The Physics
of Music

Music depends on our perception of sound, and the way that sound works
in music is a topic that has interested philosophers, music theorists, and composers since the time of Pythagorus. Styles change and musical cultures differ, but sound itself works as it did a thousand years ago and as it will a thousand years from today.

“String Vibrations,” Andrew Davidhazy, Rochester Institute of
 Technology (2008)

The photo shows modes of vibration in a single string.

For example, here’s the waveform produced by a French horn playing a single note:

The horn waveform can be analyzed as the sum of many sine waves of different frequency, all of which are being heard simultaneously. That’s why natural sounds are said to be complex; they are a mixture of many different frequencies, called partials Partials A component of a musical tone
whose frequency may or may not
be a multiple of the fundamental frequency. A natural sound is not a single frequency, but a composite of many. The lowest frequency is the fundamental, the higher ones are the partials. The overtone, or harmonic partial, is a special class of partial whose frequency is a multiple of the fundamental frequency. .

In pitched sounds – sounds with a repeating waveform –
these different frequency components have an interesting relationship to each other that can be illustrated by examining a hypothetical vibrating string. If you pluck a string that is tightly stretched between two points, it will vibrate along its whole length, creating a waveform like the sine wave above,
but it will also vibrate in halves, thirds, fourths, fifths, etc.

The figure below illustrates, in a very exaggerated way, the first three modes of vibration of a taut string:

The various modes of vibration give rise to additional tones that complicate the wave form graph. Ideally (if the string is perfectly flexible and even) each of these other tones has a frequency closely related to the original one produced by the whole string. If the whole string is vibrating at 110 cycles per second, half of it will vibrate at twice 110, or 220; a third of
it will vibrate at three times 110, and so on. Each of these
other vibrations will add to the total, so that the sound of
the whole vibrating string is actually a combination of many pitches. Partials having this relationship of 1,2,3... times the fundamental frequency are called harmonic partials, or harmonics Harmonics Most commonly used to refer to the tone produced by stopping a string at one of its nodes, which isolates the higher frequencies that happen to divide at that point in the screen. Most common is to touch the string at its midpoint, which brings out the octave harmonic. , and if you listen very carefully to a vibrating guitar string you can actually hear them.

For example, if you played a low “A” you would hear the harmonics shown below.

Performers of stringed instruments sometimes isolate a particular harmonic by touching the string at one of the division points or nodes Nodes One of the points of division along
a vibrating string or column of air.
A string or column of air vibrates simultaneously in various divisions
of its length: the whole length, then in halves, in thirds, etc. Each dividing point is a node. On a vibrating string, touching a node will block all the tones that do not divide at that point, making the “harmonic” more audible. For example, touching a vibrating guitar string at its midpoint node will produce the octave harmonic. ; this dampens all the partials that don’t divide at that point and makes the harmonic easier to hear.

Since the same phenomenon occurs with columns of air, as in the trumpet, and in fact any vibrating body, the overtone series Overtone A component of a musical tone whose frequency is a multiple of its fundamental frequency. A pitched tone differs from unpitched noise in
that it has a steady frequency of vibration with partials that are approximate multiples of its fundamental frequency. These are called harmonic partials,
or overtones. Naturally produced musical tones always will be a complex in which the fundamental frequency is accompanied by overtones approximately twice that frequency, three times that frequency, and so on, which form the “overtone series.” The pitch accuracy of these overtones depends on the material producing the vibration. Overtones from an ideally thin and flexible string are mathematically exact multiples of the fundamental frequency,
but a stiffer string’s overtones will be somewhat distorted. Bells are a good example of a musical instrument with noticeably distorted overtones.
For the same reason, a large piano with strings that are thin in relation to their length has clearer, less distorted overtones than a short spinet piano with strings that are thick relative to their length. is part of all musical sounds. In some instruments, like bells, the overtones are distorted by the stiffness of the material, but most pitched instruments make a fairly accurate overtone series at least as high as we can hear it (theoretically it goes on forever). This brings us to the importance of the overtone series for harmony.

If you play two notes at the same time they will each have their own overtone series, and if the two notes happen to have just the right relationship to each other some of those overtones may match or come close to matching. For example, suppose you play A and E together. If the shared overtones almost match, an interference pattern is created that sounds like a wavering or beating. You can, however, carefully raise or lower the pitch of one of the notes until the beating stops, which will happen when the coincident overtones match exactly. At that point the two notes will seem to be in “agreement.” We could also say that they are “in tune.”

