Greetings,
I didn't mean to have mislead any with that subject line, (what,
Foote=equality? or maybe, F==)? What follows is a posting by Dr. Kellner on
the logic for his derivation of a tuning. The particular tuning is what Dr.
Kellner states was the tuning that Bach authored. There is much to be said
for the Dr.'s research, and I think we could all find a bit of perspective
with it.
In a meantone nutshell, (which I sometimes find myself overly acquainted
with!), his premise if that the third may profitably be enlarged in the
triad, since the fifths are already beating. The limit he proposes is the
speed of the fifth, and arrives at the 1/5 comma meantone tuning as a result.
I hope this may be of interest to some, and I plan on writing a post inre
the convention as soon as I have time to sort my impressions out. The Overs
piano was VERY impressive.
We had a good time using the Disklavier in the temperament class, also.
Dan Levitan and I ended up cross tuning that piano on a daily basis for our
different classes, so my lecture was accompanied by my forty minute tuning
that not only had to change the temperament, but also had to corral all of
the demonstration pitches left from Dan's comparisons.
I think it went ok, but don't know that all my unisons were "studio"
quality. This was an interesting test of the SAT III, as I pulled up a
straight FAC for a Yamaha C3 and dumped Jim Coleman's number 11 on it as a
template. I would be grateful for any critique of the result, since being in
"convention mode" makes listening to everything a fuzzy blur).
One thing that is not fuzzy is that Laura did a beautiful job of leading
us through a mountain of education, she is to be commended for a great job.
Regards,
Ed Foote, RPT
Now, on to equality:
_____
Message: 2
Date: Fri, 13 Jul 2001 10:59:13 +0200
From: ha.kellner@t-online.de
Subject: trias harmonica - for Margot Schulter
Dear Mrs. Margot Schulter, dear members of the group,
A recent question by Margot Schulter to quote:
"I would much agree that the _trias harmonica_ is a vital
feature of the Baroque era we are discussing.
1) Here my question might be: while the _trias harmonica_ is at the heart
both of the late modal practice around 1600 and the early tonal practice
around 1680-1720 which Werckmeister helps to define, does the concept of
the triad necessarily favor any one well-temperament, given that all are
compromises?
*********** (underlining by HA Kellner)
2) Similarly, while Zarlino's concept of _harmonia perfetta_ is central to
late 16th-century practice, does it necessarily favor one specific
meantone tuning?"
I unquote the large citation of Margot Schulter.
Ad 1)
Does the concept of the triad necessarily favor any one well-temperament,
given that all are compromises?
***********
The music under consideration proceeds via the harmonies of the
chords. (except, e. g. the "bicinium", etc.). These chords comprise thus
at least 3 tones. Whereas earlier times "tolerated" even the somewhat
harsh Pythagorean thirds, tastes and preferences changed and developed in
thecourse of history. One adopted the extreme of going for PERFECT thirds in
mean-tone.
(When tuning consistently and very accurately - why not - a harpsichord,
one fell automatically at one point into a major third about
384,36 cent wide, i.e . smaller by 2 cent than pure. This discovery,
pleasant to the ear, certainly gave rise to the desire to hear more of this
sort).
One took thus PERFECT thirds for mean-tone. This, not only in
contradistinction to Pythagorean, but at the same time mean-tone is set
off from the system of natural harmony that puts basically a perfect
triad 4:5:6 (in terms of frequencies), say upon C-major. Constructing this
system further, produces the annoying fact that the tones C-D and D-E are
different and drastically reduced fifths, by one comma, show up.
Once having placed instead the perfect third on C-E and filled it up by
tempering four EQUAL fifths, equalizes the sizes of C-D and D-E.
Within the straightforward principle of mean-tone (with pure thirds and
payung no attention or concern for anything else) let us consider
the TRIAD C-E-G.
Wasn't in the natural harmonic system the triad the pertinent feature?
Mean-tone concentrates on the single and unique interval of the major third.
But the music under consideration proceeds via the harmonies of chords.
Within mean-tone, the overall purity of C-E-G can be "improved", in analogy
to
the step of improvement that went from the Pythagorean third to the "better"
mean-tone third.
Despite its pure third, the mesotonic triad will BEAT. To get things
"purer", shouldn't one attempt to reduce the beats? These triads
beat: not because of the third that is pure, but due to the fifth C-G.
This is one of the 4 fifths that has to be necessarily tempered smaller
in order to fit into its perfect third.
As the triad beats anyway, does it really make sense to insist on a pure
third? Obviously, NO! and the third may be relaxed, up to the order
of its beat-rate becoming comparable to the rapidity of beats of
the fifth. Enlarging the third somewhat will slow down the beats of the
fifth. The ideal "meeting/compromise point" will be the situation
in which - within a tempered triad - both its principal constituent
intervals third and fifth beat at the UNISON. What sufficient reason could
produce a BETTER COMPROMISE??
******************
Conclusion: the basis of musical progressions - in some sense - are chords
and not just intervals. This is borne out when basing the music upon
triads, the most elementary cords. And here, the major chord - trias
harmonica perfecta. As to the overall tempering of this basic musical entity,
the triad "wohltemperirt" is the best and most natural one to attain, derived
here heuristically, not via the detailed mathematics. Quality criterion for
purity and consonance was the behaviour of beats, i. e. their rapidity.
Having derived this triad, the step to extend this triad to a system
"wohltemperirt" follows, But this is very simple. Due to a surprising and
incredible mathematical co-incidence, the fifth of the triad "wohltemperirt"
explained above, is reduced by 1/5 of the Pythagorean Comma!!!
Therefore, given the 5 tempered fifths, one can complete the circle with the
finality of closure by the further 7 perfect fifths.Musico-theological
aspects
now enter into the play: If we insist that the "best major triad" feasible in
a
balanced system for all 24 keys just occurs just ONCE, and not more, then
B-F#
must be the 5th tempered fifth.
As regards MINOR: provided a major triad is acceptable or supportable, the
interpolating minor third following the basic major third will also be
acceptable. If the tempered fifths themselves are acceptable, the minor
chords constructed using this material will as well be acceptable. Thus,
"wohltemperirt" yields and represents a temperament for all 24 keys.
Ad 2)
Having said the above, I can see nothing in Zarlino's concept of "harmonia
perfetta", that would favor any one specific meantone tuning. The mesotonic
system definitely gives room for improvement, technologically/mathematically
and spiritually under the musico-theological aspects, perhaps best enounced
in almost all of Werckmeister's writings.
I am grateful and have very much appreciated Mrs. Margot Schulter's
excellent exposé about the trias harmonica and even more, her critical
query I hope to have answered here more or less satisfactorily.
(When pretending music progresses by CHORDS I do not attempt to offer
a treatise on counterpoint).
Kind regards
Herbert Anton Kellner
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