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Tantrum: A Tool for the Study of Temperament Anomalies

Tantrum Description

Because musical pitch classes repeat at the octave, the gamut of pitch classes may be conveniently arrayed around a circle. Where, precisely, on the circle should the pitch classes be placed in order to accurately represent their tuning? This depends on the temperament being represented. For equal temperament, the 360 degrees of a circle are divided equally into twelve 30° sections; each section represents the twelfth root of two (2 to the one-twelfth power) -- the equal-tempered semitone.

For just-tuned intervals (those in which the frequencies of the pitches are related by whole-number ratios), however, the angles are not exact multiples of 30°. A just-tuned major third (a frequency ratio of 5:4), for example, is not 120°, but about 115.8°. A just-tuned perfect fifth (a frequency ratio of 3:2) is not 210°, but about 210.58°.

Imagine a geometrical construction containing the angles represented by just-tuned intervals. Here we see three spokes radiating from a common hub. One spoke represents the root. The other spokes represent pitches at the intervals of a major third and perfect fifth from that root:


Now, let's add a second set of spokes, with the same intervals as the first, but with the root of this second set aligned with the perfect fifth of the first set:

Repeating this eleven more times, we return to the starting point -- almost. The difference between a just-tuned perfect fifth (210.58°) and an equal-tempered perfect fifth (210°) has been multiplied twelve times; we're now about 7° off. This discrepancy corresponds to the out-of-tuneness of the final perfect fifth in the Pythagorean temperament (equivalent to about 23 cents, known as the ditonic comma).

The foregoing geometrical construction is the underlying metaphor used in Tantrum, a software tool for the Macintosh, designed for studying temperament anomalies. In this program, the position of the endpoints of the spokes are shown in a magnified view, and the spokes corresponding to a root and the just-tuned intervals as measured from that root can be moved as a set, keeping the angles (intervals) constant.

In the Tantrum display, pictured below, many characteristic features of the temperament are visible at a glance.


In this temperament, for example, we can see that the major and minor triads built upon E, A, and D are all perfectly in tune. In each of the triads built upon C, F, and B-flat, the major third is perfectly in tune, but the minor third is very flat.

The tuning elements can be moved with the mouse. When the mouse is clicked on a tuning element, that element and all other instances of that pitch class are highlighted. In the example above, the mouse has been clicked on the minor 3rd of the B triad, which is the pitch class D. As a result, this tuning element, as well as the root of the D triad, the fifth of the G triad and the major 3rd of the B-flat triad, have become highlighted. If the mouse is dragged, all four instances of this pitch class will move. In this way, the entire effect of retuning D can be seen; all four triads of which that pitch class is a member will be affected.