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Prompted in part by Jim Coleman's series of articles on altering the
stretch in the SAT, I have taken a look at the more general question.
See the attached ASCII text file. It reviews some principles of
inharmonicity and suggests some directions for further research.
-Bob Scott
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Content-Disposition: attachment; filename="Inharm.txt"
The Tuning Implications of Inharmonicity
We define inharmonicity as the fractional offset (expressed in
cents) between the pitch of a partial and the pitch of the true
harmonic (a whole number multiple of the fundamental). For example
if C5 has a fundamental of 523.0 Hz and its second partial is 1047.0
Hz, then the second partial would have an inharmonicity of 1.6
cents.
Usually higher partials have higher inharmonicities. In the above
example we might find the fourth partial of C5 at 2095.0 Hz, which
would give it an inharmonicity of 2.4 cents.
In tuning an octave for zero beats, various partials can be used.
Suppose in the above example we are tuning C6 from C5 using the
second partial of C5. Then we would tune the fundamental of C6 to
1047.0 Hz. But we might also hear the fourth partial of C5 beating
against the second partial of C6. Will this beat also go to zero
at the same tuning that gives a zero beat for the second partial of
C5? In the above example, let's assume that the inharmonicities of
C6 are the same as those of C5. Since the C6 string is shorter than
the C5 string, we would probably need the C6 string to be thinner
than the C5 string to have the same degree of inharmonicities. If
C6 does have the same inharmonicity as C5, then the C6 fundamental
will have to be tuned to 1047.0 Hz to stop the beating at that
frequency. But then the second partial of C6, being 1.6 cents
higher than twice the fundamental frequency, would be 2096.0 Hz.
This would beat with the fourth harmonic of C5, which is 2095.0 Hz.
So the answer in this hypothetical case would be no. You could not
make all the partials stop beating at once.
In a more friendly example, suppose the inharmonicities of each
string are proportional to their partial number. That is, the
inharmonicity of the fourth partial is twice the inharmonicity of
the second partial. This is actually closer to reality. So in the
example above, we would have the fourth partial of C5 3.2 cents
above the harmonic, which would put it at 2096.0 Hz. Furthermore,
assume that all strings have the same inharmonicity (not a realistic
assumption). Under these assumptions, all partials will fall into
zero-beat at the same time when tuning an octave. Furthermore,
these assumptions guarantee that two-octave zero beats happen at the
same time as one-octave zero beats. That is, if C6 is tuned to C5
and C7 is tuned to C6, then C7 will automatically be tuned to C5 (at
the fundamental and at all partials).
Now that we have examined the best possible case with
inharmonicities, let's get real. With the same string size,
inharmonicities go up as you go up the scale. When the string size
jumps to a thinner string, the inharmonicites go briefly lower.
When the type of string changes (such as from wound bass strings to
mid-range) there is also a step change in the inharmonicities.
In recent messages to the pianotech list, Jim Coleman Sr. has
advised on tuning strategies to address step changes in
inharmonicity, particularly at breaks in the scale. Let's see what
can be done in a hypothetical case. Lets suppose all notes below C5
have second partial inharmonicities of 1.2 cents, and all notes from
C5 and above have second partial inharmonicities of 1.8 cents.
Furthermore, suppose higher partials have proportionally higher
inharmonicities. Tuning octaves that are completely above C5 would
best be done by using a stretch of 1.8 cents per octave. Below C5
we would want to use a stretch of 1.2 cents per octave. Assume that
all the notes below C5 have been tuned. What should C5 be tuned to?
If you listen to C4 and C5 together, the zero beat at C5 will not
happen at the same time as the zero beat at C6 (the second partial
of C5 and the fourth partial of C4). Which one you listen to
depends on what is the most prominent. Usually it will be the lower
partial. Therefore if you listen only to C5, you will tune C5 to
the second partial of C4, which will give it a stretch of 1.2 cents.
This is because you are still relying on inharmonicites of notes
below C5. The inharmonicities of C5 itself does not come into play
if you are only listening to its fundamental.
However, if you listen to the second partial of C5 (which is 1.8
cents higher than the second harmonic), you will be comparing
against the 4th partial of C4, which we are assuming is 1.2 x 2 or
2.4 cents above the fourth harmonic of C4. If you had tuned C5 so
that its fundamental was zero beating with the second partial of C4,
then it would be 1.2 cents higher than the second harmonic of C4.
This would make the second partial of C5 1.2 + 1.8 or 3.0 cents
higher than the fourth harmonic of C4. But the fourth partial of C4
is 2.4 cents higher than that same fourth harmonic, not 3.0 cents.
So listening only to the second partial of C5 in tuning the octave
C4 - C5 you would have to tune C5 a little lower than you would if
you were listening only to C5. This is counterintuitive. You are
entering a region of greater inharmonicity and you have to
compensate (at least initially) by using less stretch, not more.
If this example goes against the real-world experience, then some of
the assumptions must be false. For example, it is highly unlikely
that you would pay more attention to the beats at C6 than at C5 when
tuning the C4 - C5 octave. Also, the assumption that higher
partial inharmonicities are proportional to the partial number
needs to be checked out with research. I only used this simplifying
assumption so that I could conveniently calculate results. It is
only by studying these simple cases (even if they are unrealistic)
that one can then attack the more general distribution of
inharmonicities.
