12111098765 432 1 (or what is the frequency of middle C?)

This post is really just a collection of facts, easy to check with basic mathematics and a little bit of physics, about the frequencies of musical notes and the misguided trend of assigning importance to the fact that some note has a particular frequency rather than a different one.

We will begin with the easy part, the length of one second. Frequencies are measured in units of inverse seconds, and the universally accepted measure of a second is that of the SI measurement system. Originally a second was defined as 1/86400 of a day. This is easy to understand given that 24 x 60 x 60 = 86400, and it should also be clear that there is nothing terribly special about this number as it is the result of deciding to divide the day into 24 hours each consisting of 60 minutes each of which in turn is divided into 60 seconds. Using this definition and noting that the earth's rotation is slowing down (albeit infinitesimally) at a rate of about 1.7 milliseconds per century, we see that this is not really a precise and universal definition of a second. Given that the precision of our understanding of the world around us requires increasingly accurate measurements it was deemed necessary to find an independent and universal definition of a second, which nevertheless is as close as possible to the traditional one. The current universal definition of a second is 9,192,631,770 periods of the radiation emitted when an atom of Cs133 makes a transition between two hyperfine energy levels. This particular atom, energy levels and number of periods was chosen as it is relatively easy to measure and was the closest one could find to the traditional definition of a second to extremely high accuracy. This fixed universal definition requires that every so often a leap second must be added to our clocks so that their 24 hour cycle remains synchronised with the rotation of the earth.

 A frequency of 1Hz means that something oscillates once per second and of course musical notes correspond to oscillations in the pressure of air. As a consequence every musical note has a corresponding frequency. That sounds simple, but actually it is not really simple at all. The measure of a second is universally agreed on, the correspondence between musical notes and frequencies is not. There are two distinct sources of ambiguity in the frequency of a musical note: the reference frequency and the tuning system used to determine the frequency of all notes from the reference frequency.

Reference frequency

This is usually the frequency of the musical note A4 (or la4, in the same octave as C4 or do4) and the standard value in use today is 440Hz although various alternative definitions have been used throughout the centuries some of which are still in use, for example 432Hz .

Tuning systems

It may come as a surprise to many but the crazy fact is that there is no perfect tuning system in which all musical notes are perfectly in accord with the others. Having said this, it is also true that one can usually find the best system for a particular application.

A tuning system is a procedure for obtaining the frequencies of all notes given the reference frequency. Due to a clear consonance in wave forms the notion of an octave is unique and also very easy to hear. Two notes an octave apart are such that the higher note has a frequency precisely double that of the lower one. The main divergence, and not just of opinions but also of necessity, is the relationship between the reference tone (which can be any note one chooses) and all the other notes. Here I will outline three tunings, equal-tempered tuning, just tuning  and Pythagorean tuning - there are many more.

Just tuning. One begins with the natural harmonic series above a given base note, these are all the notes whose frequencies are multiples of the frequency of the first note in the series. This tuning is very natural for string instruments, especially fret less string instruments as the notes in a scale then correspond to placing the finger on points of the string corresponding to whole number ratios of the total string length. If the first note in the series has frequency f, then 2f is an octave, 3f is a perfect 5th above 2f, 4f is an octave above 2f, 5f is a major third above 4f, 6f is a perfect 5th above 4f, 7f is a diminished 7th above 4f and 8f is an octave above 4f (as once again it is double the frequency). As the title implies we are interested in middle C (C4 or la4) which is a minor 3rd above A3 (or la3, which has a frequency equal to half the reference frequency). The appropriate frequency ratio can be found from the above discussion by noting that the interval between 5f and 6f is a minor third leading to a ratio of 6/5. With a reference frequency of 440Hz for A4 this leads to a frequency of 6/5 x 220 = 270Hz. For a reference frequency of 432hz for A4 this would instead give a frequency of 259.2Hz. Certain schools of thought would like the frequency of middle C (C3) to be 256Hz which would require, with just tuning, that the reference frequency is 426,666....Hz as 6/5 x 213,3333.... = 256. Just tuning has problems however when transposing music due to the fact that a tuned instrument based around the harmonic series of C will not be in agreement with that based on the harmonic series of A for instance. Music that involves more than just simple harmonisation and with key changes  on an already tuned instrument will thus suffer from problems of discordance.

Pythagorean tuning. In this case, as opposed to the just tuning, one finds all notes by considering only intervals of a 5th above the fundamental note. Going through this series one finds that the interval of a minor 3rd, between middle A3 and middle C  corresponds to a frequency ratio of 32/27. Thus a middle C is 32/27 x 220 = 260.7 for A=440Hz, or 32/27 x 216 = 256 for A=432. Pythagorean tuning however also has problems as the frequency of all 12 notes is found by multiplying the previous frequency by 3/2 and following this procedure 12 times one should get back to a frequency that is a multiple of the starting note, however 3/2 to the twelfth power gives 129.74 rather than the 128 that it would give if one returned to a note precisely seven octaves above the reference tone, meaning that all notes cannot be consistently tuned using this method giving rise to problems similar to those encountered above in just tuning.

Equal-tempered tuning. This tuning was probably invented simultaneously in China and Europe in the 1500's. It has the advantage that all intervals of a semitone are identical and equal to the twelfth root of 2. It has the disadvantage that all intervals, apart from octaves, are slightly out of tune compared to the more natural harmonic tuning. In this case the frequency ratio of a minor 3rd corresponds to the 1/4th root of 2 = 1.1892. For A4=440Hz this gives C4=261.63Hz while for A4=432 we obtain C4=256.86.

The purpose of all of this discussion, which has only touched the surface of questions of tuning and musical notes, is that assigning a great importance to the fact that a note has a precise numerical frequency rather than a different one is misguided. As one can easily see from the above discussion one can always find a reference frequency and tuning system that will give you some particular numerical frequency for some given note. Music is beautiful for the sensations that it creates, of harmony or dissonance or melody, not because notes correspond to particular numbers....