In Hindemith's book The craft of musical composition. Book 1, Theoretical part, he says that (p. 97)

The lower tone of a third or a seventh (in the absence of any better interval) is the root of the chord.

If the chord contains two or more equal intervals, and those are the best intervals, the root of the lower one is the root of the chord.

However, immediately later he states that (p. 101):

Among the chords which have no tritone, there are two of which the interpretation depends on context, and which in consequence have no root, but only a root representative: the augmented triad and the chord composed of two superposed fourths.

Now, if we were to go by the criterions of p.97, wouldn't this mean, for example, the root of the augmented triad is its lowest tone? And therefore it does have a root, contradicting what's stated in p.101?

What is meant by "the interpretation depends on context"? Is it the case that in consideration of the root, one needs to check all possible transpositions (i.e. changing which octave each tone are in) of the tomes in the chord? And if the root determined from one transposition differs from that determined from another, as is the case for the augmented triad, the root of the chord cannot be determined? Or is it meant somewhat another way?

  • I think Hindemith is arguing that the chord composed of two superposed fourths "isn't real and can't hurt you", since he treats it as a lower second-class citizen than the augmented triad (at least the augmented triad and the diminished 7th both have the excuse that their inversions all preserve the exact same intervals as the non-inverted).
    – Dekkadeci
    Feb 2 at 8:17
  • @Dekkadeci But he also says that his system is supposed to make easy to analyze the chords which "any self-respecting book on counterpoint would not tolerate," and which "theorists would only analyze in their worst nightmares". In any case, my doubt isn't exactly about the superposed fourths chord, but rather the apparent discrepancy on the existence/non-existence of the roots for such chords as well as augmented triads.
    – Divide1918
    Feb 2 at 12:00

2 Answers 2


I only have the 1940 German edition of this work, so my answer will be based on that. For reference, the quoted passages are found in Part III, Section 10, of the book.

Authors of theoretical works commonly begin by stating general principles and then move on to the exceptions. The lines concerning the augmented triad and the chord composed of two superposed fourths should not be read as contradicting the previously advanced rules, but as describing a limited exception to those rules.

That exception is one of those announced by the following words, which should be found about half a page above your first quote (my translation from the German): “Thus, apart from a small number of exceptions to be mentioned later, every chord possesses a root”.

Those exceptions are then listed at the end of section 10. Hindemith first points out that if a chord contains a “strong” interval—defined as a fifth/fourth or a major third/minor sixth—the presence of a tritone does not nullify the force of the root: enter image description here He then moves on to the exceptions to the rule that every chord has a root. The first one concerns chords containing, apart from the tritone, only minor thirds/major sixths. In other words, these are the diminished triad, its inversions, and the diminished seventh chord: enter image description here Hindemith states that these chords are too weak to overcome the tritone, and are therefore just as undefined as the tritone itself, but adds that one of the two notes of the tritone has the status of a “substitute root” in the chord. To determine which note that is, the chord’s “surroundings” (the German word is Umgebung)—meaning its relation to any adjacent chords—has to be examined. Theodore has explained in his answer what that context could look like.

The final exception is the one you ask about, namely the two chords that don’t contain a tritone, but still have no defined root. Part of the problem with understanding that passage can probably be blamed on the English translation. The final paragraph of Section 10 reads as follows in the original:

enter image description here

I believe a more faithful translation might run as follows:

  • “There are also, among the chords without a tritone, two that can only be analysed by reference to their context [=the adjacent chords], and which therefore—in place of their missing root—are assigned a substitute root, these chords being the augmented triad and the chord composed of two superposed fourths.”
  • 1
    Huge thanks for the comprehensive and detailed answer! This is a hugely illuminating answer that led me to a better understanding of the text.
    – Divide1918
    Feb 3 at 15:48

I don't know whether this is what Hindemith meant, but here's an aspect of context that might at least illustrate: enharmonic spelling.

Consider C - E - G♯:

The root is C. It's a C major triad with the G raised (augmented) to G♯.

Now spell it A♭ - C - E:

The root is A♭. It's an A♭ major triad with the E♭ raised to E.

Then spell it E - G♯ - B♯.

The root is E. It's an E major triad with the B raised to B♯

The spellings are going to be dependent on the key of the piece and can give an idea of the function.

Of course the rotational symmetry of the augmented chord opens it up to ambiguous uses or puns.

Consider that same triad in the key of F minor. This is a key likely to use both an A♭ major (III) and C major (V), and might also use C aug or A♭ aug. Even if it's used as a C aug and not an A♭ aug, the spelling C - E - A♭ might be used for convenience since the key signature already has an A♭. You'll have to look at the chords before and after to see how it functions and decide what the root really is.

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