The Physics of Music

injx · Posted: Fri Dec 08, 2006 10:24 am

The Physics of Music

If one was to ask whether music was an art or a science, the answer given would probably be 'art'. However, it turns out that there is a great deal of science beneath the pleasant sounds of a piece of music that the average listener may be unaware of. Before we can look at the physics of music, we must first define what music actually is, and what differentiates it from being just sound. So what is music? A basic definition would be that music is organised sound, however some more extreme genres of music today comprise of anything but organised sounds. The dictionary defines music as "the art of arranging sounds in time so as to produce a continuous, unified, and evocative composition, through melody, harmony, rhythm, and timbre," which may seem vague, but we could spend all day debating a definition. Either way, music could not have developed into this modern definition, and today's diverse array of instruments and genres could not exist, without a strong understanding of the physics of music.
This article will take you on a brief journey through the history of the physics of music, the problems, solutions and tools involved in its development.

The Legend

The story begins in ancient Greece, around 540BC, with Pythagoras. This infamous mathematician is well known by all for Pythagoras' Theorem, but he also made a significant contribution to music – essentially kick-starting the physics of music. There are various versions of this legend – it may have included bars of metal, or blacksmith's hammers, or something completely different – but for simplicity I shall use 'bars of metal'. The gist is that Pythagoras noticed how when two similar bars of metal are taken - the second being half the length of the first – and struck to produce a sound, the two bars produce the same note, with the short bar's note being an octave higher than the long bar. He also noticed that the higher pitch produced by the short bar is also audible in the long bar's note, although much quieter. Pythagoras noted that this ratio of bar lengths sounded pleasant together and tried to identify other pleasant ratios. To do this, Pythagoras tried simple ratios of the length of his bar of metal. He found that two of these natural pitches were produced by bars 2/3 and 3/4 the length of the large bar. These sounds were therefore 3/2 and 4/3 times the pitch of the original bar's note respectively (what today we call a perfect fifth and a perfect fourth). The difference between this perfect fourth and perfect fifth is a ratio of 9/8, which is the interval of a whole tone. By combining these simple ratios, Pythagoras discovered a natural scale of seven notes.

Today this natural scale is referred to as 'Pythagorean tuning' and is an example of a Just scale (one whereby the notes are of rational ratios), and it is conventional to number the notes of the scale using roman numerals. The initial, base note of the scale (the note from which the scale is derived from) is called the 'tonic'.

The Problem

However, the story doesn't end there. If you take the key of C, for example, we label the tonic note as 'C', and the rest of the scale as D, E, F, G, A, B, and C. No problems there. However, each note of the scale then has its own scale associated with it (by using each note as a tonic), for example the scale of F or the scale of G. When calculating the notes in these new scales, you find that many of their notes do not exactly match any of the notes present in the scale of C – their pitches being slightly different to the equivalently-pitched notes in the scale of C. What results is a modern musician's nightmare. Using this Just scale, you would only be able to play in one key at a time on an instrument, and would have to either re-tune the instrument or re-write the music in order to play in a different key. As you can imagine, this is impractical and an imperfect solution – even though the actual notes of the scale are in perfect harmony.

The Solution

Unfortunately there is no mathematical trickery that can be used to make this problem go away, and we are forced to solve this problem by way of compromise. Yes, the solution is to sacrifice the perfect harmonies of Pythagoras' natural scale in order to be able to play all scales on one instrument.
If you calculate the notes of every scale, you find that, out of all the notes that don't exactly line up, many of them are only off by a small percentage, and can be 'nudged' up or down to the nearest pre-defined note without causing too much grief. For example, when looking at the scale of D, the notes II and V can be nudged up and down respectively to gain the titles of E and A. This solves half of the problem; however, the remaining 'off' notes cannot be solved so easily. Looking at the scale of D again, there are two notes that don't yet have a name and are too far off from the pre-defined notes to be able to nudge them. So, we therefore invent new note names to accommodate these pitches. In the case of the scale of D, we gain the notes of F# and C# (F# being in-between F and G, and similarly C# being in-between C and D). By following this method of nudging and inventing, we can accommodate the notes of all the scales by introducing only five of these new 'sharp' notes (the five black keys on a piano), and nudging the rest.
In fact, once we have this 12-note solution, we can refine it further. By looking at the multipliers from each note to the following note, it becomes evident that the multipliers are all roughly the same – about 1.06. So, we can refine the solution by making all the multipliers equal – we know the octave must be twice the pitch of the tonic, and that we have 12 unique notes within the octave, so the fixed multiplier, by definition, becomes

, which is approximately 1.06. And so, the Chromatic scale of 12 equally-tempered notes is born.

Tuning the octave in this way is called Equal Temperament, and is the system used by almost all western instruments and musicians. It allows all keys and scales to be played at the same time, while the small sacrifices made are not audible to the ear.

The Vibrations

The reason why Pythagoras' bars actually made sounds at pitches relative to their length is due to the way they vibrate. The loudest pitch given off from a string or long bar is the fundamental frequency, which is the simplest mode of vibration, although as Pythagoras noticed, there are other quieter octaves also present that we call overtones. Acoustic resonance is the underlying principle which determines the fundamental and overtone frequencies produced, based on the object's properties. Overtones are generally at frequencies that are integer multiples of the fundamental, and are referred to as 'harmonics'. However in real instruments, there are sometimes factors (such as the thickness of a piano string, for example) that cause the overtones to be not-exactly-integer multiples of the fundamental, which we call inharmonics. It is this combination of harmonics and inharmonics that give each instrument's sound its own distinctive texture, or timbre.

Vibrations on a string. The fundamental and first two overtones.

Vibrations in a closed pipe. The fundamental and first two overtones.

As you can see in the two figures above, the string is fixed at both ends, and so there must be a node (zero displacement) at each end. Similarly for the closed pipe, there must be a node at the closed end and an anti-node (maximal displacement) at the open end.

The Superposition

As we know, music does not usually comprise of just one pure frequency at a time – indeed even if only one instrument is playing, its sound most likely comprises of several frequencies combined, as described above.

This is called superposition and can be illustrated graphically:

This graph, despite looking complicated, is simply the sum of three pure cosine waves with the differences being the frequencies corresponding to notes I, III and V of the scale: together they form a Major chord.

However, remember that the above is made of pure notes; in real acoustic instruments each note would have its own harmonics and inharmonics to add to the mix.

The Uses

The physics of music has a wide range of important uses, not only today but throughout history. It is essential to the design and construction of instruments, not only to obvious components such as the length of piano strings, but to subtler details such as the shape of the sound-hole and soundboard on a guitar or violin.

One interesting area involving the physics of music is in electronic music. Here, the music is not merely recorded and then processed, as with traditional genres, but each instrument's sound is actually created by the artist electronically. This is usually done by combining some basic-shaped waveforms such as sine, square and sawtooth using a synthesiser, in order to achieve an appropriate sound for the part. For example, sine and square waves may be described as 'clean' sounds, whereas a sawtooth wave may be said to be a 'dirty' sound. It is also possible to emulate acoustic instruments using synthesis by choosing the right combination of waveforms. Designing one's own sounds can add an extra dimension to the creativity of music, and is often used by artists in combination with acoustic instruments.

Conclusion

Music is a ubiquitous part of society and culture, and is undoubtedly a creative art. However music could also be described as 'mathematics using sound'. So is music a science or an art? The answer is of course 'both'.

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