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Understanding Temperaments

By Pierre Lewis

Note: The introduction and table of contents for this informative article are displayed on this page, followed by links to the full text of the article, as well as other informative articles on this subject.

The purpose of this short text is to give the reader a basic understanding of the various temperaments and tunings used on keyboard instruments (harpsichord, organ) in the past. It will not give detailed tuning instructions (my next project) nor much more than general indications on the suitability of the various historical temperaments in different contexts. When discussing temperaments, one cannot avoid being a bit technical. However, I have also tried to be practical by discussing temperaments that can be useful to modern keyboardists, and by stressing their important acoustical properties (i.e. how they sound) rather then getting into some complex theories. No special skills in mathematics are required of the reader (footnotes will be used to convey some extra material).

This text is complemented by a Java applet that demonstrates tunings and temperaments (Iím also thinking of a virtual instrument one could actually practice tuning on), and by some short musical examples in various tunings and temperaments (MIDI and some .au files).

Anyone seriously interested in temperaments must read Margo Schulterís remarkable Pythagorean Tuning and Medieval Polyphony. Notwithstanding the title, her discussion of Baroque meantones and irregular temperaments is also very thorough. In addition to the physics of the problem, she also addresses the musical and musicological implications, with many references to the sources of the time. Copyright 1978, 1998 (yes, 20 years!) by Pierre Lewis, Version 1.2 (incorporates a few changes inspired by Margo Schulterís article).

Table of Contents

1.0 Introduction
1.1 The Problem
1.2 Cents
1.3 Solutions
1.4 Circle of Fifths

2.0 Equal Temperament

3.0 Pythagore's Tuning

4.0 Just Intonation

5.0 Regular Temperaments
5.1 Aaron's Meantone
5.2 Silbermann and Others

6.0 Irregular Temperaments
6.1 Kimberger
6.2 Valloti
6.3 Werckmeister
6.4 Summary

7.0 Other Resources

Full text of this article in pdf form
Full online version of this article at Pierre Lewis website
"Pythagorean Tuning and Medieval Polyphony" by Margo Schulter
"The Keyboard Temperament of J S Bach" by John Charles Francis

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