proportional beating triads...

The organizing feature of Jorgensen's first book was: he presents whatever such-and-such a temperament in "Theoretically Correct" manner, and counts up the proportionately beating triads. And then he presents the same temperament again, diddling around with it to get several (or many) equal-beating checkpoints and approximations, allegedly for greater ease for the tuner (which it really isn't)...and unfortunately wiping out the geometric regularity of 5ths and 3rds along the way. And then he counts up his PBTs and gets a higher number, and the reader draws an improper red-herring conclusion that this is somehow better. Fortunately Jorgensen's second book isn't organized in that manner, so much. Maybe somebody suggested to him between the books that it's not worth such vehement attention?

A few more remarks on the chaotic phase-shifting on acoustic instruments, now that I've had time to catch up a bit with other people's comments here.

Proportional beating is [let's say "almost"] totally worthless on the three "early" keyboard instruments where unequal temperaments matter most anyway. And it has nothing to do with practicing more diligently, to eliminate a natural feature of expressive musicianship! A healthy range of variety is a *virtue* in good playing, contributing to the ability to hold a listener's attention with the music.

1. On harpsichord, it's idiomatic to spread chords most of the time. We spread the attacks slightly for acoustical and musical reasons that have little or nothing to do with any phasing in the temperament, and it's fundamentally to give a warmer and more vocal overall tone. One could also bring in the issue of bird-quill plectra (irregular) vs plastic (more regular): the notes don't necessarily sound exactly the same from one time to the next, individually, and that's not necessarily a musical drawback! Carefully graded attacks and releases are our bread-and-butter of expression, with dozens of minute variations in timing, *in addition* to any randomness contributed by the instrument.

2. On clavichord, pitch of the individual notes is subject to finger pressure variations, plus the possibility of Bebung (vibrato). The whole tuning thing is to some extent guesswork and averaging.

3. On pipe organ, the pipes don't speak at the same speeds as they get going and therefore some chords automatically spread themselves (slightly), randomly. That's true both for chords and for playing individual notes with multiple stops drawn. The player on a tracker-action organ can even do some of this *deliberately* for special effect, pressing the keys sometimes more slowly to get more spread, and for a more Furtwanglerian build-up of the tone.

That plus weather variations. On two different days, or even an hour apart on the same day, any carefully set-up layout with PBTs will have drifted somewhat, either with the whole instrument going up or down somewhat in pitch, or on some individual notes. Change the pitch of the whole instrument by (say) 2 Hz, and any internal proportional relationships will already be seriously out of whack, in various regions of the keyboard.

So, for someone like me (being a specialist in playing these other instruments, and caring almost zero for any electronics) the whole issue of proportionally beating triads is little but impractical theorizing on paper. Careful counting of the beats or their relationships can help in setting up temperaments accurately, but beyond that the musical application in normal repertoire is something near zero.

Brad Lehman

Not that I endorse the proportional-beating business, but there are
one or two remarks arising.

If one thinks of beats as rhythmic pulsations (which seems to be part
of the proportional-beating philosophy) then it follows fairly
directly that small integer proportionalities ought to be musically
preferable, just as recognizable musical rhythms follow small integer
proportions.

However, it does *not* follow that all the beats in a chord should be
in sync or in phase. They would in general be syncopated. Perhaps some
of the trouble with the sound sample 'A' is that there is no
syncopation and the effect is too plain.

If you are OK with syncopated rhythms - and particularly if you
consider them an essential part of musical expression - you ought to
be OK with dephased but proportional beats. One cannot in any case
control the phase relationship of an acoustic instrument to the extent
required for all beats to be in phase anyway. Whether the result is
'chaotic' or not... is hardly a question that can be objectively answered.