In my article in June 1993, Piano Technicians Journal (pg 48), I
cited some inharmoncity measurements made on a 6'8" Kawai Grand.
Those measurements show that in the lower ranges of the scale, the
fourth partial has generally less than double the inharmonicity of
the second partial, while in the upper ranges, the fourth partial
has much more than double the inharmonicity of the second partial.
Therefore the real world is more complicated than the simple models.
In this discussion on tuning, I have looked only at octave tuning so
far. But what about other intervals? What affect does
inharmonicity have on them? Since other intervals are not expected
to zero beat, even if there were no inharmonicity, then the analysis
can not be done from the point of view of zero beats. It is
worthwhile to remember at this point that with pure (non-stretched)
equal-tempered tuning, a fifth is flat from zero-beat by 1.9 cents,
a fourth is sharp by the same number of cents, and a major third is
sharp by 13.3 cents. Before we introduce stretched tuning, let's see
what the effect on these beats would be by just bringing in
inharmonicity. If we go back to the simplistic assumption that
inharmonicity in a single string is proportional to the partial
number, then all intervals will beat as if they had been tuned
closer together. The fifths would beat more and the fourths and
major thirds would beat less. But the octaves would also beat as if
they had been tuned closer together. And so we would employ
stretched tuning to stop the octave beating, which would reverse the
effect on the other intervals, making them beat more like they did
in the pure equal-tempered totally harmonious case.
At this point I would like to define a "uniform" tuning to be one
where the stretch (as measured at the fundamentals) is constant
(such as 1.6 cents per octave). A uniform tuning would be the one
to use if the inharmonicity were also uniform (the same for all
strings).
Going back to the other intervals (besides octaves), let's see what
a departure from a uniform tuning does to them. As I said before,
these other intervals are expected to beat. If the beat rate were
to rise higher than expected, this might be perceived as a bad
tuning. And if the beat rate were to fall on the major third, this
might also be perceived as a bad tuning, based on what listeners
have come to expect. Each note can form many intervals with the
other notes on the piano. Changing that one note will make some
intervals better and others worse.
As Jim Coleman Sr. pointed out, a gradual adjustment is the safest.
If you have a particular stretch that is fairly uniform, and you
have a very locallized jump in the inharmonicity (such as a break in
the scale or in the stringing material) you really can't do anything
about it in a locallized fashion. The departure from a uniform
stretch has to be gradual or some interval is going to sound really
bad. It is no consolation if a great number of other intervals
sound really super. That one bad interval will determine the
quality of the tuning.
Perhaps what is needed is an automatic "smoothing" function that
takes a complete list of inharmonicities and generates a stretch
function that is locally uniform. The degree of smoothing could be
adjustable. The more you want to pay attention to higher partials
the more uniform your stretch has to be. No significant change in
stretch can happen in anything less than an octave, because this is
the minimal span of the octave tuning checks.
The FAC method used by the SAT does not even look at inharmonicities
outside of the three strings specified. The fact that this method
works at all is an indication of the acceptability of a generally
uniform stretch. But if the particular F, A, or C strings used in
this method don't happen to be typical of the other strings, then
the stretch that you get might not be the best for that piano. You
might get a better stretch by using the known characteristics of the
piano in a lookup table. A nearly uniform stretch funtion does not
need 88 numbers to be specified. It should be possible to
approximate a fairly uniform stretch formula with only a few
(perhaps four) parameters. These could be found in the lookup
table.
Here is another open area for research. When does the lookup table
method do better than an empirically measured method (such as the
FAC)? If one were to emprically measure inharmonicity in detail for
a large range of pianos, could such a lookup table be made that
correlates well with future measurements?
Here is a proposal for a lookup table format. Suppose the lookup
table is defined by four numbers. These four numbers would be the
stretch in cents per octave at C2, C4, C6, and C8. For any given
note, the stretch would be calculated from these four numbers using
the formula:
stretch( n ) = a * (n-49)**2 + b * (n-49) for n > 49
stretch( n ) = c * (n-49)**2 + e * (n-49) for n < 49
where n is the note number (1...88), so 49 is A4. The numbers "a"
and "b" define one parabola used for notes above A4 and "c" and "d"
define a different parabola used for notes below A4. The second
parabola would be concave down and the first one concave up. The
two parabolas automatically coincide at A4 (n=49), and they give a
stretch of 0. This ensures that A4 = 440 Hz. Finally, they could
be arranged so that average stretch near C2, C4, C6, and C8 is as
specified from the lookup table. I believe that for such
constraints there is only one solution for a, b, c and d. This could
easily be done by a tuning machine based on the four numbers
supplied from the lookup table.
One advantage to this method is that the tuning machine itself does
not have to store the parameters for each piano. Those four numbers
can come from a printed lookup table. Also, the four numbers used
are in understandable units - cents per octave. Therefore, if data
is not available from the lookup table for a particular piano, the
technician should be able to invent his own close approximations to
those four numbers. There could also be generic parameters sets for
approximations for whole classes of pianos (6'8" grands, for
instance). There could even be a fairly simple manual procedure to
derive the four stretch numbers from empricially measured
inharmonicity, as in the FAC method.
There is much more that could be said about inharmonicity and
tuning, but this article is too long already. Responses are
welcome.
-Bob Scott
Real-Time Specialties
Ann Arbor, Michigan
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