Pure 5ths Using the Accutuner

[Jim]

To Murray:

Thanks to you for pointing out some ambiguity in my last article.
I am enclosing a revised version which should clear up that problem.

              PURE 5ths Using the Accutuner  Part III

It was while playing around with altering the A4 stretch numbers
that I discovered the possibility of having equal temperament
with pure 5ths. For sometime it had been noticed that when
the temperament octave was stretched a little more than before
that the 5ths would be slightly slower and the 4ths would be a
little faster in beat rate.

The original experiments were done on my Steinway L which had
FAC stretch numbers of 7.0, 8.5, 8.6.  By adding 1.0 cents to
the A4 stretch number, the center octave was stretched enough
to make the 5ths come out pure in almost every case. Now this
was at the expense of making the 4ths faster as well as the single
octaves. The smooth curve ability of the SAT kept the 3rds and
6ths well in line, but the 10ths were a little faster, but the
progression was smooth for all intervals.

At first one may object to the wider octaves, but after tuning the
piano this way, and playing music, it was not objectionable and
the wider octaves prepared the way for slightly wider double
octaves than the normal readings would provide. While playing two
handed block chords up and down the piano, there was a solidity
noted which far outweighed the slightly stretched single octaves
in the center of the piano (up to 1.5 beats).

By the time I reached the 5th octave in the tuning process, and
making the 1.0 cent reset, (necessary as explained in previous
articles on Altering the Stretch Numbers), the test for 3rds,10ths
and 17ths showed a very decided increase in the 17ths over 10ths.
But the payoff was in the fact that the 5ths and 12ths continued
to be pure.

For those who have not read the previous articles on stretching
the the stretch numbers, it is necessary to reset the SAT on the
sharp side approx. the same amount as the alteration of the A4
stretch number in order to make a smooth transition from B4 to C5
during the actual tuning process. Just don' forget to erase that
reset value when you go back to check anything below C5, or to
tune the Bass.

By the time I reached the octave 6, the 5ths were actually stretch-
ed, but the 12ths continued to be pure and the triple octaves were
also pure. Eventually the 12ths (octave-5ths) gave way to the pure
19ths (double-octave-5ths). The top half of octave 7 may have been
a little too much as single octaves, but the triple octave sounded
so good, that it was tolerable. The octave-5ths began to be on the
wide side too.

There is a difference between our melodic sense of hearing and our
harmonic sense of hearing. One can prove this to himself by
playing middle C with a short staccato blow and while remembering
the pitch, playing the C7. Most of us would rather tune the C7 a
good bit higher in satisfying the melodic sense than we would tune
it as a triple octave sounded together. This test has been done in
many of my tuning classes over the years. Most classes preferred
to hear the C7 somewhere around 17 to 30 cents sharp when C4 and
C7 are played melodically (one note after the other). This gives
a stretch on most pianos which would make even a triple octave
beat on the wide side (ie too sharp).  By progressively sharpening
the single, and double octaves as proposed by this series of
articles, there is less difference in beats between the single,
double and triple octaves as viewed from octave 7. At this point,
I am beginning to think that this is what Virgil Smith is doing or
at least approaching when he says he can get pure single, double,
triple, and sometimes quad octaves. He definitely does make the
5th and 6th octaves sharper than most of have been doing.

Another interesting discovery during Bass tuning was that in going
down below the break, I was able to maintain pure 5ths.  Most of
you have noticed that the normal Bass stretch using the FAC
numbers, provided Bass octaves which are usually stretched to make
the 6-3 octave relationship a little wide. During this greater
stretch, it was noticed that the 5th above the note being tuned
was almost always pure. This should not have been surprizing to me
when I consider that for example when tuning C2, the machine is
listening to the 6th partial which is also the 4th partial of G2.
I already knew that the normal curve of FAC tunings attempts to
keep the double octaves fairly clean, so since the partial which
the machine listens to is two octaves above a note a 5th higher
than the note being tuned, the match is very good. Of course, one
should expect some slight variation in all Bass tuning tests,
because the inharmonicity of Bass strings usually jumps around
quite a bit (ie you can't have everything coming up like roses)
during the tuning process. There is almost always a jump as you
cross the break between the 2 string unisons and the single Bass
Strings. Usually the 17ths and double-octave 7ths slow down or
else one is widening the 5th, the octave and the double-octave
more than normal.

Since beginning this series of articles, I have tuned 7' pianos,
6' foot pianos, spinets, consoles and studios. Pianos with low
inharmonicity do not seem to need the 5ths as pure as the others
do to have good matching of single, double and triple octaves,
which is the ultimate purpose of this series. The mid portions of
this series attempt to lay out an approach to widening the
octaves farther than normal using only aural principles. Perhaps
the 5ths only need to be 1/10 beat per second, and the 4ths less
than 1.5 beats per second, and the center octave perhaps only will
beat 1.0 beats per second at the second coincidental partials.

Experimentation will be needed to arrive at the ideal proximity
of pure 5ths to accomplish the purpose of balancing out the
various types of octaves. I hope others are stimulated to experi-
ment further and share with all of us what works for them from the
strictly aural perspective.

Jim Coleman, Sr.