Background to the musical scale, continued . . .

This aspect of natural phenomena can be concisely described through the concept of frequency. 'Frequency' applies to vibrations and waves, and in acoustics refers to the number of vibrations that take place in one second, or the number of waves that pass in one second. If we are dealing with sound waves or with the vibration of something that is producing sound waves, then we can speak of frequency.

Suppose a first sound source (say, a tuning fork) vibrates with 100 vibrations per second. We can say it has a frequency of 100 cycles per second  or 100 cps or 100 Hertz or 100 Hz.

If we were to listen to a second, similar source now vibrating at 200 Hz, we would usually hear it as a note an octave above the first source. The frequency ratio of the second note to the first note is 2:1. Thus it is said that the frequency ratio of the octave is 2:1    

If we hear a note whose frequency is f, and call it say, the note C, then another note whose frequency is 2f will sound another C an octave above. Another note with a frequency f / 2 (half f), will sound an octave below.

There is thus a correlation between perceived musical pitch and frequency ratio. This applies to other musical intervals also. The ratios for some of the principle musical intervals in Western music are as follows:

Octave - 2:1

Perfect fifth - 3:2

Perfect fourth - 4:3

Major third - 5:4

The ratios for all the musical intervals are whole number ratios (integer ratios).

The association of integer ratios with musical intervals is supposed to have begun in the West with Pythagoras (c. 582 - 497 BC), but not because Pythagoras knew anything about frequency. The ratios he is said to have discovered are those pertaining to the lengths of tensioned, musical strings.

If the uniform string of a monochord is stopped half way along its speaking length, each half of the string (given bridges the same height at each end) will produce a note an octave above that of the unstopped string. Also, if its speaking length is divided in the ratio 2:1, by stopping it 1/3 along the length, the note produced by the shortest section (1/3 of the unstopped speaking length) will sound an octave above that produced by the longest section (2/3 of the unstopped speaking length). Thus, as portions of a string's speaking length, the ratio 2:1 is still associated with the octave. The other ratios cited above, will also be confirmed by string length ratios. Obviously, there is a correlation between frequency and string length. 

As it turns out, the same ratios apply to the lengths of organ pipes. However, all these 'Divine ratios' are not quite what they seem . . .