what I hear in harpsichord tritones 45/32, 64/45, 36/25, 25/18, 7/5, and 10/7

Carl pressed:

> Brad- "pure" is a common synonym for "beatless".
> Are you claiming these intervals are beatless?
> Yes or no will suffice.
> To those of us using "pure" to mean beatless,
> maybe we shouldn't. Maybe we should say beatless.

OK, here goes. I went through it again this morning, setting all these up on a Flemish-style harpsichord and listening closely to them at a variety of positions. And as I mentioned before, it matters a great deal WHERE the listening is done. Therefore, you're not going to get a "yes or no will suffice" answer. The same intervals sometimes give beats, or not, at a different position of only a few centimeters.

Here are my notes.

45/32 (augmented 4th). If I stick my head as far forward over the strings as I can (while sitting at the keyboard, leaning over the pinblock), I can pick out a subtle beat. This is an abnormal listening position for music, but normal enough for the process of tuning by ear. BUT: when I sit in a normal playing position, upright, centered, I hear ZERO beats in this interval. None. Beatless. Pure. Dead. Whatever. That said, I can lean about 10cm to either side, still at the same distance, and hear the beats again: less strongly than at the closer spot. Moving from side to side I can find nodes where the sound quality is considerably different (so, there's a stereo/directional component in here too). But again I'll emphasize: sitting in normal centered playing position about half a meter back from the keyboard and above it, ZERO beats in 45/32. This tritone also gives me the strongest urge to resolve it outward to a major 6th, since (in any position) I am also hearing the "missing" tone at 9/8 above or 16/9 below the lower note. That phantom root has to resolve downward by a 5th or upward by a 4th. The 64/45 diminished 5th does the same thing.

36/25 (diminished 5th). Up close, leaning over the strings, I hear a strong blur but no beats. The "missing" note that is a 6:5 minor 3rd between the two notes is exceptionally strong whether I play it or not. In playing position, or moving side to side at a similar distance, still no beats. The 36:25 interval quickly seems much less intense as I lean backward toward playing position. It all seems "pure" in effect, wherever I am. The interval seems static, not really needing to resolve. And if I happen to play all three notes, filling in the "missing" 6:5, I don't hear any blur or any beating; only a solid diminished triad of two pure minor 3rds.

25/18 (augmented 4th). It's a blur up close, but less intense than with 36/25. When I lean back toward and into normal playing position, it's still only a blur...but at small changes of distance I can make the blur start or stop altogether. Having found a no-blur point, I can get a blur again by leaning back yet farther. It never becomes beats. When the blur stops, I only hear a dead pure interval ringing out. Moving side to side I can change the relative intensity of the two played notes, but I can't hear any beating. I hear a phantom pitch a minor 3rd below the lower note and a major 6th below the upper (e.g., the D below an F-B), and I can change its intensity by moving my head to different positions from side to side.

7/5 and 10/7. Both of these sound dead/pure/beatless whether I move in close or sit back, or move side to side. They're especially static, needing no resolution. I hear a phantom 4 in them making 4:5:7 or 4:7:10.

The harpsichord seems exceptionally loud with ALL of these six different tritones. I suspect that that perception also contributes to my sense of "pure" when I hear them, seated in normal centered playing position.

All the tempered tritones that I hear everyday (in various temperaments, almost always unequal) are less strong than any of these. The SQRT(2) tritone from 12TET sounds the weakest and least centered of all. It just sits there sounding atonal. If I play or hear Baroque music on harpsichord in 12TET, it seems directionless and bland, and the tritones don't do their job in contributing any dynamic interest.

The augmented 4th of regular 1/4 syntonic comma doesn't do a lot for me, either. It's obviously tonal, and better than 12TET's, but it's so far off the spots of the above-mentioned "pure" tritones that it loses some focus.

Everyone's mileage will of course vary. So will everyone's concept of "pure", too, probably. Some of the same intervals sound variously beatless or beating at different spots of only a few centimeters, either side to side or front to back, ON HARPSICHORD. To me, the concept of a pure interval is apparently that a just-intonation scale (I suppose either 5-limit or 7-limit) contains both notes. When I hear two such notes on harpsichord, it orients me immediately to place other phantom notes that are in simple ratios with both of them, and to hear tonally/geometrically.

Brad Lehman

--- In tuning@yahoogroups.com, Brad Lehman <bpl@...> wrote:
> And as I mentioned before, it matters a great
> deal WHERE the listening is done. Therefore, you're not going to get a
> "yes or no will suffice" answer. The same intervals sometimes give
> beats, or not, at a different position of only a few centimeters.
>
[snip]

> If I stick my head as far forward over the
> strings as I can (while sitting at the keyboard, leaning over the
> pinblock), I can pick out a subtle beat. This is an abnormal listening
> position for music, but normal enough for the process of tuning by ear.
> BUT: when I sit in a normal playing position, upright, centered, I
> hear ZERO beats in this interval. None. Beatless. Pure. Dead.
> Whatever. That said, I can lean about 10cm to either side, still at
the
> same distance, and hear the beats again: less strongly than at the
> closer spot. Moving from side to side I can find nodes where the sound
> quality is considerably different (so, there's a stereo/directional
> component in here too). But again I'll emphasize: sitting in normal
> centered playing position about half a meter back from the keyboard and
> above it, ZERO beats in 45/32.

