Stretching, beating, dissonance, and perfect JI

}Also, regarding Gregg Gibson's assertion that the octave should be
stretched
}a few cents, wouldn't this be a problem for guitar players (and steel
}guitarists) who are accustomed to sounding harmonics to get an octave
above
}the note they are fingering (or barring)?

Guitarists commonly sound the 5th harmonic, even though it is 13.7 cents
off 12TET. The 2nd harmonic being 2.7 cents off will not be a problem.

}I also suspect that there would
}be a lot of beats happening around double- and triple-octave pairs. If
the
}bass player plays an A, and my A three octaves higher adds 10 cents to
the
}equation, I wouldn't expect it to sound good.

An 8:1 off by 10 cents would be no worse than a 5:1 off by 14 cents, and
we accept the latter. Certainly a 7:1 off by 31 cents is much worse, and
yet the harmonic seventh has been used as a consonance even in 12TET.
Gamelan orchestras purposely detune octaves by far more than 2.7 cents;
although their instruments have inharmonic partials, they can tune pure
octaves using second-order beating and difference tones, and yet they
choose highly distorted (usually stretched) octaves. They even tune
their unisons so that they produce beating.

Beating is not the same as dissonance. Beating occurs when two pure
tones are so close in pitch that they are perceived as one tone of
changing amplitude. Roughness, one contributor to dissonance, occurs
when the two pure tones are partially resolved from one another but are
less than the critical bandwidth apart. Beating may be incidental to
roughness, but changes in amplitude are not what constitutes dissonance
(or else we would consider a tremolo effect dissonant).

For an interval in perfect just intontation, whatever phase difference
was present at the beginning of the sound would persist for the duration
of the sound. Let's say two instruments are producing tones, both of
which have an equally loud partial at 1000Hz. For someone standing in a
particular location with respect to the two instruments, there is some
delay time between the beginning times of the two instruments for which
the 1000Hz components will interefere constructively, resulting in four
times the energy at that frequency than what one instrument alone would
produce. For a delay time just 1/2000 of a second longer, you get
destructive interference, or no energy at that frequency. Obviously no
human instrumentalist can control their onset time to within 1/2000 of a
second relative to the onset time of another instrument. So you end up
with a random number between zero and four to describe the energy of the
1000Hz component -- probably not the musical effect you were looking
for. Even an individual performer playing both instruments can't do it
-- try tuning two synth keys to the same note and notice how repeatedly
playing both keys "simultaneously" leads to random fluctuations in the
loudness.

Now if the onset times are controlled really accurately by MIDI, and
let's say a 90-degree phase shift is chosen so that the energy at 1000Hz
is twice that from one instrument, then you're doing okay, right? But
for someone standing in a different location in the room, though, the
differece in the path lengths from the two instruments, and thus the
phase, will be different. There are nodes of destructive interference
and antinodes of constructive interference in the room no matter what
the original phase shift was. So some members of the audience may be
receiving no energy at 1000Hz, while others may be receiving four times
what they would from one instrument. Again, probably not the effect you
were looking for.

If both instruments are electronic, and their signals are mixed into one
speaker, there will be no spatial nodes or antinodes. So finally you can
control the musical effect. But other than monophonic electronic
MIDI-sequenced music, perfect JI can lead to big problems. In the real
world, thankfully, acoustic instruments are slightly out-of-tune with
each other, so that whatever the onset delay and whatever the position
in the room, one gets a signal that frequencies common to both tones
have an average energy twice that of one instrument. There will not be
noticeable, let alone startling, differences in the musical effect
depending on minute variations in the onset delay or position in the
room.


SMTPOriginator: tuning@eartha.mills.edu
From: "Brian M. Ames"
Subject: Semiconductor analog to phonons
PostedDate: 01-01-98 08:42:01
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