It’s difficult to hear overtones beyond the first six or so, since they become faint, but the first six are enough. The sixth is a convenient place to stop for other reasons as well: the seventh partial would create an interval that had no possibility of agreement with any of the first six, and the eighth partial is merely an octave, so the next practical partial is the ninth,
which is so quiet that it can’t have much effect on the harmony. The first six partials give us all of the musical intervals that have the potential, if adjusted carefully, of being in agreement or being “consonant.” In fact, our ear accepts these as being basically in agreement even if they are slightly out of tune;
our tolerance seems to extend to the point where the beats become too fast to be easily counted. However, not all intervals have this potential for consonance, because not all of them have matches among the most easily-heard partials. The intervals with a clear potential for agreement are the unison, octave, fifth, fourth, major and minor thirds, and major and minor sixths.
The other intervals, such as the second, have no possibility of matched partials among the first six, and so we regard them as being comparatively unrelated, unstable, or “dissonant.”

Instrumental Color

Different musical instruments give varying emphasis to each of the partials, which accounts for a large part of each instrument’s

characteristic “color.” The clarinet, for example, emphasizes
just the odd-numbered partials, whereas the oboe is rich in all
of them. The highest partials of the piano are quieter than those of the harpsichord, and they are also more distorted, since the strings of the piano are thicker in relation to their length. The superior sound of a full-size grand piano is largely due to the longer and relatively thinner strings, which allow more accurate partials and clearer tone.

Changes in partials are also an important part of instrumental color: plucked strings like those of the guitar begin with many partials and lose the higher ones as the sound dies away; a trumpet note, conversely, can begin with relatively few partials and gain more as it progresses. Extraneous noises are significant for some instruments, though we may notice them only when they are missing. These include the wind or “chiff” at the beginning of a note played by a large organ pipe, the thunk of
a piano hammer against the string, and the scraping sounds produced by a violin bow as it attempts to start a string moving. There are also various tones that come from the resonance of other parts of the instrument: the tone of a violin’s wood mixes with the tone produced by the bowed string, for example.
The result is that the sound of an acoustic instrument can be extremely complex, which is why it’s so difficult to imitate electronically.

You might think that the ideal way to tune your piano would
be to make each of the consonant intervals sound exactly in
tune, so that no beating could be heard. Indeed, many musicians have long regarded this as a sort of ideal, but it doesn’t work
out very well on a keyboard with only twelve keys to the octave.
For example, you can tune your A so that it will make a very
nice sixth above your C, and the G as a good fifth above the C, and the D to make a good fifth below A, but then you’ll find
that the G will be out of tune with the D. In other words, it doesn’t come out even; you can’t tune all the consonances to
be simultaneously exactly right unless you have two different keys for some of the notes, such as one D tuned to agree with
G and another D made to work with A.

This problem was solved in many different ways over the
years. The first solution was to tune the fifths exactly right and not to worry about the thirds and sixths, just as Pythagoras, the semi-legendary discoverer of musical mathematics, had done. This Pythagorean tuning was the basis of organ tuning and music theory until around the late 14th or early 15th century. Thirds and sixths were regarded as dissonant intervals, or at least imperfectly consonant.

But when musicians began to want to use the consonant third a compromise had to be found, and they responded by tempering

(to “temper” an interval means to adjust its tuning a little bit away from perfect) that troublesome D to the mean (average) of the two that were needed, which produced a temperament Temperament One of the various ways of tuning an instrument to compromise acoustical precision. It’s not possible to tune a piano in such a way that all its intervals are acoustically exact. “Temperaments” adjust a keyboard’s tuning to make a compromise between acoustical exactitude and the limitations of having only 12 keys per octave. “Equal temperament” is the modern standard, in which all half steps are the same size, all major thirds a little larger than acoustically exact major thirds, and all fifths very slightly narrow. in which most of the thirds were very good and most of the fifths were only slightly off; this we now call meantone Meantone A traditional way of tuning the notes
of the keyboard to make the most common thirds sound more like the acoustically exact third. temperament.

Meantone was used for several centuries and is still enjoyed by connoisseurs of early music, but it had one problem: since four of the less-common thirds were very discordant (and one fifth, the wolf fifth, was really awful), meantone didn’t allow the free use of all the intervals in all the keys. So musicians began to alter the compromise a little with the hope of making the wolf and
the bad thirds less objectionable. The Vallotti temperament is a variant of meantone in which the wolf is gone, the bad thirds are not quite so large, and the good thirds are, in trade, not quite so good as they were in normal meantone. Others invented schemes in which some chords have both perfect thirds and perfect fifths and others have varying shades of inexactness; the Kirnberger temperament is an example. In these tunings that grew out of meantone there begins to be some variety in the quality of the acceptable chords, which may have contributed to the notion that different keys have different characters – for example,
C major was thought to be pure and cheerful, while F sharp major was harsh and dark.