I hereby dedicate this diary to Chili peppers and ice cream. Oh, how well they go together! The one too hot, the other too sweet.
This diary will be like our previous diary, The Physics of Music, but we'll go deeper to prepare us for more modern music.
What is an interval? It's two different notes at different pitches. Later, we'll use Bartok's Concerto for Orchestra to illustrate intervals.
We've already talked about fifth intervals. Let's use the "Do re mi..." scale for illustration. From Do up to Sol is a fifth. Or from Re to La. You can count it out on a keyboard using white notes. "Do re me fa sol!" Five! Do and Sol constitute a fifth interval; Re and La likewise.
It shouldn't be hard to understand what a fourth is then, eh? Let's start out at Mi and go up a fourth. "Mi fa sol la!" Mi and La is a fourth interval.
There are two types of thirds. There is a major third -- like Do to Mi in the major scale -- or a minor third -- like Do to Mi in the minor scale. We talked about the difference between major and minor in our previous diary, Major Versus Minor. There are also minor thirds within the major scale. For instance, Mi and Sol, in the major scale, are a minor third apart. You can tell the difference by counting the half-notes, a minor third being just one less.
Likewise, there is a major sixth, and a minor sixth interval, a major seventh and a minor seventh interval.
Of course, it matters where you start at, too. For instance, if you start at Fa and you go up a four notes, you do NOT get a fourth interval. "Fa sol la ti! It's four notes, yes, but if you count out the half-notes in between on a keyboard, you see that you lost a half-note somewhere because of the asymmetry of the major scale. Fa to ti is just a half-note wider than a fourth interval, and just a half-note closer than a fifth interval. And it's a royal pain in the ass.
What do we call this interval, from Fa to Ti? We can call it an augmented fourth, augmented being a musical term meaning the top part is sharpened a half note. Or we can call it a diminished fifth, diminished being a musical term meaning the top is flattened a half note. Or we can use the sexier name for this interval: tritone.
Obviously I'm going to a lot of trouble to explain something that you may never need to know for any other endeavor in your life. I promised a long time ago, I wasn't going to try to teach you how to be musicians, just how to appreciate the music you hear. So bear with me!
We KNOW by now that a fifth is a powerful interval. From our previous diaries, we know that the sound frequency Do and Sol are related by 3/2. If Do is middle C on the piano (262 hertz, i.e., 262 vibrations per second), then Sol is middle G on the keyboard, and we can be sure that it's going to vibrate about 50% faster, 392 Hertz.) You may not be able to count each of those vibrations, but through some magical voodoo, your ear does, and some low-level part of your brain processes that and says, Hmmm, Do and Sol are in a nice ratio relationship; that sounds pretty.
Likewise, there's a nice ratio between two notes of a fourth interval, like Do and Fa, 4/3. Which shouldn't be very surprising -- If you start from Do and go backwards to Fa, you find it's just a backwards fifth! That sounds pretty too.
Major and minor third intervals are pretty, 5/4 and 6/5 respectively. Major and minor sixth intervals are pretty, 5/3 and 8/5, respectively. But a major sixth interval, like Do to La, if going backwards, is just a minor third, and we already knew that was pretty.
All of the above, except the tritone, make pleasant musical sounds. They are consonant rather than dissonant.
But we still have some intervals left. A second interval comes in two flavors, major and minor. A full note interval, like Do and Re, is a major second. A half-note interval like Mi and Fa, is a minor second. And they are very dissonant. They sound nice enough when following each other closely, in a melody, but when played simultaneously, they can be ear-grating, especially the very close minor-second. Looking at a table of ratios, we see that a minor second has the complicated ratio of 25/24. That's tough arithmetic! A major second's ratio is better, 9/8.
A major seventh interval (Do re mi fa sol la ti!) like Do and Ti is about 15/8. A minor seventh, (Re mi fa sol la ti Do!) like Re up to Do is 9/5, not so shabby.
But the nastiest interval of them all is the tritone, that interval halfway between a fourth and a fifth. The table shows it as 45/32, but that's fudging. Since the tritone is exactly halfway up the scale -- either one, major or minor -- and since the musical scale constitutes a quadratic function, doubling at each octave, the tritone relationship is actually the square root of 2. Which is an irrational number. Irrational, in the mathemtical sense, meaning there is no whole number ratio possible. Since Fa and Ti constitute a tritone, we can calculate that the frequency of Ti is about 1.41421356 times the frequency of Do... Aw shit, let's just pretend it's 45/32.
But what if we go backwards? Well, since it is exactly midway between the octaves, a tritone backwards is still a tritone.
All of this was my way of convincing you that a tritone is a difficult bitch. If a fifth or a fourth interval is sweet comfort food, then a tritone is a raw habanero pepper.