One big spanner for proportional beating in organ tuning is the
'drawing together', which I am surprised Brad did not mention. This
means that each pipe gives a different pitch depending on which other
pipes it is sounding with. In particular major thirds tend to 'draw'
towards purity. [Is this true of the Taylor & Boody??] So any beat
rates you calculate on the basis of each pipe's frequency sounding
alone, are probably quite wrong when two or more are sounding
together. If this effect was taken to extremes an organ might give a
good approximation to adaptive JI!!

It is, though, just possible that proportional beating might have some
aesthetic effect on the harpsichord, due to two facts: one can easily
retune to obtain various proportional-beating triads if desired; and
the instrument has strong overtones which are virtually exact harmonics.

~~~T~~~

--- In tuning@yahoogroups.com, Brad Lehman <bpl@u...> wrote:
>
> A few more remarks on the chaotic phase-shifting on acoustic
> instruments, now that I've had time to catch up a bit with other
> people's comments here.
>
> Proportional beating is [let's say "almost"] totally worthless on the
> three "early" keyboard instruments where unequal temperaments matter
> most anyway. (...)
>
> 1. On harpsichord, it's idiomatic to spread chords most of the
> time. (...)
>
> 2. On clavichord, pitch of the individual notes is subject to finger
> pressure variations, plus the possibility of Bebung (vibrato). The
> whole tuning thing is to some extent guesswork and averaging.
>
> 3. On pipe organ, the pipes don't speak at the same speeds as they
> get going and therefore some chords automatically spread themselves
> (slightly), randomly. (...)
>
> That plus weather variations. On two different days, or even an hour
> apart on the same day, any carefully set-up layout with PBTs will
> have drifted somewhat, either with the whole instrument going up or
> down somewhat in pitch, or on some individual notes. Change the
> pitch of the whole instrument by (say) 2 Hz, and any internal
> proportional relationships will already be seriously out of whack, in
> various regions of the keyboard.
>
> So, for someone like me (being a specialist in playing these other
> instruments, and caring almost zero for any electronics) the whole
> issue of proportionally beating triads is little but impractical
> theorizing on paper. Careful counting of the beats or their
> relationships can help in setting up temperaments accurately, but
> beyond that the musical application in normal repertoire is something
> near zero.
>
>
> Brad Lehman
>

> One big spanner for proportional beating in organ tuning is the
> 'drawing together', which I am surprised Brad did not mention. This
> means that each pipe gives a different pitch depending on which
other
> pipes it is sounding with. In particular major thirds tend to 'draw'
> towards purity. [Is this true of the Taylor & Boody??]

Yes. I noticed that especially last week while playing BWV 572, the
way the 5-part harmony so often locked in, with full principal chorus
and mixtures.

I also heard about it recently when another organ-building expert
visited that organ and then had some remarks for me. He could not
believe that Taylor & Boody had tuned it correctly, because (to him)
the A-flat major triad (he didn't tell me which octave or what
registration) sounded even "purer" than C major did. [But the C major
is definitely more nearly pure: C-E 3 schismas sharp, while Ab-C is 8
schismas sharp and E-G# is 10. Equal would be 7.] I replied that the
A-flat major is simply a lot better than *his own expectation* as he
usually favors temperaments that have Ab-C 11 schismas sharp.

Even the E major triad has a startling resonance to it, and that's the
one we would most expect to be vinegary.

The major 3rd sizes are at the bottom of the first chart, on this
page:
http://www-personal.umich.edu/~bpl/larips/math.html
...but that's of course *without* any pipe-drawing factored into it.

In my recording there's a movement in C major played on 4-foot flute
alone, and the drawing is especially noticeable when the C-E major 3rd
is on top of the texture. Melodically it starts to sound like 1/4
comma meantone at that point, where it's really set up as 1/6.
Likewise when I've played that same movement for fun (not on the
recording), entirely transposed into C# major and then again in Cb
major, it all sounds "pure" but differently so, like an equally
plausible yet distinctive color.

I believe that effect comes principally from the fact that all the
scales make logical sense melodically, more than any momentary drawing
