Musical Offerings (Oct 2018)
The Battle Between Impeccable Intonation and Maximized Modulation
Abstract
Equal temperament represents a way of completing the musical circle, and systematically compensating for the Pythagorean comma. Pythagoras discovered this acoustical problem around 550 B.C., and since that time music theorists have debated how to deal with it. The problem is that no perfect solution exists—something must be compromised. As musical styles developed, specific factors and harmonic tendencies led to the gradual adoption of equal temperament. Early in music history, theorists preferred systems which kept acoustical purity relatively intact. Pythagorean intonation and just intonation serve as two examples. However, the move from modality to tonality decentralized the melody as the dominating feature of a composition. Correspondingly, this raised the importance of harmonic structure, and introduced the idea of modulation. Not all tuning systems allow a performer to easily change keys; most systems contain some type of wolf fifth. This interval sounds exceedingly dissonant, due to its distance from an ideal frequency ratio. Thus, composers had to avoid certain keys, like F# Major, or Bb Minor. During this time, hundreds of different meantone temperaments arose, all of which deal with the Pythagorean comma in slightly different ways. These temperaments attempt to balance pure acoustics and modulatory freedom. Eventually, as chromaticism became increasingly common, so did equal temperament. Musicians traded true intonation for the ability to play in any key at any time. While equal temperament is now universally hailed as the standard tuning system, it is not perfect. Rather, it represents a compromise designed to best accommodate the needs of tonal music since the Baroque Era. I will mathematically show the problems encountered when creating a tuning system, and discuss the various known solutions. I will then use historical documentation to show how musicians eventually landed on equal temperament as the most complete solution.
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