Re: F_minor...

[Bradley P Lehman <bpl@umich.edu>]

On Tue, 23 Feb 1999, Michael Arnowitt wrote:

> A well-tempered tuning (as in Bach's "Well-Tempered Clavier") is just a
> tuning system that allows you to play reasonably well in tune in all keys --
> but this doesn't mean it's all symmetrical; the distance from C to E, for
> example, wasn't the same as from E to G#, even though they're both major thirds.
> 
> Today we use "equal temperament," where all half-steps, etc. are
> mathematically equal, so now all keys are identical except for starting
> pitch level.  But back then a well-tempered tuned piano really would give
> different keys different "colors."  The tuning system was generally set up
> so that simpler keys (C major, for example) would have purer, more
> natural-sounding intervals.  Keys with large numbers of sharps or flats were
> spicier (think of some of the music in Well-Tempered Clavier, Book I - Eb
> minor and Bb minor - intense music; it's no accident Bach set them in those
> keys with lots of flats.)  In some historical tunings I have heard, you can
> really tell a difference between sharp keys and flat keys due to the
> asymmetries of these "well-temperaments."
> 
> So it wasn't just Beethoven that associated C minor with stormy music - it
> may have been inherent in the tuning system of his time.  Music that was
> "chromatic" (had more funky notes tuning-wise) - the word chromatic itself
> comes from Greek "chroma," color.  The different keys had different colors.
> I have heard a piano tuner tell me of hearing a performance of Bach's WTC on
> a historical tuning, and how it was an extraordinary eye-opener for him
> about this music.

This is a good summary you've written.  There are also entire books about
this phenomenon of keys and colors; the one I remember most is a doctoral
study by Rita Steblin where she goes through what composers and treatise
writers themselves said about key colors, comparing effects in different
times and places.  Her book is _A history of key characteristics in the
eighteenth and early nineteenth centuries_ in the series _Studies in
musicology_, #67.  (Ann Arbor, MI: UMI Research Press, c1983)

I did one of my own doctoral projects on this topic, also, but from a more
mathematical angle.  That project included a spreadsheet that analyzes the
various unequal temperaments by beat rates of the intervals.  Then I tried
to assign a subjective "tolerability" scale to the keys and intervals,
giving simple numbers to indicate how far out of tune the keys are, moving
chromatically or in a circle of fifths.  Then there was a lecture/recital
where I played illustrative pieces in various temperaments.  The
spreadsheet and explanations are at
http://www-personal.umich.edu/~bpl/temper.html

Here's an example (view in an equally-spaced font):

Harmony Chart: tension of common chords when playing in the given key 
1/4 Syntonic Comma Meantone 

Key I  i Neap ii iio III iii IV iv  V V7  v VI vi VII viio viio7 
C   1  2   3   2  2   1   2   1  1  1  4  2 31  2   1   2    2 
G   1  2   6   2  2   1   2   1  2  1  4  2  1  2   1   2    1 
D   1  2   1   2  2   1   2   1  2  1  4  2  1  2   1   2    1 
A   1  2   1   2  2   1   2   1  2  1  4  2  1  2   1   2    1 
E   1  2   1   2  1   1  31   1  2 17 10  2  1  2   1   2    2 
B  17  2   1   2  1   1   1   1  2 17 10  2  1 31   1   2    2 
F# 17  2   1  31  1   1   1  17  2 17 10  2  1  1   1   2    2 
C# 17  2   1   1  2   1   1  17  2 31 16 31  1  1  17   2    2 
G# 31 31   1   1  2  17   2  17  2  1  1  1  1  1  17   2    2 
Eb  1  1   1   1  2  17   2  31 31  1  1  1 17  2  17   2    2 
Bb  1  1   5   2  2  17   2   1  1  1  4  1 17  2  31   2    2 
F   1  1   3   2  2  31   2   1  1  1  4  2 17  2   1   2    2 

By comparison, in equal temperament all the lines are:

    3  6   5   6  6   3   6   3  6  3  3  6  3  6   3   6    6

The "well temperaments" generally have values that range from 0 to 9 in
all the columns.  I calibrated all these charts such that values greater
than 10 are the chords that are typically not musically usable (and this,
of course, is subjective): some interval within the chord beats too
rapidly.  

In the example given above, 1/4 comma syntonic meantone, the triad of Ab
major is unusable because the interval G#-C (there is no Ab) doesn't sound
like a major third (it isn't!), and G#-Eb doesn't sound like a fifth (it
isn't).  They are "wolves."

Interestingly, in many of the historical temperaments that are based on
"meantone" techniques, F minor is a special key.  Usually the note G# is
used instead of Ab.  The interval F-G# is so far out of tune as a 6/5
ratio minor third, so narrow, that it's almost *in* tune as a 7/6 ratio
minor third...a startling sound.  And the ear gives so much weight
(subjectively) to this interval that the "wolf" G#-C in that same triad is
not very noticeable.  

So, the key of F minor is more musically usable in a practical sense than
it might first appear on paper.  And this key was available to keyboard
composers and improvisers: in these temperaments that to them were the
normal daily standard, F minor was the place to go for that special
effect.  It is the first flat key that has that 7/6 interval in it; the
others are the much less used Bb minor (Bb-C#) and Eb minor (Eb-F#).  
(This 7/6 interval is the reason why those three keys get a 1 in the "i"
column on the chart above, instead of getting a 17 or 31 due to the wolf
between the triad's 3rd and 5th.)

Another interesting thing about F minor is that the dominant (C major) is
always among the best-in-tune chords, but the subdominant (Bb minor) has a
character very similar to F minor's own.  The dominant minor (C minor) is
generally much better in tune relative to either Bb or F minor.  The
subdominant major, Bb, is also very good, as is of course the tonic major
of F.  So, F minor has some particularly dramatic possibilities within a
circle of fifths modulation: it's at the threshold where the temperament
gives especially sudden shifts of character.  To some extent this also
happens in the well temperaments: F minor is always uniquely poised on
that edge where anything could happen.

All this is easier to demonstrate at the keyboard than to describe in
words.  Simply tune a harpsichord in 1/4 comma meantone, and then
improvise through those modulations into F minor and back.  The effective
difference in sound quality is obvious.

I don't know whether or not GG was aware of any of this in his own choice
of F minor as a favorite key.  Is there any evidence that he ever messed
around with unequal temperaments?

Isn't there some anecdote about Brahms liking F minor, also?  As I recall
hearing it, somebody asked him what character his new D major symphony
(#2) has, and he compared part of it to the effect of sitting at a piano
and playing F minor chords repeatedly.  (I could be mis-remembering this
anecdote, but I think this is how it went.)

> That aside, even on a modern piano with equal temperament many of us have
> psychological leanings to certain keys.  My favorite is A major.

Yep.  Mine are usually Eb major and G minor.  That's probably more from
association with their use in specific pieces than with interval
characteristics, though.

Do you know Haydn's symphony #65 in A?

I remember once I heard a harpsichord recital where all the pieces were in
either G minor or D minor, for a total of at least an hour.  Everything
was in a temperament appropriate to those keys.  And then the player gave
an encore that was in E major.  Sharps!  The psychological effect was
really surprising, like suddenly opening a window shade and seeing that it
is daytime.

Bradley Lehman ~ Harrisonburg VA, USA ~ 38.45716N+78.94565W
bpl@umich.edu ~ http://www-personal.umich.edu/~bpl/