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I mentioned ghost tone testing in an earlier post and a few people =
emailed me to ask huh? What? How? So, post the following. I am also =
attaching this as a Word document, if anyone cares to print out a =
cleaner copy with better layout.
GHOST TONES IN PIANO TUNING
Based on Information and Ideas Stolen from Various Reliable Sources
By Alan R. Barnard
Introduction
I'll try to outline the science of strings, briefly, and their ability =
to generate "ghost tones" at specific partials for accurate tuning =
checks. Most readers will already know some or all of this information, =
but it really helps me to understand the theory behind our practices so =
I assume others will appreciate having it presented. If you are pretty =
new to all of this or limited in knowing the physics of strings and/or =
music theory, please sit at a piano as you read this. Experiment with =
the relationships I describe; I think you'll "get it."
I'm writing this all out of my head. If I get anything wrong, email me =
pronto, please, and I will post a corrected version of the whole thing.
Intervals for "ghost tuning" are ones that occur naturally in one string =
so this is not a discussion on temperament or evenness-of-scale testing =
(running 3rds or 10ths, for example) for which ghosts are not =
appropriate. The reason for this is an "energy" problem (more on this =
below).
Also, my presentation of theory is based on "perfect string" science. No =
string, and especially not a heavy steel string, can begin vibrating =
precisely at its termination points due to the stiffness of the =
material. This generates, of course, the "inharmonicity" that provides =
job security for professional tuners and is never quite grasped by =
amateur "tooners." Inharmonicity, its relation to tuning, string length, =
size of piano, unwanted noise in single bass strings (longitudinal =
inharmonicity), and other topics, are all very interesting but beyond =
the scope of the present discussion.=20
So we will assume perfect string harmonicity in this discussion =
although, ironically, it is inharmonicity that makes "ghost tuning" =
useful and interval testing necessary in the first place!
The Science Behind It:=20
Every string vibrates in a complex way including ...=20
a.. End to end, at its fundamental pitch (1st partial).=20
b.. In halves, with a node (dead spot) in the center of the string, =
one octave above the fundamental. This is the 2nd partial (or, to make =
it confusing, some call it the 1st overtone)=20
c.. In thirds, with two nodes, for the 3rd partial (2nd overtone). =
This interval is a 12th and is an octave + fifth above the fundamental.=20
d.. In fourths, with three nodes, for the 4th partial (3rd overtone). =
This is the double octave.=20
e.. In fifths to produce the 5th partial. This interval is the 17th, =
or two octaves + fifth.
For a brilliant and impressive demonstration of this, ask a brilliant =
and impressive guitarist to play "harmonics" for you. Especially at the =
2nd partial, you can actually see the string vibrate in two halves with =
a node at the twelfth fret (1/2 the string length). While it is thus =
vibrating, you can lightly rest your finger on the string at this node =
without dampening the sound in the least. The 7th fret hamonic is the =
3rd partial, the 5th fret is the 4th partial, etc. Do you see the =
implications of this for hammer shaping? Damper placement and length? =
Hammer strike line? Scaling, in general?
A cellist or bassist could also demonstrate this very well. Violins you =
could hear, but not see, what is happening. Did I mention that I am a =
guitarist (classical)? The jury is still out on "impressive," and =
definitely found me not guilty of that "brilliant" thing.
Here is a chart of these intervals and their musical tone equivalents =
for the note C28. The key numbers listed under "Note" refer to the key =
that plays, at its fundamental, the same pitch as the partial of C28.=20
Note (Key)
Partial
Interval
=20
C28
1st=20
Fundamental
=20
C40
2nd=20
Octave
=20
G47
3rd
Octave + Fifth (12th)
=20
C52
4th
Double Octave (15th)
=20
E56
5th
Double Octave + Third (17th)
=20
G59
6th
Double Octave + Fifth (19th)
=20
A#62 (B()
7th
Double Octave + Seventh
=20
C64
8th
Triple Octave
=20
D66
9th
Triple Octave + Second
=20
E68
10th
Triple Octave + Third
=20
Important or Interesting Comments on the Chart Above
Please note distinctions between the terms partial, interval, and chord =
member. The note G37 in relation to C28, for instance, is the 3rd =
partial of C28, a chord fifth in a major or minor C chord, and an =
interval of a 12th from C28.
It is also interesting to note the natural science of music theory that =
a single string demonstrates. In our example, the first 10 partials =
include 4 C's, 2 E's, 2 G's, a B(, and a D. So we have a C major chord =
in the natural harmonics of the string, i.e., C-E-G. Add the B( and you =
have a C7 chord; add the D and you have a C9 chord. The same =
relationships hold for any string played.
