I would like to get some response on the following series of articles
on tuning equal temperament use pure 5ths.
AURAL EQUAL TEMPERAMENT BY PURE 5THS
Jim Coleman, Sr.
Impossible you say? That's what I always thought. The early organ
and harpsichord tuners soon learned you can't tune a circle of
pure fifths to complete an octave so they began various schemes
to go as far as possible with the pure 5ths. This was the basis
of Pythagorean tuning. Of course it would have been nice to have
pure Maj 3rds 4ths 6ths and minor 3rds and 6ths as well. But alas,
it was not to be. Actually, the 3rds and 6ths were quite busy. And
watch out for those wolf 4ths or 5ths.
Next came the Meantone tunings where it was possible to have as
many as 8 pure 3rds and a number of pure 4ths and 5ths. But, alack
and alas, there were those wolves. Then came the various Modified
Meantone schemes. Now they were really cooking. It was possible to
play in all keys with just a little careful attention to the
writing and voicing of parts. Mini wolves were created to lessen
the bite of larger wolves. This gradually developed a popularity
for what we now call Well Temperament.
Finally so many little wolves were created that it was decided to
just go all the way. After all, so many other good intervals had
already been compromised, so equal temperament seemed to be the
next logical step. Equal temperament was found to be more of a
struggle than at first expected. Not every musician had the
patience to learn the techniques and skills to be able to tune
equal temperament. This helped to more fully develop the
professional piano technician. Here were people who were willing
to devote their lives to practice and learn the art of equal
temperament. Finally after about 75 years the art was well
developed and standards of acceptable tempering were established.
Yet, there remains even to this date some disagreement as to what
constitutes an even balance of octaves. Since the advent of higher
tensioned scaling in comparison to the lower tensioned scaling of
the early Fortepianos and Harpsichords, it was discovered that
octaves could not be tuned purely either. This seemed not to
be a problem with the earlier instruments. But now, in order to
make a piano have good balance in the octaves, some tempering is
necessary. The main question today is: How wide can the octaves
be and still not be an irritant? Also, is there an ideal trade
off between the width of the 4ths or 5ths and the width of the
octaves?
Originally, the standard approach to setting equal temperament was
to first establish the octave and then by using a circle of 4ths
and 5ths, one could divide up the octave evenly by contracting the
5ths each by 1/12 of the 24 cent comma. The early attempts
involved tuning two 5ths upward and then dropping down one pure
octave and repeating the procedure until the circle of 5ths was
completed. Later it was decided that steps could be saved if one
tuned up one 5th, then down a 4th, up a 5th etc. until the circle
of 4ths and 5ths was completed. There have been many schemes used
to balance out a one octave scale.
One special scheme was to divide the octave into 3 contiguous Maj
3rds then tuning down a 5th from the second one and then building
two more Major 3rds, then dropping down a 5th etc. The earliest of
this system was popularized by Faust and later made more popular
by John Travis.
Eventually a scientific system was developed where one could tune
the circle by 4ths and 5ths and check with 3rds and 6ths. This was
made very popular by Wm. Braid White. Later a system called "Both
ways from the middle" was developed by Bill Stonaker where early
on one had some checks and balances for the 3rds and 4ths. Several
variations of this then have been used and made popular by such
people as George Defebaugh, Don Morton, Bill Stegeman etc.
Back in the early 70's, while working for the Baldwin Piano
Company I tuned many Acrosonics. I discovered it was quite easy
to set a temperament from A3 to A4 by using 3 contiguous 3rds as a
basis from which to tune 4ths and 5ths to complete the scale and
avoid the break area of the Tenor section. Later Dr Sanderson and
Rick Baldassin extended that idea to covering two octaves and
utilizing some bullet proof procedures to assure excellent results
regardless of the difficult scaling challenges of some pianos.
