Music primer, part two

Half-steps and whole steps

As I mentioned in my last post, the major scale has seven notes in it: if you start on middle C and play up the keyboard, you'll play a total of seven notes before you reach the next C. As I also mentioned in my last post, you will also skip five black keys.

Each of those black keys lies between two white keys. But obviously, if you have five black keys distributed among seven white keys, there are a few white keys that don't have black keys between them. The relationship between adjacent white keys that aren't separated by a black key is different from the relationship between adjacent white keys that are. Adjacent white keys separated by a black key are a whole step apart. Adjacent white keys that aren't separated by a black key are a half-step apart.

The difference between the vibrational frequencies of notes a whole step apart is greater than the difference between the vibrational frequencies of notes a half-step apart. As you might imagine, there is a sense in which the distance between notes a whole step apart is twice that of notes a half-step apart. But it's a rather technical sense that I don't want to get into here. (If you're interested in reading more on the subject, you might start with this Wikipedia entry.)

The relationship between whole steps and half-steps is easier to see on the neck of a guitar than on the keys of a piano. The metal frets embedded in the guitar neck mark off consecutive half-steps. To play the melodic interval of a whole step on the guitar, you have to jump across two frets; one fret will take you only a half-step away.

Finally, the distance between a white key on a piano and the black key next to it is a half-step. So you can see that the octave (eight-note span) from C to C -- on a keyboard or on a guitar -- is actually divided into 12 half-steps. On the keyboard, most of those half-steps are between white keys and black keys, but two of them -- from E to F and from B to C -- are between white keys. (There are in fact good mathematical reasons that Western music divides the octave into 12 equal half-steps.)

The major scale revisited

Armed with the notion of half-steps and whole steps, we can make a little better sense of the notion of a major key.

Play up the major scale from middle C to the C above it (all white notes). There's a black key between C and D, so the first step of the scale is a whole step. Same with D to E. But there's no black key between E and F, so that's a half-step (can you hear the difference?). Whole step to G, whole step to A, whole step to B -- then a half-step back to C. So the pattern of whole and half-steps that gives you a major scale is

WWHWWWH.

That's why you need black keys if you start your major scale on any note other than C. Start on D. Your first whole step takes you to E. But E to F is only a half-step, so your second whole step takes you to F-sharp. Then comes a half-step: G. A, B, no problem -- but now you've found the other white-note half-step, B to C. So your next note has to be C-sharp, not C. And a last half-step will bring us back to do.

Make sense?

The same procedure, of course, applies to the neck of the guitar. You can start your scale on any fret you want. The next note will be two frets (a whole step) up. The one after that will be two more frets up. But the one after that (the first half-step) will be only one fret up. Etc.

The procedure works in exactly the same way no matter what fret you begin on. That's why pop guitarists tend to be less clear on the theoretical differences between keys than pianists: changing key on the guitar is just a matter of starting the same pattern on a different fret; each key on the piano has its own distinctive pattern.

Intervals revisited

The concept of whole and half-steps also lets me clarify some distinctions I elided in my last post. I mentioned that in any given major scale (I hope that the principle of the major scale is now clear enough to you that I can stop using C major as a reference point; C is, after all, just one of 12 major scales, none of which should, in principle, be privileged over any other), the distance from 1 to 6 is a sixth. The distance from 3 to the 1 above it is also a sixth. But they're not the same sixth. The sixth from 1 to 6 is a major sixth; that means there are nine half-steps from 1 to 6. But there are only eight half-steps from 3 to 1, making it a minor sixth. Here's the mapping of total half-steps spanned to intervals:

1: minor second
2: major second
3: minor third
4: major third
5: perfect fourth
6: augmented fourth/diminished fifth
7: perfect fifth
8: minor sixth
9: major sixth
10: minor seventh
11: major seventh

The interval of the diminished fifth (6 half-steps) is commonly called the tritone, or less commonly, the diabolus in musica. For a long time, it was considered a gross dissonance, to be avoided. In the 20th century, it was to some extent rehabilitated, but melodies that emphasize tritones can still sound sharp and spiky to the modern ear.

Other scales

So the major scale goes WWHWWWH. But you could, if you wanted, make your own scale out of some random sequence of whole and half-steps -- WHHWWHHW, or whatever. Twentieth-century jazz and classical composers experimented widely with the whole-tone scale (WWWWWW) and the octatonic scale (WHWHWHWH), but by far the most common scale other than the major is, unsurprisingly, the minor.

The minor scale

The basic minor-scale pattern is WHWWHWW. (It has variants, but I'm not going to get into them.) The distinctive thing about it is that the interval from 1 to 3 is not a major third; it's a minor third, the interval between "dead" and "and" in the schoolyard incantation "pray for the dead, and the dead will pray for you". Music written in minor keys tends to have a more melancholy, or brooding, or ominous, or menacing feel than music written in major keys: recall Nigel Tufnel's sage observation that D minor is "the saddest of all keys." Play any peppy tune you know on the keyboard with the third scale degree knocked down a half-step, and it will come out much less peppy*. The Christmas hymn "O Come, O Come, Emmanuel" spends a lot of time in a minor key -- as does Britney Spears's "Oops I Did It Again". Both songs occasionally slip into major keys, however, for reasons that I hope will come clear in the next section.