Note: It is only a coincidence that the chord seventh, here, is the 7th =
partial and the 9th partial is a chord ninth. In fact, the 3rd partial =
is a chord fifth, the 5th partial is a chord third. Don't be confused.
It is the relative strength or weakness of each sounding partial (and =
the higher ones not here named) that, mostly, account for the string's =
timbre (pronounced "tam=B7br"), that is to say, it's particular =
character of sound. Middle C has different characteristic sounds in =
pianos, harps, guitars, and cellos because of the way string materials, =
lengths, masses (weights), & tensions-as well as differing soundboard =
and resonating chamber constructions-impede or encourage the strength of =
each partial. Similarly, it is the reason an oboe, a clarinet, a flute, =
and a saxophone can play the same note (fundamental) yet sound =
distinctly different. This likewise explains why two human voices =
singing the same note sound different, especially if there are =
significant physical differences between the two persons, e.g., a tenor =
and a soprano each singing middle C. Further examples include organ =
"stops," reproduction of sound through two different hi-fi speakers, the =
same instrument played in two different concert halls, etc.
Demonstration
At a tuned piano, hold down the C28 key (without sounding it). Use no =
pedals. Strike and instantly release C40 so the damper immediately stops =
this note. You will continue to hear C40 because, through what is called =
"sympathetic resonance," the energy you put into the piano has caused =
C28 to vibrate at it's 2nd partial (1st overtone)--the C40 pitch! To =
prove that this is so, strike and release the C40, as above, then =
release C28 and note that the middle C tone of C40 stops instantly. =
Instantly, that is, if your C28 damper is working properly.
Just for fun, hold down C28 as above and quickly strike and release any =
or all of the notes above C28 listed in the chart above. They will all =
sound until they decay or until you release C28.
Now, holding down C28, try striking and releasing a key not in the above =
chart. Any tones that are picked up by the C28 strings are from =
overtones (partials above the 1st) of that struck key, not from its =
fundamental. These overtones, have less energy than the fundamental of =
that key. This explains, in part, why sympathetic vibrations picked up =
by the C28 strings are relatively weak and somewhat out of tune, in this =
case. The other reason is that the C28 strings are naturally sympathetic =
only to their own partial series, as per the chart above.
Try the experiment with G35. This is the first natural fifth of the C =
major chord built on C28. So one might expect it to produce a loud =
sympathetic ring in C28. But it doesn't. In fact compare it to what is =
generated by G47 at the 12th. G47 is in the partial series (chart) and =
G35 is not. The energy put into C28 by G35 is at G35's second partial, =
not at its strong fundamental, and therefore has the pitch of G47-but =
not with the amount of energy the G47 key will impart-at its fundamental =
pitch-if it is struck.
The beats between strings of different intervals (or out-of-tune =
unisons) are caused by the interactions of closely--but not =
identically-aligned partials of the respective strings. Two strings in a =
unison provide the easiest example: When they are in tune (and presuming =
they have identical mass, tension, and length-with no nicks, kinks, =
rust, or other flaws), every partial will, in theory, line up exactly. =
Each partial will be exactly in phase, meaning the highs and lows of =
their generated sound waves will occur at precisely the same instant. If =
the strings differ in any way, something will change, some of the =
partials will be out of phase.
When two tones are near each other but not in phase, their respective =
sound waves will sometimes reinforce each other (we hear louder) and at =
other times interfere with each other (we hear softer). These are the =
proverbial "beats" we tune with or try to tune out. Beats are a real, =
physical phenomenon-a change in the way sound reaches our eardrums-not a =
psychological trick of the brain, as some believe.
As an interesting aside, multi-engine aircraft can also be tuned with =
beats. When the beats get slower, the prop speeds are closer; when the =
beats stop, the two propellers are spinning at exactly the same speed =
("rotational velocity," to be precise). If the plane has more than two =
engines, you have to "tune" two of them, then start and "tune" the =
third, etc. The same principle applies to tuning strings: you can really =
only tune one at a time. Even if you aren't muting out other strings, =
you are only listening to and tuning one string at a time. Also, if the =
propellers differ in some way (not a good idea), would you have trouble =
tuning them to beats? Yes. It would be like tuning two badly mismatched =
strings in a bass unison (Oh, Mercy, don't you know: Been there, done =
that, got the souvenir T-shirt.).
If you want to see this effect, take some stones of differing mass or =
size and find some still water (a pond, not a river). Start dropping =
stones near each other and watch how the ripples interact.
