Is it wrong not to resolve sevenths and suspensions?
Question: In tonal music, Why would it be wrong to leave unresolved 7ths or unresolved suspensions? - M.

Answer: Well, it wouldn't be morally wrong. Really this is a matter of style. There are certain expectations that form part of the style of tonal music, and one of those is that we expect when we hear a minor seventh that its upper note is going to move downward. Or that when we hear a harmonic second the lower note is going to move downward.

One can find acoustic justification for this; the idea didn't just arise arbitrarily. The interval of a seventh, or its inversion, the second, is acoustically unstable in the sense that none of the more audible partials (overtones) of each pitch has any possibility of matching another. You can find some discussion of the acoustics of harmony in the appendix of Exploring Theory with Practica Musica, but the essence of the thing is that every sound produced by a vibrating string or a column of air is not just a single frequency; the sound is made up of a number of different frequencies all sounding together, and in pitched sounds these have a simple mathematical relationship with each other: there's the basic pitch corresponding to the full length of the string or column of air, and added to that is the sound of half that length, one third that length, one fourth, and so on. In music terms this means that if you play the note C below middle C you are actually sounding a sort of chord composed of the fundamental, C, its octave, middle C, its fifth, G, then another octave, then (getting fainter and fainter with each one) E, G, Bb, another C, D... The actual frequencies of these pitches don't quite match those on the piano because the piano is adjusted to make everything come out even in 12 steps, but they're close.

Anyway (are you still there?), this means that if you play the notes C and G together, the third partial of the C (G) will coincide with the 2nd partial of the G (another G):

If the C and G are tuned just right they'll seem to have a special strength; if slightly off you'll hear a "beating" as the overlapping partials go in and out of phase with each other. So we say that this interval, the perfect fifth, is strongly consonant. It can be tuned in a way that makes it acoustically self-reinforcing and stable, and even when it's slightly off the reinforcement is there (it's very slightly off on a piano). The same is true of the thirds and sixths. But the seventh lacks that quality: none of the easily audible harmonics really line up, and the best one, the 7th harmonic itself, is not only faint but also a little lower in pitch than the one on the piano. So the minor seventh seems unstable, and yet it's only a half step away from a nice stable interval, the major sixth. We seem to find some kind of satisfaction when the seventh moves (resolves) to a sixth. Same for the second resolving to the third.

The fourth, often heard in suspensions, is a special case. It's not as unstable as the others, but to our ears accustomed to the sound of triads it feels up-side down. It's an inverted fifth, but just a half-step away from being a nice stable major third. So it, too, seems to aspire to becoming a third.

These are all just conventions, but you can see that they have natural origins. That is why I'm quite certain that when we finally get a chance to hear the music created by creatures in distant galaxies we'll find that it makes sense to us. Their vibrating strings, or columns of whatever gas they breathe, will work just like ours do. The main differences will be whether they choose to concentrate on the melodic and rhythmic elements of music (as in traditional Indian or Chinese or African music) or whether they get interested in the harmonic aspects, which would inevitably lead them to triads, seventh chords, etc. Some of them will have counterpoint, and those that do will likely have a style in which they resolve sevenths. But they'll also have their own version of Miles Davis or John Coltrane, and in that style they won't necessarily resolve sevenths but might just let them hang out there, floating - it all depends on what you're after: sometimes what you want is to defy expectation. The thing is to know what you want; then you find out how to get there. If you want the tonal-music effect, you eventually resolve those sevenths and suspensions.

Bonus followup question! Q: Thanks! I understand what you mean perfectly. But, I was wondering how would Bach explain it? You know, in a more historical context.

A: Oh. Thing is, Bach the father didn't write much in the way of instruction. However, his son C.P.E. Bach did write a very well-known book on music, and we can assume that a lot of what C.P.E. knew and believed he got from Dad. C.P.E.'s book is titled, in translation, Essay on The True Art of Playing Keyboard Instruments, and while that would appear not to touch the subject of composition it actually does, because in Bach's day a keyboardist was expected to be able to improvise accompaniment, and this improvisation followed the same rules as in written composition.

In particular, Part Two of his book is a composition primer, discussed in the context of improvisation from "thoroughbass." Thoroughbass (discussed in a previous answer) was the practice of indicating harmony with a numeric shorthand to be freely "realized" by keyboardists. So C.P.E. tells the keyboardist in great detail how to handle and resolve each dissonance in the suggested harmony.

C.P.E. Bach was, however, not inclined to provide a reason why a dissonance should be resolved. Here are some quotes from William Mitchell's translation of Bach's essay:

"The basic characteristics of dissonances are suggested by their name, which expresses the fact that they sound bad...[my emphasis] Their natural harshness must be mollified by preparation and resolution; that is, the dissonant tone must be played, previously, as a consonance and it must succeed to a consonance."

"[The bass of a second] always resolves by stepwise descent."

"[The seventh and the fourth] resolve by stepwise descent."

That is to say, C.P.E. just tells us what is correct in the style of which he is a master; he isn't going to provide us with any justification other than to observe that "The observations are not speculative but rest on experience and wisdom. With no desire to boast, it may be said that this experience can hardly be rivaled, for it has grown out of many years of association with good taste in a musical environment which could not be improved."

You could find more of philosophy in Bach's English almost-contemporary, Roger North, who wrote many essays on music that have been collected into a fascinating book, Roger North on Music (Novello, edited by John Wilson). North is somewhat more generous to those dissonances,

"...of the more inferior accords the 2nd and 7th, which of themselves are not accounted good sounds, yet so [used, i.e. resolved properly] become excellent music."

North was aware of the partials and was curious enough to investigate the concept of dissonance as arising from the lack of agreement among partials:

"I find it needfull to examine stricktly the nature of Discord, because the word is of ambiguous use; for even among concords there are sounds gradually more or less pleasing, of which the limits in Nature are not defined. I say that consonance depends on the frequency of coincidences [i.e. among the partials]; but where begins discord?"

He goes on to reckon that dissonant tones are not fundamentally different from consonant ones, but both exist along a sliding scale of consonance so that the "dissonances" are simply less consonant than other combinations, an idea that finds support in the study of the partials:

"The authors of musicall composition harp much of intermixing discord to set off concord, but they mistake their termes, for it should be accords better and worse, harsh and smooth, and the like."

North reserves his scorn for notes that are simply out of tune and therefore have no trace of relationship to each other; these

"are not to be heard in consort, upon no account whatsoever."

So one could talk forever about this, and North almost did in his 2000 pages of essays. But the resolution of these intervals still ultimately comes down to a matter of taste, though one with a real relationship to the natural world.

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