Triad triangles

Since ancient times, the monochord has been used as a medium for conveying music theory knowledge. A string with a fixed fundamental pitch is stretched over a resonance box. If the string tension remains unchanged and the string is divided into two parts with a bridge, the dependence of the pitches on the lengths of the string parts can be investigated.

According to tradition, the discovery that simple length ratios such as 1:2, 2:3, 3:4 also correspond to elementary musical intervals must have inspired the Pythagoreans to far-reaching interpretations. Some commentators see in this discovery the birth of science par excellence.

So, while the theoretical and practical study of musical intervals can look back on a venerable history spanning thousands of years, major and minor triads are fairly recent items that have only been discussed for 400 years, at least in theory. In 1612, the theologian and music theorist Johannes Lippius coined the term trias harmonica. He argued that a series of harmonics, which we now call triadic inversions, should all be regarded as appearances of that abstract object. Lippius saw the Trias Harmonica as nothing less than a musical expression of the Holy Trinity.

The triads developed by Bernhard Ganter combine in an original way the concept of triads as independent music-theoretical objects with the didactic tradition of the monochord. In each of the six triads, a string stretched around the entire triangle is divided into three sections of different lengths. Freely rotatable cylinders in two of the three corners ensure that the tension of the string on the sections is more or less balanced. As a result, the lengths of the triangle sides in question are inversely proportional to the frequencies of the notes played with them. Each of the six triads conveys geometrically and acoustically the intervals of a triadic inversion of a major or minor triad. The concrete ratios can be taken from the following table 1.

DesignationLength ratiosFrequency ratios
Major triad in root position15:12:104:5:6
Major triad in first inversion (sixth chord)24:20:155:6:8
Major triad in second inversion (fourthsixth chord)20:15:123:4:5
Minor triad in root position6:5:41/6:1/5:1/4
Minor triad in first inversion (sixth chord)5:4:31/5:1/4:1/3
Minor triad in second inversion (fourthsixth chord)8:6:51/8:1/6:1/5
Table 1: The length and frequency relations to the different triads

To explain the prominent role of major and minor triads for the music of harmonic tonality, their consonance is often invoked, which in turn is associated with small number ratios. In the context of psychoacoustics according to Hermann von Helmholtz, this view even receives some confirmation, at least for sounds with harmonic spectra.

However, the psychoacoustic properties of sounds are again only very conditionally suitable for an explanation of their musical use. Here we may refer to a relevantly known problem case in which psychoacoustic and musical criteria clearly diverge in the use of the concept of consonance or dissonance. With respect to its frequency ratios, the major fourth-extave chord 3:4:5 is obviously simpler in construction than the major triad in root position 4:5:6. In counterpoint, however, the major triad in root position is considered a consonance, whereas the fourth-extave chord as a leading dissonance or passing chord requires sensitive treatment and, in particular, must be resolved into a consonance.

Figure 1: The exhibit

Notes on other expositors:

  • Two of the six triad inversions can also be studied on the “Galileo” exhibit: the major triad in root position 4:5:6 (pendulum 1, 2, 4) and in first inversion 5:6:8 (pendulum 2, 4, 5).
  • The combinatorics of major and minor triads with common chord tones is illustrated by the exhibit “Triad Polyhedron”.
  • The combinatorics of conductor triads with common chord tones and their inversions are illustrated by the “voice gyro” choreography on the exponant “tone gyro”.