This little phenomenon also explains why string "level" is important to =
tone quality in unisons. If the strings are not in exactly the same =
plane with respect to the hammer face, one string will start sounding =
slightly ahead of the other, i.e., out of phase. So if the hammer shape =
is uneven, the hammer "travels" and strikes at an angle to the string =
plane, or the strings themselves are not exactly level, it is not =
possible to align all the partials of those two (or three) strings. Even =
if each string is precisely, stop-the-lights, in tune at the fundamental =
pitch, out-of-phase higher partials will make the note sound whiney, or =
worse. (Hello Betsy Ross and all your cheap little friends.)
Ghost Tuning ("Finally," Many Will Say) or, More Precisely, Ghost Tone =
Interval Testing
This part, I will just present as "how to's" and examples. The idea, in =
each case, is to isolate exactly the partials we are trying to match in =
the two notes of the interval without having to listen to-or through-the =
fundamentals or any other partials of the notes.
Octaves
Please note that this is not a treatise on tuning, per se, nor on what =
size octave to use in what part of the piano or anything like it. This =
is simply a way to isolate the pitch of string interactive beats-for =
training purposes, if nothing else. Often times, the "best" tuning for a =
bass note, especially in poorly scaled or tired pianos, will be =
somewhere between a perfect this and a perfect that-a process known as =
"evening out the growls."
To test a 6:3 octave, hold the two octave keys down, silently, =
(remember, no pedals) while striking and releasing the 19th (two octaves =
+ fifth) from the lower note. This will excite the lower string of the =
octave at it's 6th partial and the higher one at its 3rd. The resulting =
tone will have a beat or a slow roll unless the two octave notes are =
exactly in tune at the 6:3 interval or so far out of tune that they are =
not picking up the vibrations of the "energy" key, the 19th.=20
If our octave sounds like an octave and this test produce no beat or =
roll, you have a perfect 6:3 octave. For many pianos, especially smaller =
ones, this makes a fairly clean, nicely "stretched" octave in the bass. =
The double octave, in this case, will have about a =BD second roll that =
you can isolate by using the double octave note as your energy key.
In our chart example, the C28-C40 octave 6:3 test will use the G52 key. =
Don't forget to immediately dampen the "energy" key after playing it. We =
don't want to listen to the sound of the G52 key at all! We just want it =
to put some energy into the other strings, via the soundboard and =
bridges. Also, if your "energy" key is above the dampers, be prepared to =
mash your right thumb or, better, a chunk of felt onto that unison as =
soon as you release the key.
To hear the double octave in our example, use C52 as the energy key. If =
the octave is "in tune" and you hear no beat using C52 for energy, guess =
what! You have a perfect 4:2 octave. The fundamental of C52 is the 4th =
partial of C28 and the 2nd partial of C40. This is, of course, the =
approximate octave size we want for the temperament and on up into the =
treble.
I will note, at this time, that it gets harder to hear octave ghost =
tones as you go higher in the piano. Sometimes the lowest bass notes are =
hard to excite with "ghosts," also. I don't know why, but I expect it is =
because of their mass. In fact, my theory is that "tubbiness" in big and =
fat and/or tired and dirty strings is just a reflection of their =
inability to generate or sustain higher partials. (Comments welcome.) =
Anyway, ghost tones are really most useful in the lower tenor and =
bichord bass regions, in my experience.
In the "To err is human, to really screw it up takes a computer." =
department: I had to correct my own error in the last paragraph. My =
spell checker wasn't the least bit bothered by the phrase ". to hear =
octave ghost towns ...")
Tuning a Steinway D or a B=F6esendorfer Imperial? Try C64 with the =
C28-C40 octave. See how close the piano will let you stretch for 8:4 =
octaves.
Unisons
When string pairs are very noisy, especially in the bass, it is =
sometimes helpful to tune or test with ghosts. The usual culprit or "bad =
boy" in the mix is the 15th. The energy key for this is at the octave + =
fifth. In our example, if we were tuning two strings of the C28 unison =
we would strike G47 and try for no beat, no roll. Not that this always =
produces the "best" unison: As always, listen carefully-then judge, =
nudge, or fudge, as necessary.=20
Other Intervals
Though not as useful as for octaves and unisons, other intervals can be =
tested with ghost tones. The beat we listen for in fifths is found one =
octave above the higher note of the interval. The beat of the fourth is =
found two octaves above the lower note of the interval. For example: The =
F33-C40 (fifth) roll can be isolated and energized by striking the C52 =
key. For the F33-A# (fourth) beat/roll, strike F57.
Ghosts for other intervals are, in my opinion, hard to hear, hard to =
remember (where they are), and not any help to the tuner.=20
Well, that's it for now, boys and girls. This is Uncle Al, the kiddies' =
pal, saying: So long, and thanks for listening.
Disclaimer
If this information is correct, coherent, and useful, I wrote it. =
Otherwise, it's someone else's fault, entirely.
Alan R. Barnard
Salem, MO
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