Equal Temperament is not a myth

The principle of Equal Temperament is that it 'divides' the octave into 12 equally sized semitones. Mathematically, the 'size' of the equally tempered semitone is normally expressed as the twelfth root of 2 :

where this is the frequency ratio of two frequencies separated by an equally tempered semitone. A set of 12 semitones steps of this size will create an octave with a frequency ratio of 2:1.

There is of course nothing 'mythical' about the twelfth root of 2. The alleged myth appears when we claim to actually tune an instrument according to this principle, or play or sing using an 'equally tempered scale' defined in this way.

The above mathematical definition is in itself an idealised model, because it assumes that a musical interval can be entirely defined by a single frequency ratio.

The tone of a piano string does not consist of a single frequency. It consists of a large spectrum of partial frequencies. So a musical interval on the piano cannot be fully described by a single frequency ratio. It cannot therefore be entirely defined by a single frequency ratio, except in an arbitrary way, by say, conventionally referring to only one particular partial frequency in each tone's spectrum, for example the fundamental. The use of such an arbitrary convention cannot be objectively justified, unless of course the relationship of partial frequencies in a spectrum, is the same for every spectrum. This is not the case for piano strings in situ. Not only is this due to inharmonicity, but also because of the effect of the bridge-soundboard system on string behaviour.

Nevertheless, all this does not mean that the idea of Equal Temperament becomes a myth on the piano. What it does mean is that the idealised mathematical definition of Equal Temperament, does not suffice to describe what it is.

The principle of 'dividing' the octave into 12 equal semitones still applies, but not as a quantified, mathematical definition. What should matter in defining Equal Temperament in the actual acoustical situation, on the actual piano, is not frequency ratios, but the musical reasons for having Equal Temperament in the first place.

The whole point of dividing the octave equally into 12 semitones, is:

1. To have every scale in every key exhibiting the same set of interval sizes, as they are perceived musically.

2. To have, within any octave, the same degree of audible tempering in every interval of one kind.

Good artist tuners achieve both the above as a matter of course.

It is the notion that artist piano tuners cannot achieve Equal Temperament, that is really the myth.

It is true that a slight imperfection tips the balance and turns what was supposed to be Equal Temperament into 'pseudo-equal temperament', a 'random' unequal temperament with no real musical reason for being. But on a good instrument an artist tuner would not let such imperfections pass.

Alexander Ellis may have concluded in the late nineteenth century that tuners failed to tune Equal Temperament within desirable limits of error, but this hardly applies to the modern artist tuner tuning on a top quality modern instrument.

Equal Temperament is no myth. It definitely does exists on pianos tuned by tuners who are sufficiently skilled. The 'real thing' (which is quite distinct from pseudo-equal temperament) is the most difficult to tune of all the temperaments. It is like balancing tempering principles perfectly on a knife edge. But good tuners can do it. Those who can tune precision Equal Temperament really understand what it is and what it means, whilst theoreticians have their own criteria that in reality, have very little to do with it.