Intonation
A
Musician’s Scientific Exploration of Overtones
Created
by John Shortridge
Science and Art. Commonly considered complete opposites, yet it is often overlooked that there is one area of study where the two come together: Music. From a musician’s point of view, it is harmony, triads, intervals, melody, and counterpoint; from a scientist, music is frequencies, waves, beat frequencies, and overtones, but most importantly, the fundamentals of sound itself. My objective, as a scientist and a musician is to be able to understand music from both sides.
I’ve chosen to focus on overtones and their relationship with
the way we play in tune. It is common knowledge that the overtone series
resembles pitches in the tempered scales (scales adjusted in order to
sound in tune) of western music however; somebody with good ears, or a good
knowledge of the science of sound knows that there are slight differences
between the two. This area of music has been subjective to much debate by
scientists and musicians.
There are basically two different tunings debated
among musicians, just tempered tuning, and equal tempered tuning. Equal tuning
requires every note to be evenly spaced by 100 cents. Just tuning has is very
similar to equal tuning, but it the notes have been adjusted slightly in order
to sound in tune. A more detailed explanation and comparison of both
temperaments can be found in Kyle Gann’s “Just Intonation Explained”
and Michigan Tech’s “Just vs. Equal Temperament” comparison.
Problems of involving the equal tempered scale were first noticed when the scale
was recreated using a different method. By taking one note through a clever
system called the circle of fifths, every note can be achieved, yet when the
system brings that note back to the original note it has been raised 24 cents.
This method and its downfall is explained to a greater extent in “The Physics
and Psychophysics of Sound and Music” by David Worrall. My hypothesis is that
the when notes are played and adjusted as in the familiar Just temperament, they
sound more in tune because they match those notes in the overtone series.
To do this, I’ve chosen to focus on diads (A
pair of different notes in sounded simultaneously), and what is required to make
them sound in tune.
The
first step in this process was to get a good idea of what the overtone series
is. The overtones occur when one pitch is sounded, at any frequency. The
air vibrates in such away that it creates more pitches in a special order at
high frequencies than the fundamental pitch (the original pitch sounded). These
higher frequencies are called the overtone series. Mathematically, I
mapped out the overtone series up to 16 pitches above a fundamental A at 440hz.
Now I have the frequencies for the overtone series above the fundamental A at
440hz. Yet to continue I needed to derive a formula to convert hertz
(measurement used my scientists) to cents (the measurement used my
Musicians). This was a lot more difficult than I had planned, and I ended
looking for the answer on the Internet. I finally found the formula from Bruno
Putzeys of Philips
ITCL. The
formula for cents deviation is:
C=1200*(Ln(Fnew/F
old))/Ln2
Solved
to get a new frequency using cents deviation:
Fnew=Fold
*2^(C/1200)
Using these new formulas, I can now predict a more exact hypothesis.
After
taking the frequencies of the overtone series that I found algebraically and
converting them into cents, I can compare those to the cents deviation of one
note using the equal tempered scale (western scale with each note an equal 100
cents apart. How a piano is tuned). For example a major third should be sounded
400 cents above the fundamental tone according to the equal tempered scale. On
the overtone series however, the major third is 386 cents above the fundamental,
a difference of 14 cents. Therefore I hypothesize that a good musician would
sound a major third 14 cents flat making it in tune with the overtone series. I
concluded a hypothesis using the same method for 5 other tones:
|
Diad |
Cents
Dev. |
|
M2 |
#3 |
|
m3 |
#15 |
|
M3 |
b14 |
|
m6 |
#13 |
|
M6 |
b16 |
|
m7 |
#17 |
In order to test my hypothesis, I used a very simple method. I had three
subjects, Tim, Ian, and Jay play the top note of a diad. I provided only one
instruction: play in tune. One pitch steady C or F was produced by my computer,
and fed thru a pair of headphones as to not interfere with the tuner (see figure
1). The subject then played the prescribed note to complete the diad. The tuner,
in cents deviation from that same note, recorded this note, as it would be in
tune with the equal tempered scale. Finally, I compared the average of these
trials to my hypothesis.
Figure
1:
My
results for this experiment oscillate as would a sine wave (HA!).