The modes

Take a look at the minor-scale pattern of whole and half-steps. The minor scale, like the major scale, starts over again at the octave. So two octaves of the minor scale will look like this:

WHWWHWWWHWWHWW

Trace out the eight-note pattern beginning on the third scale degree of the minor scale instead of the first. That is, knock off the first two and the last five letters:

WWHWWWH

Look familiar? Yes! It's the major scale! The minor scale is just the major scale begun on a different scale degree and wrapped around on itself. The converse is also true: the minor scale is the major scale begun on a different scale degree and wrapped around on itself.

Another way to say the same thing is, if you play an A on the piano, and play up the next seven white keys, you will have played the minor scale. (The white keys give you the major scale only if you start on C; they give you the minor scale only if you start on A.)

Note that starting the major scale on a different scale degree and wrapping it around will not give you the whole-tone or the octatonic scales: they have fundamentally different patterns (the whole-tone scale has no half-steps at all, so of course it can't give you the major scale). But if there are seven notes in the major scale, there must, perforce, be seven different "wraparound" scales (2 to 2, 3 to 3, 4 to 4, etc.). These wraparound scales are called modes. Two of them -- 1 to 1 and 6 to 6 -- are our familiar major and minor scales. Of the remaining five, three have minor thirds between 1 and 3, and two have major thirds between 1 and 3. The ones with minor thirds partake of the minor-scale melancholy; but the ones with major thirds have distinctive flavors all their own, and it's on those two that I will concentrate in this blog.

The two major-third modes (other than the major scale) are the lydian and the mixolydian. These are the scales that arise when you start on F and G, respectively, and play up the next seven white keys. They are also the modes that follow the following permutations of the major-scale sequence of whole and half-steps:

lydian: WWWHWWH
mixolydian: WWHWWHW

Each scale differs from the major scale in only one respect. The lydian mode has a raised fourth degree relative to the major scale. That is, in the lydian mode, 4 is a half-step higher than the 4 of the major scale. This makes the interval between 1 and 4 a tritone, which gives lydian melodies a piquant sound. The mixolydian mode has a lowered seventh degree relative to the major scale, a similarly fateful alteration. In the major scale, you'll recall, 7 is only a half-step below 1. That close proximity gives the 7 a feeling of kind of leaning toward the 1. It's hard to describe but very easy to hear -- it's the sense in which it instinctively brings us back to do. Widening that interval removes that leaning feeling, which drastically changes the color of the scale.

Because all the modes consist of the major scale wrapped around on itself, it requires a certain amount of effort on the part of the composer to keep modal melodies from simply drifting back into the major: our ears, conditioned by so much major-key music (and possibly predisposed by evolutionary adaptations), tend to pull as back to the familiar (or perhaps the instinctive).

Okay, I think that's gonna do it. I may want to say more about harmony at some later point, but then again, I may not. The distinction between different keys may be as much of a harmonic distinction as I'm going to need to make.

* I'm sure that the popular Boston band the Dresden Dolls has some tunes in minor keys. I don't really know their music, but years ago, when Amanda Palmer was still busking in Harvard Square as the 12-foot bride, I saw her play some of her songs, solo, and without makeup or carved eyebrows, at the original Zeitgeist Gallery on Broadway in Cambridge. After she'd played about four or five songs in a row in minor keys, I yelled, "Play something in a major key!" She thought for a minute and said, "Hm, I don't think I've written anything in a major key since I was 17," and I said, "Back when you could still believe in major keys."

Music primer, part one

Before I make any music-themed posts on this blog, I want to explain a few technical terms that I expect I'll occasionally want to invoke. They're not difficult, but some readers may be unfamiliar with them or have only a vague notion of what they mean. I assume a passing familiarity with the layout of the piano keyboard. If you don't have a keyboard handy and find any of the descriptions below difficult to visualize (or "auralize"), try playing with the little Flash keyboard here. (If you don't have Macromedia Flash installed, there's also a Java piano here.)

The major scale

Most people, I think, know how to find middle C on a keyboard and know that, if you play the next seven white keys in sequence, up the keyboard, you'll spell out the do re mi scale familiar from The Sound of Music ("Do, a deer, a female deer, re, a drop of golden sun," etc.). The last note in that eight-note sequence is another C -- not middle C, but the C an octave (a span of eight notes) above middle C. That is, the do re mi scale -- a.k.a. the major scale -- has only seven notes in it; with the eighth note, you're starting the scale over again, only higher ("that will bring us back to do").

