Much has been written about Wagner's "Tristan chord" in terms of musical analyses, and the assessments of it's tonal reference reach from almost every step in the chromatic scale to outright atonality. It is therefore remarkable that the analysis by Ernst Terhardt, an acoustical communications engineer by profession, has found worldwide acclaim. Based on his virtual pitch theory, Terhardt claims that the Tristan chord's point of tonal repose is C#, so that what we perceive is C#-F-B-d#-g#, rather than the F-B-d#-g# chord that is actually played (compare with the score below).
But there is a second, entirely different way that leads to practically the same result. It is obvious that Richard Wagner was out for something thoroughly "unheard of" when he wrote that composition, and it is possibly wrong to suppose from the way he noted it down that he meant 12-tone equal temperament. If we assume, for instance, that he was thinking, at least originally, in terms of just intonation, then the chord F-B-d#-g# suddenly reads as 5:5 - 7:5 - 9:5 - 12:5, if we choose the closest and simplest just intonation intervals to represent the notes. This is a chord of high gestalt value that in practice is further enhanced by more members of the form n:5 through combination tones. It is tonally unambiguous and points to a clear root, namely 1:5, which translates into C# in the second octave below F, supporting Terhardt's analysis.
we now look at it from Terhardt's angle of view (virtual pitch)
or from the one invoked in the previous paragraph (gestalt enhancement
by combination tones), it is in both cases surprising how extensively
intervals that are now elements of the Bohlen-Pierce scale are
involved in the Tristan chord:
The just intonation version of the chord is clearly based on 5:7:9, the "narrow" triad of the BP scale.
In Terhardt's approach, we find that
- the chord's frame interval C#-g# is a twelfth (3:1), or, in Bohlen-Pierce (BP) terms, a tritave, the frame interval of any BP scale;
- two more intervals, C#-B and F-d#, are minor sevenths which, when derived from the 9th harmonic in the sense of Terhardt's analysis, represent 9:5, another BP interval;
- the tritone F-B, when under the same aspect derived from the 7th harmonic, can be considered as the BP interval 7:5;
- B-g# is a major sixth (5:3), an interval shared by the conventional scale and BP;
- all other intervals are traditional (two major thirds, a forth, a ninth, and Wagner's original frame interval, a minor tenth).
the Tristan chord presents itself as an effectual assembly of
traditional intervals and unconventional ones that now have found
a place in the Bohlen-Pierce scale. One might argue that this
heterogeneity is the reason for its unique and haunting appearance.