# 6 Frequency and note identity

Western musicians often assign specific frequencies to the notes of the gamut. In the most widely used system for allocating frequencies, the A below middle C vibrates at 220 hertz, or 220 vibrations per second. Table 1 lays out the frequencies conventionally associated with the twelve notes above that note (called A220 in the table and located in the lowest row).

## Table 1 Frequencies for A below middle C and the twelve notes above it (after Suits, 1998)

Note name | Frequency in hertz | Distance from A220 in hertz | Distance from the note below in hertz | Distance from A220 in cents | Distance from the note below in cents
--- | --- | --- | --- | --- | ---
A440 | 440 | 220 | 25 | 1200 | 100
G♯/A♭ | 415 | 195 | 23 | 1100 | 100
G | 392 | 172 | 22 | 1000 | 100
F♯/G♭ | 370 | 150 | 21 | 900 | 100
F | 349 | 129 | 19 | 800 | 100
E | 330 | 110 | 19 | 700 | 100
D♯/E♭ | 311 | 91 | 17 | 600 | 100
D | 294 | 74 | 17 | 500 | 100
C♯/D♭ | 277 | 57 | 15 | 400 | 100
C | 262 | 42 | 15 | 300 | 100
B | 247 | 27 | 14 | 200 | 100
A♯/B♭ | 233 | 13 | 13 | 100 | 100
A220 | 220 | 0 | — | 0 | —

Two important points emerge from this table. The first involves the octave relationship. Notice that the frequency (the number of vibrations per second) of A440 is exactly double the frequency of A220. This doubling is evident in all octaves. The A below A220 has a frequency of 110 hertz and the A above A440 has a frequency of 880 hertz. Similarly, the G below G392 has a frequency of 196 hertz and the G above it has a frequency of 784 hertz. (A110, A880, G196, and G784 are not shown in the table.)

This helps explain why notes in an octave relationship sound like the same note: their respective rates of vibration exist in simple ratios such as 2:1.

A second point, related to the exponential octave relationship (110, 220, 440, 880, 1760, etc.), is that some semitones are further apart than others, at least in terms of vibrations per second. As the fourth column in Table 1 shows, the number of vibrations per second increases as notes get higher and higher. A220 and A#233 are 13 hertz apart, but G#415 and A440 are 25 hertz apart. At the same time, to the ear, these intervals sound like they are of the same size.

Because the figures involved when discussing frequency can jar with the musical experience of hearing intervals, intervals are often discussed in terms of ‘cents’ instead. The cent is a unit of perceived intervallic distance, first proposed in an article by Alexander J. Ellis (1885). In Ellis’s measurement system, an octave is made up of 1200 equal cents, such that each Western semitone comprises 100 cents and each Western tone comprises 200 cents. This measurement system accords a little better with musical perceptions of intervals, in which tones sound like tones and octaves sound like octaves, no matter how many vibrations per second are involved.

The fifth and sixth columns of Table 1 give the breakdown of the twelve notes/intervals above A220 in terms of cents. The remainder of this course’s discussions will employ Ellis’s 1200-part measuring system.

## Intervals and ratios

Intervals have been described here in terms of hertz and cents, but in many music cultures, intervals are understood in terms of ratios. For example, the octave is widely associated with the ratio 2:1. This is consistent with what you learned in the discussion of frequency: A440 vibrates twice as fast as the note an octave below it, A220. Similarly, a plucked string sounds an octave higher when you shorten its length by half. Meanwhile, the Western tone or whole step has sometimes been associated with the ratio 9:8 because a string sounds a tone higher when you shorten its length by a ninth.

OpenLearn - Introduction to music theory 2: pitch and notation
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