7.2.5 Chord value

The seconds appear in various places in the harmonic row, for example as 7:8, 8:9, and 9:10. This would result in different root tones, but Hindemith opts for a practical solution: as the dominant seventh chord occurs quite frequently and the second is an important element of it, we hear the highest note of the second as root tone.122

With the seventh this is the lowest tone, as it is the inversion of the second. - The tritone presents a bigger problem. Here too the dominant seventh chord comes to the rescue. The difference tones, together with the interval, invariably form a dominant seventh chord. Hindemith posits that the tritone itself does not have a root tone, but that it can still be determined by proxy by looking at how the interval is resolved. He assumes that the interval is resolved traditionally, i.e., through leading tones. We can now determine series 2, the series that orders the degree of strength of the root tone function of an interval, its ‘value’: octave – fifth – fourth – major third – minor sixth – minor third – major sixth – major second – minor seventh – minor second – major seventh – tritone.

7.2.5 CHORD VALUE Chords can also be made of other intervals than from the stacking of thirds; all possible intervals may be used. In Der Unterweisung Hindemith presents a system to bring all chordal structures together by way of determining the root tone. To this end, he looks at how the intervals of the chords are structured.

The method is as follows: in the chord, one looks for the most valuable interval according to series 2; the root tone of this interval is the root tone of the chord. The tones of this interval do not have to immediately follow each other. In c-e-g and in c-g-e the fifth c-g is the strongest interval and is therefore c is the root tone. In g-c-e the fourth is the strongest interval, and the root tone is again c.

The results of this system and those of the traditional system are usually the same, but things become interesting when we determine the root tone in complex chords. The problem is that here, the method of establishing a root tone is not convincing because the outcome, the final root tone, is inaudible as such. This brings us back to my thesis in Chapter II: a dissonant chord – that is not built from thirds like the dominant 7 and 9 chord -– does not have a root tone, and therefore music built from dissonant chords cannot have tonal functions. Hindemith fails to

122In c-e-g-b , c is the root tone, as in b -c-e-g.

demonstrate that this would not be a correct proposition, even though he denies the existence of atonality! He ignores the problem by having the more complex harmonies hardly play a role in his system.

His system departs from the assumption that there are two main groups: chords without a tritone (stable) and with a tritone (less stable or unstable). And then there is the category of ‘undefinable’, under which he groups some peculiar chords. Here, in addition to chords that are traditionally used for modulation, he includes the fourth chord, which, according to his own method, should have the highest tone of the lowest fourth as its root tone.

Ex. 35: Chord tables from Unterweisung in Tonsatz.

A Sounds without tritoneA Sounds without tritone

I Without seconds and seventhWithout seconds and seventh

1. Root tone and bas tone are the same1. Root tone and bas tone are the same

2. Root tone and bass tone differ 2. Root tone and bass tone differ

B Sounds with tritoneB Sounds with tritone

II Without seconds and seventhII Without seconds and seventh

Tritone subordinated Tritone subordinated a. only with minor seventh (without major second). Root tone and bas a. only with minor seventh (without major second). Root tone and bas tone are the same tone are the same

b. With major second and minor seventhb. With major second and minor seventh 1. Root tone and bas tone are the same1. Root tone and bas tone are the same

And similar… And similar…

2. Root tone and bass tone differ2. Root tone and bass tone differ

And similar…

And similar… 3. with more than one tritones3. with more than one tritones

And similar… And similar…

III III with seconds and seventhwith seconds and seventh 1. Root tone and bas tone are the same1. Root tone and bas tone are the same

And similar…And similar…

2. 2. Root tone and bass tone differRoot tone and bass tone differ

And similar…And similar…

V V UndefinableUndefinable IV With major second and minor seventhIV With major second and minor seventh One or more tritones subordinatedOne or more tritones subordinated 1. Root tone and bas tone are the same1. Root tone and bas tone are the same

And similar…And similar…

2. Root tone and bass tone differ2. Root tone and bass tone differ

And similar…And similar…

VI VI Undefinable, tritone subordinatedUndefinable, tritone subordinated

A few remarks about the two schemes: - The more to the right or to the bottom of the scheme the chord is, the lower its value. This value only concerns stability, it does not reflect an aesthetic appreciation. - Hindemith says that chords that are not or only partially built from thirds lose their specific character when they are inversed; sometimes they even acquire a different root tone. By this he admits that chords with a complex interval structure have a value of their own.

In case of inversion or repositioning they are transformed into essentially different sounds. - From his proposition that the minor third is derived from the major third Hindemith concludes that this is also true of the major and minor triad. The minor triad is nothing but a clouding (Trübung) of the major triad. In his own experience, when playing a slow glissando from c to g over a sustained c, one actually cannot hear exactly when e changes into e. Therefore, c-e -g is actually the same chord as c-e-g, only coloured differently. - Finally, he posits (following the theory of root progression) that all harmonies can be placed on all scale degrees of a key and that only the triad of the first degree indicates whether one is dealing with major or minor. With ‘key’ he means the ordering provided by series 1. In doing so he greatly expands the definition of ‘key’.