Choreboy wrote:
John Popelish wrote:
(snip)

That is Fourier analysis.

The rest is about making the math more efficient.


That's easy for you to say!

I think you've shown me something. When I hear "sine wave" I imagine
one cycle. I guess that's wrong, and a wave is a train of cycles.

True mathematical sine waves extend from infinite negative time to
infinite positive time. Practical sine waves last long enough for
things to respond to their frequency. How long that is, depends o
what is reacting to it. A frequency counter operating in period mode
needs only a single cycle to make its measurement. An ear needs
several cycles to several dozen cycles, depending on exactly what part
of the audible spectrum being detected (this property of ears is part
o the MP3 music encoding scheme). A quartz lattice filter may need
thousands of cycles to of a pure frequency before it develops a nearly
steady state output.

Musical harmony is in a sustained interaction between trains of cycles.
The interaction won't be simple enough to hear unless the quotient
between the frequencies is a small integer.

Something like that. Each frequency component in the signal has to
last long enough for the time response of that frequency of the ear's
sorting system to respond to it. If two frequencies fall within a
single reception band, they are not heard as two tones, but as a beat
addition and cancellation) as a single tone at about the average of
the two frequencies and an AM modulation at the difference of the two
frequencies. Obviously, if the beat is very long period, you have to
hear the two beating tomes for a cycle or two of the beat period to
detect that effect. Harmonically related tones just produce a
repeating pattern at some integer multiples of each of the component
frequencies. This can produce a very pleasing effect. You hear sound
from one musical source as a fundamental and several harmonically
related frequencies. If a second musical source (a harmonizing voice,
for example) has its fundamental at one of the harmonics of the other
signal, your brain recognizes this simple multiple relationship as a
pleasing musical harmony. For some ratios. This page shows some of
the approximate ratios between notes that sound interesting together:
http://www.jimloy.com/physics/scale.htm

When they talk about harmonics in an electrical wave, I guess they're
talking about the potential for energy transfer.

Not really. since linear circuit components react to many frequencies
by the addition if the effect of each frequency, it is a very powerful
analytical procedure to break a signal down into its harmonics and
evaluate the response of a circuit to each of those harmonics, and add
all the effects together to get the total response.

In that case, only odd
multiples of the fundamental will stay in phase to tap the energy from
the distortion.

Symmetrical distortion of a sine wave (shape of positive half cycle is
a mirror image of that on the negative half cycle) can be shown to be
made up of only the fundamental and odd harmonics (3 times. 5 times,
etc.). If the distortion peaks up one half cycle and flattens the
other or shifts the zero crossing so that one half cycle lasts longer
than the other, there are even harmonics in the wave shape. There may
also be odd ones, too. Got to do that Fourier analysis to quantify that.

Where a wave is flattened it may resemble part of a
sine curve with a longer period than the fundamental, but that doesn't
count because you can't tap energy from the flat part.

You can with a resistor. From a Fourier perspective, that flat part
just represents a time when the curve of some frequencies is nearly
canceled by the curve from other frequencies. You need an infinite
number of harmonics to make a truly flat square wave with perfectly
square corners.

If there's any truth in what I've said, I'll forget in a flash. In 1975
I was working in a repair facility. We'd use Bird Wattmeters to see
forward and reflected power in antenna feeds. We knew the jargon and
how to use the meters, but one day it struck me that none of us
understood why they worked. I had a flash of insight and everybody
stopped work to listen to me explain. Their faces lit up with
comprehension. I felt pretty smart. The next day I couldn't remember
whatever it was I'd figured out.

I hate it when that happens.