Paul Hindemith & Omitted Fifths

Question

Three note voicings of seventh chords where the perfect fifth above the root has been omitted are common in jazz piano because the ear does a good job interpreting the function of the chord with or without it. In Paul Hindemith's 'Craft of Musical Composition' he expresses that because it is limiting to think of chords as being constructed in stacks of thirds, the root may not be determined by doing so and simply seeing what tone ends up on the bottom. Instead we look at the root of whatever the "best interval" within the chord is.

If there is a fifth in the chord, then the lower tone of the fifth is the root of the chord. Similarly, the lower tone of a third or aseventh (in the absence of any better interval) is the root of the chord. Conversely, if a fourth, or a sixth, or a second is the best interval of a chord, then its upper tone is the root of the chord. Doubled tones count only once; we use the lowest one for our reckoning. If the chord contains two or more equal intervals, and these are the best intervals, the root of the lower one is the root of the chord (Hindemith 97).

In a (major or minor) 7th chord with an omitted fifth, the "best interval" would be the perfect fifth between the 3rd and the 7th, the 3rd being the lower tone, then, would be the actual root of the chord. Why then would the ear hear the root of a CΔ7, for example, as C and not E? This book has been extremely illuminating so far & I'm sure there is an explanation for this inconsistency that I'm missing.

Answer 1

The theories of harmony of Hindemith, Schoenberg and Janacek were attempting to make sense of the music of the time. They came from a heritage where thinking about lines in counterpoint was more dominant than it is in the music of today. The music of Debussy and other composers was challenging old ways of understanding harmony... unresolved dissonances etc... and they tried to understand these changes through the movement of lines that form re-occurring vertical arrangements (chords).

Jazz music, at least how it seems to be taught in the academy, takes the chords much more as units in their own right, to be assembled and re-assembled into progressions. The function of the chords are often (not always) "in" the chords, rather than between the chords...

So... C Eb Bb in jazz might be heard as dropping the fifth... whilst in the music of Mahler, it might be heard as Eb add 6 without the third, all depending on what else was going on in the music with motives or whatever.

This is a quick response with some hunches. Of course, there was a good amount of cross-fertilisation between jazz and classical music. One area in common is lines/voices that move symmetrically via chromatic motion. The key is that classical musicians play parts... and jazz musicians charts...

My understanding is that dropping the fifth in jazz piano can be done because the bass player might play the fifth... the ways chords are used comes from the different kinds of musicianship expected - comping vs doing a solo etc.

PS have you read Persichetti?

Answer 2

Hindemith wrote textbooks on harmony, then proceeded to ignore much of what he wrote in his own compositions. Or so my old music professor used to tell us! And his 'best interval' theory certainly falls down when confronted with a (5th omitted) maj7 chord. But there's good ideas in his books. Use them as a resource, not a bible. (Same, but even more so, with George Russel's 'Lydian Chromatic Concept'. Definitely had a bee in his bonnet, but some of the buzzing is interesting.)

To sidetrack a little into the practicalities of jazz piano playing: Yes, you'll find yourself playing 3rds and 7ths a lot in your left hand. Not so much because of any harmonic theory, but because you want to keep the texture light and un-muddy, and you want to keep out of the way of the bass. There's another technique when comping behind someone else's solo where you avoid playing the 3rd of a major chord, because the soloist is very likely going to play it 'blue' or elaborate it with suspensions. So you keep out his way.

Answer 3

The explanation is that Hindemith (not surprisingly!) believed in the theory he had invented. But don't forget Bertrand Russell's definition: "belief" is "that for which there is no evidence".

From the paper Hindemith's Contribution to Music Theory, William Thomson, Journal of Music Theory, Vol. 9, No. 1 (Spring, 1965), pp. 52-71:

The population of speculative theorists is split like that of other ontological realms into those who are "believers" and those who are not. The faithful, in this case, hold that music operates within a closed system, its basis unchanging through the ages and potentially demonstrable. Those who entertain such immutable "truths" are known as natural theorists, for a usual concomitant of their speculations has been the derivation of all manner of "laws" from the known, assumed, or merely the fancied "facts" of the natural world.

If natural theorists were organized in a manner befitting their allegiance to dogma, Paul Hindemith would be recognized as the most recent addition to their Pantheon; his life as a theorist was dedicated to an attempt to demonstrate music's participation in natural laws, ...

If you like this sort of stuff, google for terms like "density degree theory". The nicest things about pure speculation are (1) anybody can do it, and (2) nobody can be proved wrong.

Answer 4

The answer is easy - because C is the lowest note. Hindemith talked about the correlation between tone register and functional importance. C, being the lowest tone and a fundamental, provides a relevant fifth twice (3rd and 6th overtone). Harmonically speaking, the bass tone determines functionality, it gives context to the other tones (even if it's not the root). Also chords are naturally build by thirds, and the only way you can get these three notes by stacking 3rds is in Cmaj7 chord. Sure, you can have non-chord tones too, but cloud the function of the chord and the ear is tuned to recognise triads best. There is no way anyone can claim that C-E-B sounds like Eadd6 (if not preceded by anything). This analysis is concerned with the chord as a separate unit, but in music this is rarely the case.

In the bigger picture it depends on context - where the chord stands in a progression. I'm sure you know that in jazz you can easily outline a chord with two notes (especially 3rd and 7th) and the ear can fill it out based on the logic of the progression, which in most cases is very common and familiar. In cases where the progression is more complex and unfamiliar (try outlining Giant steps for someone who is not that into jazz) this effect is lost though. A more trivial reason for omitting the fifth in jazz is that it sounds too stable, consonant and doesn't need a resolution, making it less appealing and stifling forward momentum (and you could use that finger to play a non-chord/altered tone).

Answer 5

Hindemith used the theory of “the lowest strongest interval to name the root of chords and clusters based on combination tones resulting from intervalic frequencies combining to produce an audible sub octave note. The entire ensemble would need to be considered not just the piano in this instance.