These
are the average recorded values of raw data recorded:
|
Diad |
Jay
Abolifia |
Tim
Archer |
Ian
Tornay |
Hypothesis |
|
M2 |
#2 |
0 |
b1 |
#3 |
|
m3 |
#13 |
#7 |
#14 |
#15 |
|
M3 |
b10 |
b15 |
b12 |
b14 |
|
m6 |
b7 |
0 |
#5 |
#13 |
|
M6 |
b10 |
b7 |
#3 |
b16 |
|
m7 |
#16 |
#20 |
#11 |
#17 |
As you can tell, some samples were right on the money with my hypothesis and as for others, their results were not supportive of my hypothesis. Conclusively, I feel that my experiment supported my hypothesis of the major second, minor third, major third and the minor seventh. I don’t completely reject my hypothesis on account of the minor and major sixths because there is a large amount of uncertainty involved. This can be noted on the table above, it can most clearly be seen by the graph on the next page. The results and my hypothesis are compared to the accepted note value based on the equal tempered scale. In this case, zero cents deviation is the value according to the equal tempered scale. Bars reaching above zero are how many cents sharp the interval was played. Bars reaching below zero correspond to how many cents flat the interval was played. Please note that the Major and Minor sixth intervals show considerably less structure when compared to the similarities of the other intervals. On a quantitive basis, my uncertainty could be agreed upon as the furthest tilt left and right of the needle on my tuner. Surprisingly, I found that the uncertainty was less on the accepted tests (M2, m3, M3, and m7), within 10 cents of the average, excluding outliers. The results for the sixths were much more spontaneous; they didn’t seem to follow any guidelines for intonation for any subject. Basically, the total variance in these cases was from –30 to +30 cents deviation. This leads to a more qualitive explanation of the uncertainty involved. First of all, the equipment, the tuner, involved was not very accurate as far as for scientific purposes go. Besides that, there is a lot of responsibility placed on the actual player in this case, who, in some cases, may not agree on what is “in tune.”
And here is the word banana:
banana
The scientific and artistic communities have traditionally been separate as far as their ideas of “truth” in this world. Realistically, I think that this is the most important part of that relationship because it keeps both in rigid tow of each other, which ensures that neither field ever becomes “stale.” That said, it is impossible for me to take a completely unbiased side to my argument of intonation. From a scientist’s standpoint, there is definitely a strong argument for my hypothesis. Yet also, there is reason to believe that an experiment such as this has so much uncertainty involved due to opinions of different players as what is “in tune,” and the personal judgments and ability of the performer, that it has no place in the scientific community. From a musician’s standpoint, I can tell you that there is a connection between the overtone series and playing in tune. I know this from experience, and the ability to listen closely and analytically. I can also say that there are some opinions that are less varied among musicians and those that are. For instance, another source of uncertainty as far as the major and minor sixth diads, is that the sixth can be considered somewhat foreign to many players, which means that Jay’s opinion of what an in tune minor sixth diad can be completely opposite of Ian’s. Or perhaps the uncertainty comes from malpractice. I think most musicians hear the sixth as an inverted third. Many people would hear the top note as the root of a chord, and the bottom as the third. That said, tuning the top note of a sixth diad, as I attempted, would be like trying to tune the root of a chord, which is in essence, changing the entire key. Very difficult.
Scientifically, using more sophisticated equipment, and perhaps using one human being for judgment of in tuneness could improve this experiment. Perhaps having two frequency generators play the two frequencies and one being adjusted by a listener until the pitches were acceptable would be a better approach to the problem. If a tuner were to be used, digital inputs would be more accurate than the built in microphone. If a note was recorded onto a computer, it could be purified digitally, by eliminating background noise, and overtones so a single steady pitch could be fed into the tuner.
There are a lot of people who believe that the answer to the bigger question of playing in tune has been answered. These people are wrong and this is why: Ever since man first sung, or hit a rock with a stick, music has been about individuals. Every musician is unique, and therefore an agreement on what is in tune and what is not will never be made. If there is one thing that my experiment did support it is that there is no one definition of playing in tune, but that doesn’t mean that this type of research is not important. Our goal as artists in scientists is not necessarily to define, but rather just to simply understand.
Just vs. Equal temperament (Michigan Tech Univeristy)
A good comparison between the advantages and disadvantages of equal and just temperaments.
Just Intonation Explained (Kyle Gann)
Just Intonation explained using Fractions to notate intervals (not discussed in my paper)
Re: What is the operation to convert Hertz to cents deviation in music pitch? (Bruno Putzeys)
The operation that converts hertz to cents deviation.
The Equal Tempered Scale and Peculiarities of Piano
A close study of the equal tempered scale and its importance to the tuning of pianos
An expansive network of those who call themselves "intonation masters."
Solution to the Rubiks Revenge (4by4by4)