of individual notes higher or lower. The temperament refuses to draw
much attention *to itself* as there aren't any notes that ever stick
out as grossly too high or low, either melodically or harmonically.
It just sounds impossibly "pure", on that organ, all the time, in all
keys.

This made such a gentle and un-aggressive effect on my colleague that
he reported being even a little bit disappointed: he had expected
something more outlandishly colorful, overall. Again (I believe) this
was from bringing in his own expectations from some other more extreme
temperaments (Werckmeister, and various 1/5 comma schemes), and then
being surprised at how gentle and innocuous this is.

It sounds surprisingly colorful to equal-temperament enthusiasts, and
surprisingly tame (virtually un-noticeable) to unequal-temperament
enthusiasts. Right down the middle, saying more about the listener's
expectations than about the actual sound.

The more interesting comment I got last week was from two people,
separately, who are also frequent listeners to another nearby organ
(Notre Dame University's) that is in Kirnberger III. They both said
that the K-III organ makes a thrilling and exciting sound that wears
them out after an hour or so (well yeah, all those Pythagorean major
3rds!), but this one by comparison is so much more mellow and
continually refreshing: they could keep listening to it for much
longer without ever feeling tired. Both of these instruments are of
comparably excellent craftsmanship, and the rooms are of similar size
and acoustic clarity. And I would factor out the performer part, too,
because Notre Dame has a much better organist than I am. When I heard
him play this instrument, in May, I had the same reaction as these two
people who talked to me on Sunday: this organ and its tuning together
simply don't sound tiring. Nor is it dull. The builders just caught
a remarkably good balance, all around.

>
> It is, though, just possible that proportional beating might have
some
> aesthetic effect on the harpsichord, due to two facts: one can
easily
> retune to obtain various proportional-beating triads if desired; and
> the instrument has strong overtones which are virtually exact
harmonics.

Yes, but only for a few seconds, and only at very close distances, i.
e. the player himself/herself seated to play. I did some
experimentation with this, yesterday and today at three different
harpsichords: set up triads with proportional-beating in them and then
listen to them repeatedly, with the head in different positions above/
below/beside the instrument. The beat structure sounded qualitatively
different in every position, whether I was trying to hit all the notes
exactly in phase or not. The listening position itself mattered at
least as much as that. The simplest illustration is to hit a chord,
and then two seconds later while holding it, drop the head to listen
below the plane of the keyboard...any beating pattern goes away and
it's just some qualitative tone-color of the whole. Or vice versa,
start with the head low and then sit up during the chord...suddenly
there are beats. All of this is probably inconsequential to any non-
playing listener during a performance, of any of the standard
repertoire. Reminds me of the old viola joke: how is playing viola in
an orchestra like wetting your underwear? Both are a warm personal
experience that nobody else is aware of as it happens.

Brad Lehman

> A few more remarks on the chaotic phase-shifting on acoustic
> instruments, now that I've had time to catch up a bit with other
> people's comments here.
>
> Proportional beating is [let's say "almost"] totally worthless
> on the three "early" keyboard instruments where unequal
> temperaments matter most anyway.

Why do they matter most there?

> And it has nothing to do with practicing more diligently, to
> eliminate a natural feature of expressive musicianship!

Did anybody suggest this?

> 1. On harpsichord, it's idiomatic to spread chords...
> 3. On pipe organ, the pipes don't speak at the same speeds...

Actually, the staggering of attacks even when one is
playing normally on, say, a MIDI keyboard is probably
enough to cause significant phase differences through
most of the audio range.

> Change the pitch of the whole instrument by (say) 2 Hz,
> and any internal proportional relationships will already
> be seriously out of whack, in various regions of the
> keyboard.

Not to mention the rest of the tuning (you're implying
shifting all the notes by constant Hz.). Many instruments
that go flat, though, do it proportionally, keeping the
tuning and the beat ratios more or less intact.

-Carl