In all of the later developments, it was discovered that the 4ths
would beat slightly faster than the theoretical rates. The 5ths
would beat a little slower than the textbook values. As these
later systems became more popular, the desire to stretch the
octaves a little more to accommodate the effects of inharmonicity
became more popular. At the present time it is common to find the
octaves of the better tuners stretched by from 1/4 to 1/2 beats
per second at the second coincident partials of an octave. ie
when one listens to an octave at the partial one octave above the
upper note of the octave there will be up to a 1/2 beat per second
widening. Another way of checking would be to compare F3-A3 and
F3-A4 where the beat of the latter would be faster than the former
by up to a half beat per sec.
The result of this causes the 5ths to beat even slower. But of
course this is at the expense of the 4ths beating a little faster.
The main purpose of this first article of the new series is to
show that there are trade-offs when one attempts to favor one
interval above others in stretching equal temperament.
Many have held tenaciously to the idea that octaves should be
pure.
Can we talk about this? Is there a law of nature that says that
octaves must be pure? What about double octaves, must they also be
pure? What about triple octaves, or quadruple octaves? Must they
also be pure? Who is to say?
Contrary to what my good friend Virgil Smith says about always
tuning pure octaves, we have been able to show that when comparing
coincident partials, it is impossible to have all pure single,
double and triple octaves. If one starts in the center of the
piano and tunes a pure 2-1 type octave and then another contiguous
pure 2-1 type octave, and then another contiguous 2-1 type octave,
the resultant triple octave will be terribly flat on any typical
modern piano. As a result of this the modern tendency is to
stretch octaves judiciously in order to minimize this discrepancy.
In spite of Virgil's explanation of his method of tuning, his
results are very good and he actually is able to tune to greatly
minimize the discrepancy created by inharmonicity. After studying
Virgil's tuning style, I believe I have a clue as to what he is
really doing. He is not the only one. Brent Fischer of Arizona
State University is an excellent tuner who has also been able to
realize this ideal type of stretch. Recent study of the aural
tunings of Tom Kinney of La Crosse, WI have convinced me that for
a long time we have ignored a very helpful interval in controlling
our stretch tunings. This interval as you may have guessed is the
pure 5th.
As we have noted above, each effort to stretch octaves a little
more has resulted in the 5ths being slower and the 4ths becoming
faster than the theoretical values we all started with. Brent
Fischer is the first one to bring to my attention the idea that
the octaves should be stretched to the very edge of the limit.
That is, as far as you can stretch without the octave beginning to
sound bad. Perhaps others have tried to suggest this to me and at
that time it just went over my head.
The next logical question becomes: "What is the limit to the
stretch of an octave?" Can single octaves be stretched to the
point where triple octaves will sound good. I believe Virgil Smith
does this. I also believe that this is why his tunings sound so
good.
Can octaves be stretched as much as 3 beats in 5 seconds? We've
been listening to 5ths which are compressed that much for a long
time and it has been tolerable. Worse yet, we have been listening
to 4ths which theoretically beat 4 beats in 5 seconds.
Now when you realize that we have been listening to 4ths that are
more than 1 beat per second due to the recent trend to stretch
octaves a little, just how far can we go in that direction? If we
stretch our octaves just a little more, can we still tolerate even
faster beating 4ths? I believe the answer to this is YES. After
all, since the advent of equal temperament we have been condition-
ed to listening to 3rds which beat more than 7 beats per second.
What is the practical limit of beats which we can tolerate in the
sound of the single octaves and the 4ths? Could they be balanced
out equally? Should they be balanced out equally?
I believe the answer to that question can be found in the 5ths.
If one sets a temperament using pure 5ths, the 4ths will be
faster and the octaves will be wider. In the early experiments
thus far the results of this type of tuning has by the strangest
coincidence provided just the right kind of octaves to produce
tolerable triple octaves and the quadruple octaves are not bad.
The overall sound of the piano is quite exciting.
Next we will show some very simple practical ways of accomplishing
this type of pure 5th tuning and at the same time keeping all the
other intervals equally spaced (ie equal temperament). This next
installment should appear tomorrow showing step by step tuning
procedures.
Jim Coleman, Sr.
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