Bartok's Concerto for Orchestra
Bartok composed his Concerto for Orchestra (strange name, I know) in 1943. His second movement is the one I want to play now, sometimes called "The Game of Pairs." The woodwinds, in pairs, take turns playing themes based on intervals. It makes a nice illustration of things so far. They sometimes use this to teach children about the instruments of the orchestra.
The little drum motif acts as a separator for the different parts. First up, at 0:11, the two bassoons play their part in minor sixth intervals. Next, at 0:40, the two oboes play their part in minor third intervals. At 1:05, the two clarinets play in minor seventh intervals. At 1:25, two flutes play in fifths. And then there's some other stuff.
Intervals are the building blocks of chords.
Back in The Physics of Music, we talked about the three basic chords any crappy garage-band guitar player needs to learn if they want to take it -- the tonic, the dominant, and the subdominant. If we are playing in C major, then that means C major is our tonic chord, F major is our subdominant, and G major is our dominant. If you know those three chords, any others you learn are just showing off. C, F, and G. F is a fourth up from C; G is a fifth up. Fourths and fifths. What else do you need?
What is a chord, these funny things we have been talking about? A basic chord consists of three notes: a root note, a note a minor or major third up, and a note a fifth up. A C major chord consists of C (root), E (a major third up) and G (a fifth up). Change the middle note to a minor third and you have a minor chord instead. All basic review, yeah.
But why stop at three? A four-note chord might be even spicier! Let's monkey with our C major chord. C (root) + E (major third) + G (fifth) + B (major seventh) = the C major seventh chord. Tada!
C Major chord
C Major Seventh chord
Spicy! Like a chili pepper, only, well, maybe not a very hot one. Let's look at the problems it introduces. We have C+E+G+B. C+E+G is a C major chord. But E+G+B is an E minor chord. Ooooh, there's a conflict here! Plus, we have that seventh relationship in there, C+B, and we already talked about how dissonant that is. We have the delicious mindfuck of two different chords atop each other, fighting for control, plus the added dissonance of C+B. It shouldn't be a surprise then that seventh chords took some time to be fully adopted into western music.
The seventh chord (as a deliberate choice, not an accident) came on the western musical scene during the Renaissance. The little history I can dredge up associates it with Ren composer Monteverdi. Very rarely used during the Ren period, it saw increased usage during the Baroque era, more usage in the Classical era, and yet even more in the Romantic era.
It changed the game of music. And music is certainly a game. Some of its rules emerge from the inherent nature of sound waves and our ears, but another big part of it is cultural, the result of the evolution of western musical style over the centuries.
Ninth chords! Eleventh! Thirteenth!
But why stop at four notes? Why not five! Okay. Let's take our C major seventh chord and stick another note on the end of it. We can't add an eighth onto it (that would just be another C) so we have to go to a ninth interval, which is an octave and one note up. So we get C+E+G+B+D = C major ninth chord! And with it come a whole new set of ambiguities and dissonances. If we cannibalize the parts, we can make a C major or an E minor or a G major chord. Plus we have dissonance of C+D+E in there, three very close notes, all scrunched up like the Three Stooges trying to get through a revolving door. It's a habanero with mace on it!
But why stop even there? C+E+G+B+D+F = C major eleventh chord! And C+E+G+B+D+F+A = C major thirteenth chord! And about at that point, we run out of notes, because a thirteenth chord uses all seven notes of the scale. I suppose you could try to make a fifteenth chord, but you would just be repeating something. And I wouldn't blame you if you thought this was all very silly and with limited usefulness. Certainly, no garage-band guitarist would play such a monstrosity. They only have six strings on their guitars. Of course, they could always shell out for a Gibson Seven-String Guitar. "It's better because it has one more string."
Although ninth chords were rarely used before the 20th century, they have proven very useful. You can't have jazz without ninth chords. They're everywhere, like cockroaches. They define the sound of jazz and blues.
T. Bone Walker, "Mean Old World." I did some searching to find a good example of liberal 9th chords. Which chords are they? The ones that sound dissonant, a bit shriller than the others, which are all seventh chords.
For eleventh and thirteenth chords, the best example is still Maurice Ravel's La Valse, a waltz he wrote to parody the heavy-handed, over-orchestrated Germanic style of the period, well-personified by Richard Strauss. The thick elevenths and thirteenths in La Valse give it a slightly disturbing, hysterical edge.
It takes a minute before the music really takes off. And, by the way, this is one of those pieces of music you just cannot really hear right on shitty gear or with a shitty recording. There is too much stuff going on. But we must make do.
Part 2 is here.