In this blog, I will refer to the notes of the major scale by number. So do is 1, re is 2, mi is 3, etc. Ti ("a drink with jam and bread") is 7, which brings us back to do, or 1.

If you play middle C, and then play the next four white keys, up the keyboard, in sequence, you'll get to G, or 5. But if you play middle C and then play the next three white keys down the keyboard, you'll also get to G, or 5. For every 1, there's a 5 above and a 5 below. There's also a 4 above and a 4 below, etc. And for every 5, there's a 1 below and a 1 above. Etc., etc.

Intervals

An interval is the distance between two notes. We call the distance from 1 to the 5 above it a fifth: the total number of white keys you have to press to get from middle C to the G above it is five. The distance from 1 to the 5 below it, however, is a fourth: the total number of white keys you have to press to get from middle C to the G below it is four. Conversely, the interval from 1 to the 4 above it is a fourth, while the interval from 1 to the 4 below it is a fifth. The interval from 1 to 6 is a sixth, from 1 to 3 is a third, etc.

What's the interval from 5 to the 3 above it (from G to E)? Well, if middle C is 1, how many white keys do you have to press to get from the 5 below middle C (G) to the 3 above it (E)? If you can't figure the answer out in your head, try actually pressing the keys, and then check your answer against the one at the end of this post.

Key

It so happens that the first line of the Christmas carol "Joy to the World" traces out the major scale -- from 1 back down to the 1 below it. Sing it to yourself: "Joy to the world, the Lord is come." You sing the word "joy" on 1, "world" on 5, "lord" on 3, and "come" on 1 again. If you play the eight white keys from the C above middle C back down to middle C in the right rhythm, you'll play the opening line of "Joy to the World."

But let's say that, instead of starting on a C, you start on the next white key above C -- i.e., D. Now, if you just play down the white keys, the tune will sound completely wrong. In order to make it sound right, you'll have to throw in some black keys -- specifically, at "to" and "lord".

If, instead, you started playing "Joy to the World" on E, you'd need four black keys to make it sound right, and if you started on B, you'd need five!

There's a fundamental principle here, one that I've found is not intuitive for nonmusicians. Of everything I've said on this page, it's the most important thing to remember (if you don't know it already): no two major scales use the same notes. If you start your major scale on C, you can use all white keys -- but that's not true for scales begun on any other note. If you start your scale on G or F, you need only one black key -- but it's not the same black key. That is, if you play "Joy to the World" starting on F, you'll need a black key at "the" -- B-flat; but if you play "Joy to the World" starting on G, you'll need your black key at "to" -- F-sharp. (Try it.)

So, a few definitions:

a melody is a set of notes played in sequence;
a chord is a set of notes played simultaneously;
music written in a particular key is music all of whose melodies and chords use the notes of a single scale.

If you play a piece that's entirely in the key of C on the piano, you'll use all white keys. If you play a piece that's entirely in the key of G, you'll use one black key: F-sharp. If you play a piece that's entirely in the key of F, you'll use one black key, but not the same black key: B-flat. Etc., etc.

If you play middle C, then play the next six notes up the keyboard (stopping just shy of the next C), you will have played seven notes total: each of those notes determines a unique major scale, so each of those notes determines a unique major key. You will also, however, have left out five black notes. Each of those notes also determines a unique major key. So there are 12 major keys total.

An experienced musician can tell from a handful of notes what key a particular piece is in. Indeed, she can tell from only three notes what key a piece is in, if they're the right three notes. For instance, there are seven different major keys that contain the note C: C, D-flat, E-flat, F, G, A-flat, and B-flat. But five of those -- the ones with "flat" in their names, plus F -- contain the note B-flat instead of the note B. So if a melody begins on C and moves to a B (not a B-flat), it is in one of only two possible keys: C or G. The C and G scales, in turn, differ by only one note: the C scale contains an F, but the G scale contains an F-sharp. So if a melody contains only three notes, and they're C, B, and F, then the melody must be in the key of C. (Note that, by contrast, if the melody contains the notes B, C, D, E, G, and A, it could be in either C or G.)

This post has taken me a lot longer to write than I anticipated, because I'm trying to be both accurate and accessible. I'll be back with more Music Theory 101 in the coming days.

Answer key: a sixth

Brief dispatch from the front

What did you do with your Friday night? I spent mine drinking the port that my friends Arkadi and Nancy gave me and composing a reply to Charlie Greenbacker's comment on my "Gelernter wrapup" post. If you stopped back here expecting more ruminations on consciousness, you might want to look at what I wrote -- although I should warn you that it's a little, uh, looser in both diction and argument than an official blog entry would be. (It seemed, however, not quite in the spirit of the blogosphere to go back and delete the obscenities and tipsy hyperbole.)

I also think I should mention that the next issue of Technology Review will feature an essay by Daniel Dennett, so both the anticognitivists and the cognitivists are getting a fair hearing in its pages.