Thommy, the harmonica player for the Clam Daddys,  and I were discussing Equal Temperament and came to the conclusion that equal temperament doesn't exist.

The formal theory of musical scales in Europe started at lyre tunings. Lyres were tuned in something like this order: C c g G d D a A etc., that is, starting low and tuning in fifths (and octaves).

It was discovered by the time of Pythagoras that if you leap  twelve times tuning by fifths (transposing back into the original octaves) and come back to C you're off by almost a Half Tone. That's because the result of twelve iterative divisions of a vibrating length by 2/3 (Perfect Vth) doesn't equate to an integer multiple of 1/2 vibrating lengths (Perfect Octave).

 The gap is called the Pythagorean Comma and was accounted for in various ways by various Just Intonation tuning systems which were tried over the centuries, including, for example, the (particular) Well-Tempered system used by Bach for the Clavier.

By the 19th century math, musical instrument construction and the symphony had matured enough that a perfectly even Equal Temperament emerged in which the octave is divided into 12 Half Steps, each of whose frequency is  2^(1/12) times the frequency of the previous Half step. Equal Temperament was adopted around the same era by both European and Arabic classical music, with the difference that the Arabic system recognizes 24 evenly spaced quarter steps.

But are any instruments ever tuned to equal temperament?

 The answer is "No". Just Intonation is the system where intervals are tied to ratios, e.g, 3/2, the ration between the frequencies of a Fifth and its Fundamental.  Such intervals can be found on a stringed instrument by touching lightly at appropriate nodes, growing weaker in force as we  is ascend the Harmonic Series. It's easy to tune a string instrument, notably the Dobro slide guitar, to a just intonation system of choice because it reflects the natural physical properties of a vibrating column

Equal Temperament, which offers perfect key modulation, does so by"cheating" and assigning the intervals to irrational number values. Digital tuners exhibit (tiny) rounding errors. Non-digital tuning is done by approximation.

 Benoit Mandelbrot in his book The Fractal Geometry of Nature pointed out that squares and circles don't really exist in nature. An analogous critique seems to apply to the mathematically ideal Equal Temperament system.