The Diminished Seventh Chord
But let's back up to the seventh chord. In a way, the ninth chord and eleventh chord are just more extreme forms of the seventh. "Well, it's one more note, isn't it?" to paraphrase Spinal Tap. But let's back up and look at a specific form of the seventh chord now, the diminished seventh chord.
C Major Seventh chord (for comparison's sake)
C Diminished Seventh (Cdim7)
What do we see here? A C major diminished seventh. (Or, as it would be written in a guitar chord/tab book, Cdim7.) C + E flat + G flat + A. The fifth note, that so important fifth note, the note that keeps us in our harmonic comfort zone, has been substituted with G flat: the dreaded tritone! It's interesting in other ways. Like, it's symmetrical. C to E flat is a minor third. E flat to G flat is a minor third. G flat to A is a minor third. How very cute, however dissonant!
Ask yourself this: Could you make a diminished ninth chord? Add another minor third on? No, you can't! Because going up another minor third gets you back to C, where you started from. In fact, a Cdim7 is exactly the same notes as an EflatDim7 and a GflatDim7 and an Adim7. Same notes, same chord, different names. In fact, there are only three possible combinations of notes in the whole 12-tone scale that you can use to make a diminished seventh chord. So it is just chock-full of cuteness on paper.
But, ew, that tritone makes it very dissonant. More than that, it destabilizes the sense of a home key. It's a hot chili pepper, very tasty, very exciting, but when used in conventional music, it makes us yearn for some homemade vanilla ice cream to soothe our palate! It's a powerful dramatic device in music, helping to set up a minor crisis that requires resolution back to a home key, some home key, any home key, something the music-processing part of our brain stem can listen to and go, "Ah, thank God! A 3/2 ratio interval! Now we can chill!"
The diminished seventh was one of the most powerful tools in the toolkit of the Romantic composers. It was around before -- you can hear diminished sevenths in many of Bach's pieces, for instance. Mozart, too, at times. Beethoven, though, was the guy who went bananas with dim7 chords. They gave him a powerful tool for creating drama, for creating violence, for creating a sense of the mystical, for achieving climax. If you can remember any particularly tense, scary, climactic chord in a work by Beethoven (like the final thunderbolt in the storm from the Pastorale, odds are that it's a dominant dim7.
You can find dim7's in all his sonatas, but I chose as an example this one, Beethoven's Piano Sonata 32 in C minor, because the beginning of the first movement is so rich in dim7 chords. Sviatoslav Richter performing.
For a long time, I thought Beethoven and Mozart sounded too alike for me to tell one from the other. The only distinguishing factor I could see was that Beethoven was more dramatic, more angry, and Mozart was lighter-hearted, more precious. But more often than not, they sounded very, very similar to me. The real distinction between the two is the dim7. In fact, there are computer programs designed to distinguish the works of different composers by analyzing their chord usage. They find that Mozart used dim7 chords but not nearly as often, often opting for the simpler and less dissonant dim5 chord (three notes rather than four).
I didn't get as far as I had hoped today, but I didn't expect to cover everything. So where is this going, then? We're preparing to cover more advanced types of music, but here we are talking about basics, you say. We need to establish the ground rules for music as we know it before we can see how many ways that different composers found to cheat. Again, I don't want to teach you to be musicians. If this were a real music theory class, you'd be expected to whip out your instrument (your musical instrument, please!) and play these things for yourself and learn to identify them. I play a bunch of different instruments, most of them just okay, but honestly, I can't always tell only by listening a diminished seventh from a regular seventh or just a nastily played ordinary chord. So if you can't hear these things well enough to identify them, don't sweat it. Knowing the basic concept is good enough for our purposes.
As I said, music is a game. Games have rules. If you break a rule in a game, somebody else will usually yell foul, but since music is a creative game, the rules are of our own making. In fact, breaking the rules is what keeps it interesting. Just as Monteverdi livened things up with seventh chords, Mozart livened things up with his chromatic arpeggios, Beethoven livened things up with his gung-ho dim7 chords, later composers, even in the Romantic period, found new ways to break the rules.
Maybe you can also see a trend, developing here, of increased dissonance. That nasty tritone, for instance, being substituted for our lovely, ice-creamy perfect fifth! Some of the things done with dissonance in music can be quite off-putting if you aren't familiar with it. But with a little preparation and warning, maybe we can realize that it's just a new dish, one with less carbs and more chili peppers.
The period of western music from the Renaissance up to the beginning of the 20th century is called The Common Practice Period, because our understanding of how the game of music is played was fairly standard throughout the period. Beginning in the mid-nineteenth and then accelerating in the 20th century, rules that had been sacrosanct were violated. We'll look at some of that in coming diaries.
Next week: If I can swing it, more chords, more dissonance, chord progressions, Roman numeral notation, cadences, and the basic principle of tonality. And probably some Wagner.