GG: Equal vs. Well-Tempered

["John P. Hill" <jphill@frank.mtsu.edu>]

Re:  Equal vs. "Well"- Tempered

There *is* a difference, of course.  As Mark mentioned, equal
temperment utilises the semitone frequency relationship of the 12th 
root of 2 throughout the keyboard.  It's pretty clear that this was
*not* the case during Bach's time. The "well-tempered" system in place at
that time would allow performance in all keys without grotesque problems,
but it seems likely that different keys really would have had different
"characters" based upon their specific intervallic frequency
relationships.

I think one can find examples of this in Bach's keyboard writing.  There
are certain "emotions" and writing devices common to certain keys.
For example, if you look at pieces in B-flat major (2-part Invention,
Fugue from Book 1 of WTC and Preludium from the first Partita) you see
some real similarities in the writing:  square-cut 8th and 16th rhythms
that have a certain drive to them, a sort of "happy" and "introductory"
character, lots of contrapuntal interest and a tendency to stick pretty
closely to key centers of B-flat major and g-minor.

I think it was the original Partitas vinyl album that had one of those
great conversations that GG had with himself (or was it "David Johnson"?).
In this one, he pondered the question of whether you could transpose a
Bach piece written in c-minor to, say, f-minor and have it communicate
the same things.  GG ends up saying, "don't know...I really must try it".

In a truly equal-tempered system, the two versions of the piece should
*only* differ in one being higher or lower than the other.  If you remove
the tactile information and the issue of someone with perfect pitch having
always *heard* it in the original key, the two versions really are
identical.  So, on a modern-day synth, c-minor and f-minor don't say
anything different for the composer or the listener. This is *not* the
case in a "well-tempered" system, where one key might have very pure
perfect fifths and major thirds but larger intervallic errors in other
places.  We also have the issue of where A4 (A above middle C) really
was pitched;  it was undoubtedly lower than 440 Hz. (A440)

So.....bottom line....Bach may have exploited these differences
inherent in the tuning system of the time to match character/emotion of
the composition with certain key areas.  Performing these pieces on
a perfectly equal-tempered synth, for example, loses this "character"
because every key is identical to every other.

Incidentally, pianos are rarely tuned for perfect equal-temperment. It
is very common to "stretch" the higher and lower extremes of the keyboard
beyond where a computer would tell you to pitch the notes.  We also know
that in controlled listening tests, most people prefer octaves that are a
bit *larger* that the frequency ratio of 2:1.  They seem to sound more
musical to the human ear.

Sorry if this got technical....but I think it's kind of interesting
and GG was certainly up on all of it.

jh



On Fri, 6 Jun 1997, Mark Williamson wrote:

> 
>      Well, define equal temperament.  I am familiar to the 
>      scholarly theory to which you refer, and in fact several 
>      excellent craftsmen who have worked on my instruments as I 
>      have moved around the country have been subscribers to that 
>      theory.  But all I mean by "equal temperament" is that the 
>      instrument does not have to be *totally retuned* regardless 
>      of what key you are playing in.  That probably fits your 
>      definition of "well-tempered," and certainly Bach's 
>      definition, so I am OK with that.
>      
>      Mark
>      
>      BTW, my digital synthesizer is absolutely equally tempered 
>      unless you choose a different system.  That is, the ratio of 
>      the frequency of any tone to the tone one half-step 
>      immediately below it is exactly equal to the twelfth root of 
>      two.  The computer makes it so.  I cannot see any 
>      disadvantage to this tuning system if you are playing in all 
>      24 keys.
>      
>      One of the things that's so odd about tuning a piano or 
>      harpsichord is that no craftsman, no matter how skilled, can 
>      tune any instrument to this degree of accuracy.  And even if 
>      he or she did, there would be variations beyond this degree 
>      of accuracy almost as soon as he or she were done.  So it's 
>      questionable whether some of the differences between 
>      different systems of "well" temperament are really that 
>      significant.  What matters to me is whether you can play in 
>      all 24 keys without (1) retuning the instrument or (2) any 
>      noticeable difference between how the keys sound.  And (2) 
>      is optional:  in some styles of music you want different 
>      keys to have different sounds and in others you want a more 
>      neutral, ambulatory feel.
> 
> 
> ______________________________ Reply Separator _________________________________
> Subject: Re: the well-tempered clavier
> Author:  Jeff Dods <jdods@alchemy.chem.utoronto.ca> at internet
> Date:    6/6/97 12:58 PM
> 
> 
> > 
> > 
> >      You will certainly get a lot of responses on this; it's a 
> >      favorite subject for many.  WTC 1 was written (a) to 
> >      demonstrate the modulatory advantages of equal temperament 
>      
> There was a lot of discussion about this on the harpsichord list a few 
> months back. Let me give you my watered down version.
> My impression is that no scholors believe anymore that "well temperament" 
> was equal temperament.  Yes, it was probably a "modern" temperment in the 
> 18 century, i.e. more modern that any kind of mean-tone, but in no way 
> mistakable with equal temperament.
>      
> I *know* there are others on the list who could speak volumes on this topic, 
> and quote 18th century German...
>      
> Jeff.
> jdods@chem.utoronto.ca
>      
> 
> 
>