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# Maths & Music

Subject: Maths & Music   Sat Sep 25, 2010 9:33 am

MATHS & MUSIC

One of the earliest known applications of mathematics in music is attributed to Pythagoras, a Greek mathematician best known for his theorem concerning right-angled triangles.

According to the myth, Pythagoras was walking past a blacksmiths, listening to the sound of the hammers on the anvils. After a while, he realised that all but one of the hammers were sounding in harmony. Curious as to the reason, Pythagoras made a thorough examination of the hammers and discovered that when their masses were simple ratios i.e. 2:1 or 4:1, then the respective notes produced were in harmony. On the other hand, the mass of the hammer producing the discordant note wasn't in a simple ratio with any of the other hammers.

Although the absolute truth of this myth is questionable, it illustrates the first real application of maths in music.

Frequency

The harmonies Pythagoras was talking about were undoubtedly octaves, the same note repeated at a higher pitch.

Nowadays, this pattern can be observed in the frequencies of the various musical notes. The note musicians call Middle C has a frequency of 262Hz. The C an octave above Middle C has a frequency of 524Hz, a ratio of 2:1. In fact, all C's are successively double or half the frequency of Middle C.

The pattern continues for any octave interval no matter what interval is chosen, e.g. the ratio of the frequencies of two G's an octave apart is 2:1. This leads to a convenient mathematical way of writing the frequency of any musical note using the table below:

Frequencies of some musical notes

C 262
C#/Db 277
D 294
D#/Eb 311
E 330
F 349
F#/Gb 370
G 392
G#/Ab 415
A 440
A#/Bb 466
B 495

All C's for example can be written as 262x2n where n is an integer.

Instrument Design

Mathematics also plays an important role in the contruction of musical instruments. On a flute or recorder for example, the holes must be placed in exactly the right position in order to produce the correct notes at an accurate pitch. Similarly, the frets on a guitar must be accurately positioned so that all the notes remain in tune wherever they are played along the neck of the guitar.

For this reason, the shapes and designs of most musical instruments has changed very little over the last few centuries, and they are now made to highly accurate specifications. The tone of an instrument also depends on its dimensions. A large guitar with a deep body has a fuller sound than a smaller, shallower instrument.

The materials with which it is made will also effect an instrument's tone because each material has its own resonant (natural) frequency which could cause the body to reinforce important harmonics or buzz annoyingly.

Superposition

Instruments are designed so as to reinforce certain harmonics vital to that instrument's tone. For this reason, a violin sounds nothing like a piano and it is relatively easy to identify instruments contributing to an orchestral piece even if they are playing the same note. This is because the sound waves produced by all instruments are very different.

Sound waves are created by adding together the harmonic waves produced by the instrument which are unique to each individual instrument. This principle of adding together waves is known as superposition.

Harmony

Superposition provides us with an explanation regarding the phenomenon of harmony. For some reason, certain notes when played together create a pleasant sound, whilst other combinations just don't sound right. (Note: This is different to the Pythagorean 'Octave' harmonies described earlier.) The note pair C and G is an example of a good harmony, whereas E and F# don't provide the same effect.

To produce the sound wave of the harmony, all that is required is to add the waves of the individual notes. When scientists analyse the two harmony waves, that of the C-G pair has a more regular repeated pattern than that of the E-F# pair, and this appears to be the reason that the first pair produces a more desirable harmony than the second. The simpler the ratio between the wavelengths (and thus the frequencies) of the individual notes, the more regular and repeated the combined wave is, and thus again, the important factor appears to be the ratio between the frequencies.

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