a harmony lesson

[calimac]

A few days ago, fringefaan posted this quote, from Christopher Palmer's liner notes to a recording of Benjamin Britten's opera The Turn of the Screw:

"It is perhaps scarcely an exaggeration to claim that, in this opera, the entire 'natural' musical order of things is inverted; 'inverted' being the operative (and in this case appositely loaded) word, since the result of inverting a perfect fourth is a perfect fifth, and what is the musical meaning of 'Quint' but a fifth? I am aware that this is in effect asking the reader to accept that the entire musical structure of The Turn of the Screw was motivated by a pun; and while I feel that Britten must have been aware of the musical implications of Quint's name, I think it likely that they unconsciously, rather than consciously, influenced his choice of vocabulary. Whatever the facts of this matter, we should not be chary of recognising Britten's achievement (a) in creating a sense of all-pervasive evil through the very musical formulas normally and naturally (but there is nothing 'normal' or 'natural' about The Turn of the Screw) associated with all-pervasive good, and (b) in avoiding totally the cliché of the augmented fourth/diminished fifth, the conventional, traditional diabolus in musica."

How interesting, I thought, musical symbolism, but rather technical in nature. When fringefaan said that, while he got the gist of the point, he didn't understand the technical musical points, I undertook to explain.

Having a keyboard in front of you may be handy, and if you don't have one, here's a picture of one:

Well, "inversion" means taking a chord or a sequence of notes and doing them upside down, i.e. going down the scale instead of up and vice versa. Thus, a C and the F above it would be a fourth, but if you take that C and the F below it instead, that's a fifth, as you count four steps upward (including the note you start with) from C to get F, but five steps downward to get an F.

If C and the F above it are a fourth, and C and the G above it are a fifth, then C and the black key in between F and G on the piano are the tritone, either an augmented fourth (C and F-sharp) or a diminished fifth (C and G-flat) depending on harmonic context. Perfect fourths are traditionally a consonant interval; perfect fifths without fill-in harmonies are hollow and were once avoided but became a basis of modern post-1600 harmony (if C and the G above it are a fifth, add the E in between and you have a major chord in root, i.e. uninverted, position), but the tritone in medieval harmony was a big no-no, the diabolus in musica as it was called and Palmer refers to it. It has consequently been a favorite of composers wanting to express pain or unease ever since.

That's what I wrote, but there's more. If an F and the C above it are a fifth,¹ then what's a fourth above F? Is it B, the white key four steps (inclusive) above F?

No! F and B are the tritone! If you want a perfect fourth above F, you have to go to the black key below B, which in this context is B-flat.

Why is that? Well, it's because I was simplifying it. The reason I picked C as the note to start with is that it's the only note from which the rest of basic major-mode harmony² can be reached on the white keys only.

Though the fourth and the fifth (and other intervals³) are so-called because of the number of white keys involved in the interval from C, when you count them you're not really counting the white keys; you're counting all the keys. So a fourth isn't really 4 white steps including the one you started with, it's 6 white and black steps, and a fifth is 8. Starting with F as number one, count 6 keys up and you reach B-flat.

This is why the key of F major has a key signature of one flat. Let me explain. If you start with C and go up, you'll see that the pattern of steps taken to get all the white keys (this time excluding the step you start with) is 2-2-1-2-2-2-1.⁴ That's the major scale, because, as I said, the major mode from C equals the white keys. But you can begin with any note on the keyboard, including a black key, and if you follow the same pattern, it's still the major scale. Start from F and count 2-2-1-2-2-2-1 upward, and it's all white keys except B, which is flatted, replaced by the black key below it. Thus, one flat. Try it for all of them; the patterns are fascinating.⁵

Clear as crystal, or as mud?

