencies should be as, say, 5:3.

>>Unfortunately, this means that you must retune your instrument
>>for each different scale you wish to play. Which is impossible
>>with some instruments. In the evenly tempered scale, the octave,
>>or 2:1 relationship is divided into twelve geometrically evenly
>>spaced intervals. Take the twelfth root of two (approx 1.05946..).
>>Each tone bears this relationship to the next lowest tone.
>>That way, no matter what scale you play, it will sound harmonically
>>the same.

>>The problem is that a third interval is no longer exactly a third;
>>that is, it no longer consists of two tones in a simple integer
>>relationship. It's very close, but no cigar, and the difference
>>is noticable.

But "no longer" implies that the previous state was some Pythagorean
ideal tuning.  It wasn't; equal temperament comes as an adjustment 
to well-temperament comes after meantones comes after...and all these have
at least some non-pythagorean intervals.

>>So, an even tempered scale doesn't sound quite as good as a
>>diatonic scale. 

Diatonic?  Perhaps you mean: untempered.

>>But it makes the piano possible (you don't have
>>to retune for each different scale) 

Yet keyboards had existed for centuries before equal temperament...

>>and it makes music with key changes possible.

It makes music with *more* key-changes possible.  You can modulate quite
a bit with mean-tone tuning.

>>As I understand it, Bach wrote the Well-tempered Klavier as a
>>demonstration of the advantages of an even tempered scale.

>This has been my impression, which leads me to the idea that Bach
>indeed had equal temperament in mind, since you have to be
>able to play all the preludes and fugues at the same keyboard
>without tuning.  

*big sigh*

-- Who says you have to be able to play them all at once?  This 
was a book of studies, or of pieces to be learned and played one
or two at a time.  It was not intended as a concert suite. 

-- Even assuming that you don't retune, why does that mean equal
temperament? All you need is *any* temperament that allows you
to play in all keys, even if there are differences in semitone sizes.
 
>In practice, it may be possible to bias the
>tuning away from equal temperament to favor the more commonly
>used keys -- those harmonically related to C and the keys that
>are playable by the non-tunable instruments, which from some 
>of the posts in this thread is often done. I.e., there are
>not a whole lot of pieces in C-flat or G-sharp, so why not
>accommodate the more common ones, and especially the other
>instruments and the human voice?  

Good point, and some groups do this.  As do some organs.

>(Question:  Are there any serious orchestral or large ensemble 
>pieces written in one of the far out keys?  And are there any 
>with piano?  If so, what kinds of tuning problems do they have?)+

Haydn Farewell Symphony (F# major/minor); it plays on all the 
difficulties ofthe key.  The one that follows is in B major.

MAny arias by Handel in F sharp minor.  Generally lamenting.

Lots of Chopin.  (D flat/b flat minor were favorite keys because they
feel nice under the hands.)  

Beethoven Sonata Op. 78: F sharp.  Moonlight Sonata: C sharp minor.

The B minor Mass.  Mozart's B minor adagio.

Quite a few out there.  

>As for the viability of equal temperament, I still ask, and
>suggest, that the development of tonality based around the
>concept of key, makes equal temperament a viable possibility
>because of the fact that a tonal piece establishes its own
>harmonic framework, asserts its own tonic by surrounding it with
>harmonic swings from dominant to tonic to the subdominant side,
>etc.  This self-establishing tonal horizon and the tonal per-
>spective that emerges from it may or may not be well crafted,
>or it may be deliberately left vague, but it tends to speak
>so much more "loudly" than the slight dissonances of inexact
>tuning that those dissonances matter even less than they would
>in earlier, modal music.

Yet modal music, and all the other music from before the common-practice
period that wasn't modal, which is most of it, did those things too.  
It's hard to decide what to call some pieces from, say, 1600: tonal or
modal.  They may use a mode, but you can identify all the tonal 
procedures in them.  And they don't need a more modern tuning, either.
What needs the fancy equal temperament is the much later developments
of tonality: chromatic and enharmonic modulations as a matter of course.

Roger