Pitch and Frequency

Frequency is a precise scientific notion, the rate of vibration, which corresponds to what is commonly referred to in music as pitch, although as we shall see below, the two are not synonymous. Frequency is most often measured by the number of vibrations per second. If something vibrates 500 times a second, it is said to have a frequency of 500 hertz (abbreviated 500Hz).

Most commonly, frequency has to do with the vibration of a string or of air in a tube. In both cases there are factors which contribute to the speed at which the vibrations travel (the density of the air or tension in and thickness of the string) and in both cases there is the factor of length which, along with speed, determines the frequency.

Because the factors contributing to speed are most often fixed, it is common to focus somewhat on the relationship between length and frequency. To understand this better consider the violin which, while tuned through adjustment to the tension of strings, is played by adjustment of the length of string allowed to vibrate. This is of course, a simplification and all factors come in to play during performance, but, all other things being equal, frequency is determined by length.

The relationship between length and frequency is one of inverse propoportion. If length is halved, frequency is doubled. Our perceptions of frequency are based on the ratio between frequencies rather than the arithmetic difference. For example, we perceive the relationship between notes of 440Hz and 880Hz as being the same as that between notes of 880Hz and 1760Hz. They both have a frequency ratio of 1:2.

This is an important point which highlights the difference between pitch and frequency. The difference between two pitches corresponds to a ratio in their corresponding frequencies. In mathematical terms, pitch therefore has a logarithmic relationship with frequency. Here and elsewhere I use the term pitch to mean mathematical pitch rather than perceptual pitch.

Pitch is not the same as the letter name of a note. An A of 440Hz has a different pitch to an A of 412Hz even though they share the same letter name and octave. Likewise, the difference between two pitches (corresponding to a frequency ratio) is not the same as an interval like that of a fifth. The relationship between letter name and pitch/frequency and the relationship between interval and pitch difference/frequency ratio depend on a choice of tuning.

Pitch differences are often measured in cents where a pitch difference of 1200 cents corresponds to a frequency ratio of 2:1. A pitch difference of p cents corresponds to a frequency ratio of f = 2 ^p/1200.

Although it is pitch differences that are normally meant when talking about cents, it is possible to speak of actual pitches (corresponding to a given frequency) in terms of cents. For example, 440Hz can be expressed as approximately 10537.632 cents. In these discussions I will refer to such a pitch as P₄₄₀ and more generally for a given frequency of f hertz, P_f = 1200 log₂ f