Equal Temperament / Just Intonation

[Scarpia]

Not sure about this. I think of the issue in terms of graphical representation of the waveforms, without regard for ratio. Where the wave peaks synchronise, there is a beat. Isn't that right? Why would that not be audible?

The two representations are mathematically identical.  If the two tones have an integer ratio of frequency n:m one will complete n cycles while the other completes m and they will arrive at the same point after a fixed time interval, (time required for n cycles of the first tone) = (time required for m cycles of the second tone).  They will therefore be stationary on the oscilloscope.  The most obvious audible beat is when you have two tones that almost match, their ratio is almost 1:1 and you can hear them going in and out of synchronization. 

Two tones will have a subtle beat if they are close to one of the simple ratios because their overtones are not meshing.  For instance, you play a C and a G which is not quite in a 3:2 ratio with the C.  The third harmonic of C is G, and the second harmonic of G is G, and those two harmonics will beat with each other if the original C and G are not in the proper 3:2 ratio.

However, these "just" intervals are inconsistent.   Suppose you start with C.  Then you tune G so that it is in a perfect 3:2 ratio with C.  Then you tune D as a 3:2 ratio to G, then you tune A as a 3:2 ratio with D, continuing with E, to B, to F#, to C#,  to G#, to D#, to A#, to  F, to C.  That C you arrive at after going around the circle of fifths is not in tune with your original C.  Compounding those ratios of 3:2 you get a final ratio of (3^12)/(2^12) = 531441/4096 = 129.75.  The problem is that C seven octaves above your original C should be in ratio 2^7, or 128.  Not 129.75.  That C is way out of tune with the C you started with, and came be be called "the wolf" because it seemed to howl.    The bottom line is you have to compromise.  If you pick your intervals to make one chord sound just, chords based on other tones will sound off.

[eyeresist]

If the two tones have an integer ratio of frequency n:m one will complete n cycles while the other completes m and they will arrive at the same point after a fixed time interval

This should be true of any two tones, regardless of the ratio relationship.

[Scarpia]

This should be true of any two tones, regardless of the ratio relationship.

Not true.  Suppose one tone has frequency 100 cycles per second, it takes 1/100th of a second to complete, or 10 milliseconds.   Another tone has frequency 200 cycles per second, it takes 1/200th of a second to complete or 5 milliseconds.  After 10 milliseconds the first tone has completed one complete cycle, the second has completed two complete cycles, they are synchronized again. 

Now suppose the second tone has frequency 199 cycles per second instead of 200.  It takes 5.025 milliseconds to complete a cycle.  After 10 milliseconds, tone one has completed a complete cycle, but tone two is just shy of completing its second cycle.  They are out of synchronization.  After every 10 millisecond cycle of the first tone, the second tone falls a little more behind.   However after one full second, tone one has completed 100 cycles, tone two has completed 199 cycles.  It has fallen a full cycle behind and the two tones are finally synchronized again.  A frequency error of 1 cycle per second means it takes one second for them to come back into synchronization.

[eyeresist]

I guess the point is that notes out of ratios produce an audible beat. But it's still the same phenomenon occuring in either case.

[Scarpia]

I guess the point is that notes out of ratios produce an audible beat. But it's still the same phenomenon occuring in either case.

You will not hear an audible beat unless the tones are very close.  For instance, 100 Hz and 101 Hz will produce a very strong beat as the two tones go in and out of synchronization. 

The combination of 100 Hz and 199 Hz will not directly produce a direct audible beat.  However, there will be an indirect beat.  Any musical instrument will produce overtones at multiples of the fundamental pitch.  So a 100 Hz note contains overtones at 200 Hz, 300 Hz,  400 Hz, 500 Hz, etc.   The 199 Hz pitch will contains overtones at 398 Hz, 597 Hz, 796 Hz, etc.   So if you play 100 Hz and 199 Hz together, you will hear a beat between the second overtone of the first note (200 Hz) and the fundamental of the second note (199 Hz).  You may also hear beats between different combinations of overtones.  These beats make use of by people who tune instruments (as was mention above in this thread).
 

[jochanaan]

You will not hear an audible beat unless the tones are very close.  For instance, 100 Hz and 101 Hz will produce a very strong beat as the two tones go in and out of synchronization. 

The combination of 100 Hz and 199 Hz will not directly produce a direct audible beat.  However, there will be an indirect beat.  Any musical instrument will produce overtones at multiples of the fundamental pitch.  So a 100 Hz note contains overtones at 200 Hz, 300 Hz,  400 Hz, 500 Hz, etc.   The 199 Hz pitch will contains overtones at 398 Hz, 597 Hz, 796 Hz, etc.   So if you play 100 Hz and 199 Hz together, you will hear a beat between the second overtone of the first note (200 Hz) and the fundamental of the second note (199 Hz).  You may also hear beats between different combinations of overtones.  These beats make use of by people who tune instruments (as was mention above in this thread).

Nicely explained!  I would only add that for tuning purposes, there is little effective difference between a "direct" beat and an "indirect" beat; they're both equally useful.  The difference is that in the equal-tempered scale you have to put up with the indirect ones, except for the ones resulting from an octave.

When the "beat" reaches 16-20 Hz, it begins to sound like another note.  How audible that other note is relates to which overtones are beating.  For a "unison," if you have that many beats per second, it's very unpleasant!   But for, say. a major 3rd, it's much less obvious since what's beating is the 5th harmonic overtone of the lower note and the 4th harmonic overtone of the upper note--overtones that usually aren't even heard consciously.  In many cases, such fast beating sounds like an audible undertone.  These are usually unwanted in the Old Masters' music, but I'm sure some recent composers have taken advantage of just such effects...

[Scarpia]

In many cases, such fast beating sounds like an audible undertone.  These are usually unwanted in the Old Masters' music, but I'm sure some recent composers have taken advantage of just such effects...

I think that is the effect that gives elementary-school and middle-school bands their characteristic sound.   

[zamyrabyrd]

http://www.youtube.com/watch?v=BhZpvGSPx6w

That link is really cool. As a kid, I usually could not stand the sound of a newly tuned piano. Later on when working with string players, I found it was quite a task for them to adjust their intonation to the piano. Maybe over time we make the adjustments with our ear and brain to hear more perfect harmony, just as we can reconstruct a telephone conversation while hearing less than 50% of it.

Equal temperament seems to have been bolluxed up with just temperament at least in the 20th century. Otherwise, why would Schoenberg say that composing in tonality is like writing everything in the key of C, assuming all intervals being equal? Now, though, it is generally agreed that "well tempered" is not equal tempered.

This article was quite a revelation. http://www.eunomios.org/contrib/francis2/francis2.pdf

Apparently decorative curls or glyphs from the title page by Bach of the WTC had a more esoteric meaning, that is, the beats of tuning various notes on the keyboard. The significance is that keys like C and closely related ones were nearly perfectly tuned but others like Eb minor or B minor had an unsettled quality built into the tonality.

It's nice that electric keyboards can be tuned electronically so we can actually hear what Bach had in mind.

ZB