In article ,
John Fields wrote:

On Thu, 05 Jul 2007 13:48:04 -0700, Jim Kelley
wrote:

John Fields wrote:
On Wed, 04 Jul 2007 09:11:58 -0700, isw wrote:


In article ,
"Ron Baker, Pluralitas!" wrote:

The beat you hear during guitar tuning is not modulation; there is no
non-linear process involved (i.e. no multiplication).


---
That's not true.

But it is true.

The human ear has a logarithmic amplitude response and the beat note
(the difference frequency) is generated there.

The ear does happen to have a logarithmic amplitude response as a
function of frequency, but that has nothing to do with this
phenomenon.

---
Regardless of the frequency response characteristics of the ear, its
response to amplitude changes _is_ logarithmic.

For instance:

CHANGE APPARENT CHANGE
IN SPL IN LOUDNESS
---------+------------------
3 dB Just noticeable

5 dB Clearly noticeable

10 dB Twice or half as loud

20 dB 4 times or 1/4 as loud

---

(It relates only to the aural sensitivity of the ear at
different frequencies.) What the ear responds to is the sound pressure
wave that results from the superposition of the two waves. The effect
in air is measurable with a microphone as well as by ear. The same
thing can be seen purely electrically in the time domain on an
oscilloscope, and does appear exactly as Ron Baker described in the
frequency domain on a spectrum analyzer.

The sum frequency is
too, but when unison is achieved it'll be at precisely twice the
frequency of either fundamental and won't be noticed.

The ear does not hear the sum of two waves as the sum of the
frequencies, but rather as the sum of their instantaneous amplitudes.
When the pitches are identical, the instantaneous amplitude varies
with time at the fundamental frequency. When they are identical and
in-phase, the instantaneous amplitude varies at the fundamental
frequency with twice the peak amplitude.

---
You missed my point, which was that in a mixer (which the ear is,
since its amplitude response is nonlinear) as the two carriers
approach each other the difference frequency will go to zero and the
sum frequency will go to the second harmonic of either carrier,
making it largely appear to vanish into the fundamental.
---

When the two pitches are different, the sum of the instantaneous
amplitudes at a fixed point varies with time at a frequency equal to
the difference between pitches.

---
But the resultant waveform will be distorted and contain additional
spectral components if that summation isn't done linearly. This is
precisely what happens in the ear when equal changes in SPL don't
result in equal outputs to the 8th cranial nerve.
---

This does have an envelope-like
effect, but it is a different effect than the case of amplitude
modulation. In this case we actually have two pitches, each with
constant amplitude, whereas with AM we have only one pitch, but with
time varying amplitude.

---
That's not true. In AM we have two pitches, but one is used to
control the amplitude of the other, which generates the sidebands.
---

The terms in the trig identity are open to a bit of misinterpretation.
At first glance it does look as though we have a wave sin(a+b) which
is being modulated by a wave sin(a-b). But what we have is a more
complex waveform than a pure sine wave with a modulated amplitude.

---
No, it's much simpler since you haven't created the sum and
difference frequencies and placed them in the spectrum.
---

There exists no sine wave with a frequency of a+b in the frequency
spectrum of beat modulated sine waves a and b. As has been noted
previously, this is the sum of two waves not the product.

---
"Beat modulated" ??? LOL, if you're talking about the linear
summation of a couple of sine waves, then there is _no_ modulation
of any type taking place and the instantaneous voltage (or whatever)
out of the system will be the simple algebraic sum of the inputs
times whatever _linear_ gain there is in the system at that instant.

Absolutely correct. And as that "simple algebraic sum" varies with time,
which it will as the phases of the two signals slide past each other, it
produces the tuning "beat" we've been talking about. Totally linear.

Isaac