The theory of temperament

Some top-down facts

Temperament theory is based on the notion that musical intervals and consonance, are themselves based on whole number ratios. Temperament theory is a theory of whole number ratio arithmetic. It is about how whole number ratios "fit together", or as the case may be, do not "fit together". From this is derived a theoretical knowledge of how musical intervals, and musical scales, "fit together". 

The basic assumptions of temperament theory are approximately correct. Therefore they are useful, up to a point. However, temperament theory can become very complicated. Its complications beyond a certain point, would be swamped by other complications in the actual, acoustical behaviour of instruments and voices, and in the nature of psychoacoustics, that the theory ignores. Therefore, its usefulness in what it is saying, at any time, is limited.    

Temperament theory is an idealised  theory, but being approximately correct, up to a point, it is a useful model on which general practical tuning strategies can be based. 

Tempering is a process in which musical intervals are tuned so that they do not produce an optimum acoustical consonance. (Acoustical consonance is determined by objective, acoustical criteria, rather than subjective musical judgment alone). 

An interval that is tuned for optimum acoustical consonance is usually said to be Justly Intoned. Temperament is necessary because it is impossible to create a Western diatonic or chromatic scale in which all the intervals are Justly Intoned. The deviation from Just intonation is precisely controlled by tempering.

Temperament theory, as an idealised theory, is useful as a framework, but it is not a comprehensive acoustical theory of tuning. It is a basis for piano tuning theory, but it is not tuning theory itself.

A very great deal has said and written about music and temperaments, and about the mathematics of temperaments. Much of this ignores the fact that temperament theory and its mathematics, is in the first place, an idealised theory, and is therefore often far from being an accurate description of how musical instruments actually behave. What is said also often ignores how music happens as a process in its own right, and how it is affected by performance practice and other influences.

The intensive study of temperaments often leads to misleading conclusions:

Temperament theory is connected to the old science of "harmonics", and this originates from the Pythagorean tradition. The situation with the "scientific" study of temperaments today, is often somewhat similar to the intensive study of the motions of the planets that took place in the past according to geocentric Ptolemaic astronomy. It is largely an idealised exercise in attempting to make factual sense of an idealised theory.

Temperament theory is worth studying, especially in an historical context. But it is not a proper, modern, scientific, acoustical theory, even though it is often mistaken for one.

Theorists of the past, from the 15th century right up to the 18th century, can be forgiven for treating temperament theory as though it were a perfect description of the basis of tuning and intonation in music. For them, it was. And for many of them, the ratios it employs were as Divinely ordained as the circles and harmonic ratios assumed to present in the Earth-centered Solar system.

Temperament theory is an early simplification of something that is now much more accurately understood with modern acoustics. As an idealised, simplified description, it has its uses as a rough scientific tool. As a "scientific" theory it has now been superceded.