Scales: Music and Measurement

Across the centuries, the development of tuning methods for musical instruments has been driven by the desire of musicians to expand the variety of harmonies and increase the types of easily playable music. There are intriguing parallels with the development of social science measurement. For instance, mathematical perfections were claimed for "Pythagorean" tunings, as they are now for some IRT models. On the other hand, special practical virtues were perceived in "just meantone" tunings, as they are now in raw- score-weighting schemes.

Most music we hear today is played with equal-temperament tuning. This became the de facto standard in 1854. In this tuning, the relationships between the notes sound the same in every musical key (either major or minor), wherever they are played on the piano. In each major (or minor) key the notes bear the same mathematical and acoustic relationship to each other as they do in every other key. This is a measurement ideal. One more note is the same amount extra everywhere on the piano. It is also a musical ideal. Playing music in another key does not require retuning. A transposed piece of music has the same musical feel, no matter what key it is played in. But there are costs. This type of tuning requires advanced technology and expertise. Also, the distinctive sounds of particular major and minor keys in earlier tunings, beloved of particular composers and performers, have been lost.

Prior to equal-temperament tuning, the pianist was required to make decisions. Is the note next above C to be C-sharp or D-flat? It could not be both simultaneously, as with equal temperament tuning. Once the decision was made, the piano was necessarily harmonically optimized for certain keys, and sub-optimal for others. J.S. Bach's "48 Preludes and Fugues for the Well Tempered Clavier" was written for this type of tuning. He demonstrates how the use of notes, tuned for one key, to play music written in a different key, produces a change of sound texture, and so a change of the emotional impact of the music on the hearer. This is type of change does not occur with modern equal-temperament tuning.

"Well temperament" tuning was prevalent from 1690 onwards. Its goal was to maximize the number of harmonies while simultaneously eliminating gross disharmonies from the scale of choice. This is equivalent to maximizing the reliability of a test for a particular sample.

Prior to "well temperament" tuning were the Pythagorean tuning methods. These followed an ideal of mathematical and musical perfection that worked nicely for centuries. But there were drawbacks. A change of musical key required retuning of the instruments. Tuning was done according to the performer's preference, so there was only a rough standardization of sound. Each musician claimed that his personal tuning method was a superior realization of the Pythagorean ideal of integer ratios between note frequencies. But Pythagorean tuning was simple, and musically effective. Its limitation was that only 11 of the 12 notes of an octave could be in tune simultaneously. Yet it was so easy and familiar, just as raw scores are today, that it was still widely used as late as 1800 A.D.

The development of tuning methods continues as musicians seek to enchant their audiences with ever more exotic harmonies and discords. Unfortunately, in a remarkable parallel to the current proliferation of psychometric models, "the history of tuning is saturated with clever and original theories that have no practical application." (Jorgensen, p.33) Jorgensen, O. (1977) Tuning the Historical Temperaments by Ear. Marquette: Northern Michigan University Press.

Scales: Music and Measurement. 2001. Linacre J.M. … 15:3 p. 838


Scales: Music and Measurement Linacre J.M. … Rasch Measurement Transactions, 2001, 15:3 p. 838



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