Virgil's beatless octaves.

[Jim Coleman, Sr.]

Hi to all:

Listening to the Whole sound

This is a response to the virgil Smith article in the PTJournal.

As many of you know, Virgil and I have been good friends for many years and
we have an ongoing dicussion on this subject of listening to the whole 
tone 
when tuning intervals, octaves, or unisons. I believe that I am finally 
beginning to put this all together.

When we listen to a Symphony, we hear a common pitch from the first violins
even though each violinist is playing with vibrato around the pitch center.
The vibrato rate is slightly different for each violinist. So they are 
not 
actually playing the same pitch at the same time. But our ears synthesize
this, or put it together as one unified pitch.

A similar thing happens when we tune a unison of strings, each of which 
has 
a wild beat of its own. We are able to balance these beats against each 
other in such a way that we perceive a clear solid unison.

I believe now that a similar thing happens when we tune an octave where 
we 
know that only one pair of partials can actually match at any one time. 
In 
many of my tuning classes over the years I have demonstrated a slide rule
which shows the mis-matching of partials when one attempts to tune a pure
octave. Below, I will attempt to show in ASCII art form how this 
mismatching 
works:

Think of the upper graph as representing the partials of the upper octave.
The lower graph as representing the lower note of the octave.

2-1 Match leaves the 4-2 out of tune, the 6-3 worse and the 8-4 worst.
                    1             2         3      4                  8  
                    |-------------|---------|------|-----|----|---|---|
     |--------------|-------|------|-----|----|---|---|
     1              2       3      4          6       8

4-2 match leaves the 2-1 out slightly, but the 6-3 and 8-4 are better
                     1             2         3      4                    
                     |-------------|---------|------|-----|----|---|---|
     |--------------|-------|------|-----|----|---|---|
     1              2       3      4          6       8

6-3 match makes the 8-4 better, but the 2-1 and 4-2 worse.

                      1             2         3      4                  
                      |-------------|---------|------|-----|----|---|---|
     |--------------|-------|------|-----|----|---|---|
     1              2       3      4          6       8

8-4 match leaves all the others worse.

                       1             2         3      4                  
                       |-------------|---------|------|-----|----|---|---|
     |--------------|-------|------|-----|----|---|---|
     1              2       3      4          6       8

A slightly wide 4-2 would be the best compromise in this case. It has 
been 
my experience that the better tuners do just this, especially in the 
temperament octave. This may cause the mismatch of the 2-1 and the 4-2 
to 
beat against the 6-3 and the 8-4 mismatches, thereby giving the perception
of there being no beat as Virgil reports.

Carrying this same principle out to include the double octave and the 
triple 
octaves, there will be a compromise between the single, double and 
triple 
octaves such that there is a best match when the double is just slightly 
expanded. This causes the 12ths and 19ths to be balanced against each other.
For example, the F5-C7 will be slightly wide and the F4-C7 will be 
slightly 
narrow, resulting in the triple octave (C4-C7) being only slightly 
narrow, 
the double octave (C5-C7) slightly wide and the single octave (C6-C7) 
wider 
yet.

It just may be that the beatless octaves which Virgil speaks of may just 
be 
the best match which we mortals hear. I must remind the readers that in 
the 
Tuneoff in Chicago, there wasn't a dime's worth of difference in my tuning
which was done by carefully balancing the partials and Virgil's tuning 
by 
listening to the whole tone.

Jim Coleman, Sr.