It's called interference, Brad, or phase cancellation. It's exactly
the same thing causing the beats in the first place, except that in
this case, it is the interference of a single wave against its own
reflection.

Whenever you are fairly close to a sound source, such as when you are
sitting at he keyboard of a harpsichord, the first sound that arrives
at your ears from any source travels along a straight line, and is
called the Direct Sound. However, if there is any large, flat, and
fairly reflective surface nearby, like inside of an open lid, or a
wall against which the harpsichord is placed, you also hear a strong
reflection from this surface, which arrives at your ears delayed just
a little bit, since is has followed a path described by the two short
sides of an obtuse Isosceles triangle as opposed to the direct sound
which has come to you by the long side. The key to understanding what
happens when both sound waves arrive at your head is to compare the
distance traveled. If the difference between the lengths of the
pathways is equal to n times the wave length (n being an integer),
there will be no cancellation, and you will hear any beats that
frequency is making against its near equivalent coming from another
sound source (i.e. another note) - provided there is no cancellation
happening within radiating waves of the second note, that is. However,
if the difference in distances is equal to (or nearly equal to) n+1/2
times the wavelength, the waves will always arrive in direct counter
phase and therefore that particular frequency from that particular
source will be canceled. Naturally, if the frequency is canceled, it
cannot produce any beat with the almost identical frequency emanating
from the other source.

There are many points where cancellation takes place, often separated
by very small distances. Remember that reflected sound works just like
reflected images, in that the single source appears to be in two
places at once. The Apparent Source for the reflected sound is located
along the angle by which it arrives to the listener at an apparent
distance equal to the total distance it has traveled from the real
source. See the drawing:

http://polettipiano.com/Temporary/reflect.jpg

So to understand the cancellation, we can pretend that the sound is
coming from two separate sources, though it is being emanated from
both exactly in phase, as though they were two speakers producing the
same signal. See the drawing:

http://polettipiano.com/Temporary/interference.jpg

The black circles represent the waves of the Direct Sound, and the
gray circles the waves of the Reflected Sound. Each line of each
series of circles represents 0°, and the center of each white space
within each series of circles represents 180°. Now if you, as the
listener, are located at ANY point representing the same number of
degrees for both series of circles - for example, any point where any
black circle crosses any gray circle - the waves sum and you hear that
particular frequency somewhat louder than if you only heard the direct
sound alone. However, if you are located at ANY point where the
difference between the two waves is equal to 180, the waves cancel and
that frequency greatly reduced in volume. If the distances are
similar, the frequency is essentially canceled completely. Keep in
mind that this happens not only at places where a line crosses the
center of the white, which represents a phase relationship of 0:180,
but also at points which represent relationships such as 10:190,
20:200, 30:210, etc. In fact are lines of cancellation running outward
as diagonals of the trapezoids, connecting the points where lines
cross whites and vice versa.

Now, as if that weren't complicated enough, remember that the very
same thing is going on with the harmonic series of the OTHER note as
well, and since it radiates from a different place on the soundboard,
it will have a separate series of superimposed circles representing
phase cancellation for that source at that frequency. Remember you
need only eliminate the one or the other of the two nearly identical
frequencies in order for the beats to be eliminated. Therefore you
would have to superimpose the cancellation "topography" for both notes
to determine all the cancellation points.

That's why sometimes it takes but the smallest of movements, or even
just a slight rotation of the head to suddenly make a beating audible.
Naturally the distance will be smaller the higher you go in the
overtone series, because the wave lengths are progressively shorter.
That's also one of the reasons why it's easier to tune in the tenor
octave than the octave above middle c, because the distances between
the null points are twice as far apart.

Listeners, on the other hand, are usually far enough away from any
particular reflecting surface that they are receiving lots of
reflections simultaneously. This means that the sum total adds up in
such a way that cancellation is for all intents and purposes
"canceled". As they say, the player has the worst seat in the house,
not only for playing, but for tuning as well. When I teach tuning, I
sit in front of the instrument a meter or so away, and I'm amazed at
how often I can hear beats the student can't, at least not until he or
she moves around slightly to discover a Beat Spot. So what you are
hearing from the drivers seat is not an accurate representation of
what your listeners are hearing.

Regarding recording, the issue is the same. If the mics are close and
the lid is on the instrument, there will be trouble. If the mics are
somewhat distant (though not further than the Critical Distance, of
course), and especially if the lid is off and the instrument is not
next to any walls, there will be no cancellation distortion of the
recorded sound.

Hope this clears up at least that aspect of the issue.

Ciao,

P