1. Earlier I'd expressed it the other way around - C and the F below it - but it works in either direction. And if that's a fifth, then C and the G below it are a fourth, whereas C and the G above it are a fifth. So it's all nicely reciprocal.
2. Everything we're discussing here is in the diatonic major mode. There are other harmonic patterns or modes, though, starting with the minor mode, and continuing on through the modes that are meant when "modal" music is discussed, ranging from Dorian, the most common of them, to the delightfully-named Mixolydian, and more exotic scales like the chromatic scale and the whole tone scale ... If you think this is complicated, you don't even want to think about the others.
3. Which are also more complicated. Only fourths and fifths get to be called "perfect."
4. Actually, musicians say "half step" for "1" and "whole step" for "2", but the numbers might be easier for a first lesson.
5. And if the major mode is 2-2-1-2-2-2-1, the minor mode is 2-1-2-2-1-2-2, and the Dorian mode is 2-1-2-2-2-1-2. The whole tone scale is all 2's and the chromatic scale is all 1's. See, maybe it's not that complicated after all.

Comments

[(Anonymous)]

I guess I'm on the other side of the discussion: I follow the music lesson well enough, but I'm quite unclear on what Palmer is trying to say in the original comment: Britten uses fourths instead of fifths (quartal harmony a la early Schoenberg)? Or fifths instead of fourths? And whyever (says the person who loves tritones) would you not use tritones in an opera based on The Turn of the Screw, a work where unease and evil are the order of the day?

I also note that Palmer is one of those conservatives who regards radical elements as "conventional, traditional" "clichés," when it seems to me that "normal," "natural" "musical formulas" (note the betraying word "formula"!) "associated with all-pervasive good" (which I'm assuming means major key, but maybe I'm wrong) are far more traditional, conventional, and clichéd (i.e. boring, from my idiosyncratic point of view).

I.e., a cliché is a convention one dislikes.

Don Keller

[kalimac.livejournal.com]

I think we'd need more context to know what Palmer says Britten is doing with his fourths and fifths. I doubt it's full-fledged quartal harmony; it could be individual fourth-based chords, or it could be horizontal harmony, intervals of fourths and fifths in the melodic line.

It does seem to me that Palmer is acknowledging that the "formulas" of fourths and fifths are conventional. What he's praising Britten for is an ability to use those formulas to express a totally different emotional effect. What's cliched, in his view, about the tritone is not its existence, it's its use to generate a particular stock response. When we reach the point where (to use an example I know irked you) that Bartok sounds like monster movie music, we are in the land of cliche.

Mind you, I'm not endorsing Palmer's view of Britten's artistic achievement. As I also wrote in response to the original post, I count The Turn of the Screw as the least interesting Britten opera I've ever heard.

[(Anonymous)]

I take your point—it is of course possible to use any musical element in a clichéd manner—but I'm not sure I take Palmer's. If one avoids, say, tritones in specifically musical contexts (by listening to non-dissonant music), and only encounters them in quasi-musical contexts (like monster-movie soundtracks), then yes, one might regard them as clichés; but I feel like the stock response is as much the problem as the stock usage. (Is the usage of Beethoven's 5th and 9th, and Barber's Adagio, and Carmina Burana in film soundtracks so clichéd filmmakers should stop using them? Well, maybe.)

Perhaps I'm overreading Palmer, but I think he is in fact saying something more than you think he's saying: that if Britten had used tritones, in a context where their musical effect was entirely warranted—i.e., The Turn of the Screw is, in fact, a kind of (very subtle) "monster movie"—the result would have been clichéd, and therefore less masterly than what actually resulted; but (never having heard it, I'm only guessing), that result seems to have been a kind of blandness, so that you (more conservative than I, but less than Palmer) found it uninteresting.

Don Keller

[kalimac.livejournal.com]

I can tell you one place where the use of Orff's "O Fortuna" became so cliched that everyone except the aesthetically clueless did stop using it, for that reason, and that is as backing music for SF convention masquerade entries.

As I implied before, I'm not sure if I agree with Palmer or not, on the question of whether the tritone is cliched, and certainly on the question of what Britten achieved by avoiding it. But I understand the nature of Palmer's argument, including the part you discuss in your last paragraph. His may seem an implausible argument, and may in this particular case be completely unjustified; my point in citing Bartok and the monster movies was to show that such a state of aesthetic reaction is in fact possible, whether it exists in this case or not.

[kip-w.livejournal.com]

[chorale.livejournal.com]

I found your discussion of music theory interesting. Thank you.