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:
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.
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.