Publication Date


Document Type



Equal Temperament, Mean Tone Temperament, Pythagorean Intonation, Just Intonation, Harmonics, Music Theory


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.





Article Number




Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.


DigitalCommons@Cedarville provides a publication platform for fully open access journals, which means that all articles are available on the Internet to all users immediately upon publication. However, the opinions and sentiments expressed by the authors of articles published in our journals do not necessarily indicate the endorsement or reflect the views of DigitalCommons@Cedarville, the Centennial Library, or Cedarville University and its employees. The authors are solely responsible for the content of their work. Please address questions to dc@cedarville.edu.



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.