Approaches to tuning and temperament
Authors: Rasch Rudolf A. 
Published: October 2005

The tuning of musical instruments has kept music theorists and others busy since the times of Antiquity. Although the problems have largely remained the same, the approaches to it have differed through the ages. Oldest is the numerical approach, first proposed by Pythagoras. Antiquity and Middle Ages worked with string-length ratios solely including factors of 2 and 3, giving rise to Pythagorean systems. Since the late 15th century factors of 5 were included, leading to a large body of just intonations. Higher prime numbers, like 7, 11, and 13, were incorporated in tuning systems during the 20th century. Because of the practical defects of just intonation, a new class of tunings was introduced during the 16th century: they were temperaments, like meantone tuning and equal temperament. The description of the temperaments became only fully possible with the help of logarithmic interval measures, which were developed from the 17th century on. Some historical authors have given only practical instructions without much or even any theory. On the other hand one can find highly complex numerological speculations as a source of inspiration for ideas about tuning. In modern times, approaches to tuning and temperament should not be confined to mere paper-and-pencil computational work. Instead, the results of both physical (frequency) measurements and psychophysical experimentation should be connected to the theoretical component and incorporated with this in a broader view. We developed a descriptive model of musical tunings, in which a tuning is seen as a set of interrelated interval sizes [R. A. Rasch, J. Acoust. Soc. Am. 73, 1023–1075 (1983)]. The most important properties of the interval sizes are their temperings, the deviations from the “just” value. Many tunings comprise different sizes for a particular kind of interval, which leads to the development and use of statistical measures such as means and standard deviations. In fact, the statistical capabilities constitute the strength of the model. In the practice of tuning many sources of variability appear: inaccuracies caused by the limitations of the tuner, imperfections of the instruments, changes after the tuning, etc. The situation becomes even more complex if intonations during performance by other than keyboard instruments are involved. Due to the statistical features of the model all kinds of tuning data can be compared to each other. Examples will be given from several, widely differing tuning-and-temperament phenomena, like beat frequencies and strengths, inharmonicity effects in piano tuning, the internal tuning of wind instruments, the practical intonation of stringed instruments, applied organ tunings, descriptions of just intonations, and theories of multiple octave divisions. Modern psychoacoustical experimental methods make possible other extensions of the proposed model. Examples will be given how perceptual properties of tempered intervals, like beats and perceived mistuning, relate to properties of intervals as described in the model. It will be clear that a complete evaluation of a tuning system is only possible by considering the physical and psychophysical consequences of the system as well.