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In Opus 1, we announced our (royal we) intentions to look at the forest rather than the trees, so we could get a grasp of a whole work of music rather than let the pieces flit by, just a pretty wash of disconnected images and sounds.

Today, though, despite what I said, we ARE going to look at trees.  Okay, wait, not trees.  Pine needles!  In fact, we're going to get out a microscope and look at the beads of resin on pine needles!  We had to leave some questions unresolved after our first lesson, analyzing Mozart's Magic Flute Overture.  So let's get small.  And let's let Donald Duck lead the way!

I don't know how many times I saw that as a kid.  At Jane Adams Elementary School in Lakewood, California, when it rained, it was a traumatic event for everybody.  The outdoors lunch-baggers had to eat indoors with the tray-lunchers, but space being at a premium because of the Baby Boom, not everybody could be accommodated at once.  So school lunch, on such days, became a two hour or longer affair wherein we were all packed into the school auditorium and called out to eat lunch in shifts.  It was humid and noisy and disorganized.  When we got too rowdy, the authorities threatened to make us sit boy-girl-boy-girl, a terrible, terrible thing.  So they showed Donald Duck in Mathemagic Land while we waited.  And they showed it and they showed it!

In the film, Pythagoras demonstrates to Donald that there are whole number ratio relationships between notes of music.  Pinching the lyre string in certain ways gives us different notes, but pinching it in simple ratios, like 2/3 or 3/4 produces more interesting, musical sounding notes than just random pinches.

The Physics Hypertextbook puts it very elegantly:

The distinction between music and noise is mathematical form.

All sound is the result of vibrations in the air.  A single note of music is a single frequency of vibrations.  But musical relationships between those notes are mathematical.  They aren't random.  It's possible to make music out of random noises, like the Beatles' Revolution Number 9, but that is an aesthetics game.  What makes music magical and different is the way the hardware in our brains and auditory nerves manages to process those sounds and ferret out distinct frequencies and organize them into relationships without our having to understand any of this.  

But let's try to understand anyway, so we can increase our understanding for fun things to come in future opuses.  (The real plural for opus is opera, but let's not confuse issues, eh?)

Let's talk about Do Re Mi.  (Doe, a deer, a female deer... la la la)  If you ever had chance to sit down to a piano, you, a complete novice, just as likely to break it and get yelled at as not, you probably figured out pretty quickly that the white keys are the nice ones and the black ones are evil.  If you wanted to plunk out something simple, like Chopsticks or Mary Had a Little Lamb, you did it using the white keys, and on a piano, that means C Major.

[The Julliard grads reading this might be bored about now, but cut the rest of us some slack.  We have to start with the basics before we can explain ninth chords and Phrygian modes and whole-tone scales.]

I had hoped to make a youtube to make this easier, but I haven't mastered the youtubing arts yet, so we'll have to do it this way.  Of course, everybody knows Do-Re-Mi, so we don't need to play that.  Let's look at the C Major chord, then.  To play a C Major chord (or just C chord), you press down on the C, the E, and the G keys.  Do + Mi + Sol.  

It sounds very nice!  Why?  Mathematical ratios.  The sound frequency for middle C on a piano is 262 Hertz (vibrations per second).  The frequency for the next higher C, the one seven notes to the right on the keyboard above, is twice that, 524 hertz.  But G (Sol) has one of those nice whole-number ratios.  It is three halves of 262, 392 hertz.  3/2 is about as simple a ratio as you can get for two different musical notes.  What else has a nice easy ratio between 1 and 2?  4/3.  And that ratio gives us the frequency of F (Fa), 349 hertz.  How about one more easy ratio?  5/4 is the ratio that gives us the frequency of E (Mi), 330 hertz.  

There is a nice table here showing the ratios between all twelve notes of what we call the just chromatic scale, the one that Pythagoras was demonstrating for Donald.

So the C chord has C and E and G... why not C, F and G?  F has an even sexier looking ratio than E, doesn't it?  F was 4/3, while E was 5/4, right?  Well, try playing it on a piano, and you'll notice, it doesn't sound as pretty.  It doesn't have that same oomph.  And the problem here is that F and G are too close together.  The ratio between F and G is 9/8.  But the ratio between E and G is 6/5.  So these three notes, C, E and G, (Do Mi Sol) give us the most elegant set of mathematical ratios between three different notes.  No brilliant mathematician invented this system and foisted it upon us.  It's based on the nature of audible sound itself and the way that our ears hear it, the way our brain processes it.

In 1977, NASA launched the Voyager probe, which, after surveying the planets of our system and sending back many beautiful pictures, left our solar system forever and is now on its way into the great woowoo beyond.  Inside the Voyager, there is an LP record made of gold which carries greetings from Earth in a host of different languages, as well as samples of Earth music.  You can listen to it here.  

On the unlikely (it seem to me) chance that it should ever be discovered by another race of intelligent alien creatures, what would they think of it?  Assuming they did find it, and they determined that the grooves represented some kind of vibrating noises of importance to us, the people who created it, what are the odds that they would process sound information the same way that humans do?  The beautiful mathematics of the C Major chord would probably be beyond the capability of perception of such creatures.  They could analyze the patterns, the forms, the ratios, and they might find something interesting in beautiful in that, the same way some people find beauty in the formula for pi.  But that's not the same thing.  This ability to perceive musical beauty that we have is something basic to the way we humans are built.

How to Fake Your Way to Superstardom with Three Chords

When I gave my daughter her first guitar, I wasn't going to be around to instruct her, so I told her this:  All you really need to know how to do is play THREE CHORDS!  Three chords, and you can fake your way through thousands of songs, including most folk music.  All you have to do is strum the guitar with one of the three chords while you sing the song.

The piano is a little harder.  But since we started the discussion using piano chords in C, let's continue that way.  On a piano, the three chords that will let you fake your way to stardom are the C chord, the F chord, and the G chord.

C chord = C + E + G
F chord = F + A + C
G chord = G + B + D

Since a guitar is easiest to play in G instead of C, the three magic chords are different.  But just for kicks, you can look at this list of guitar chords and lyrics for miscellaneous children's songs.  They all use the same three chords, and some of them just two.  And the funny thing is, songs that require more than three often sound just as good with three if you smile big while playing them.

Songs are generally played according to a specific scale based on a home note.  That home note is called the key.  For instance, Do-re-mi, is in a way, a kind of song of its own, in the key of C as illustrated above, because C is the first note, the home note, the note that everything else resonates with and leads back to.  You can play other chords, like F and G, while you are in C, but the song is still C.  

When we are in C, the chords C, F, and G acquire special names, and these are important, gonna be on the test double-damn-guaranteed.  

The C Major chord (while playing a song in C major), is called the tonic.  The tonic chord means the same chord as the key of the song.  The tonic is obviously the chord that matters the most.  It's what you almost always must start with and end with, although there are rare, fancy-schmancy exceptions.

The G chord, (while playing a song in C) is called the dominant.  It is the second most important chord in music.  If there is a song that doesn't require it (I can't think of one off the top of my head), it's hardly a song at all.  Even Do-re-mi-fa-sol-la-ti-do, if you play the harmony for it, requires a dominant chord to sound right.

The F chord, (again, while playing a song in C) is called the subdominant.  And I've always thought that a curious name, too.  Is it a sub?  Is it a dominant?  Why can't it make up its mind?  The subdominant is a little less important than the dominant.  There are plenty of songs with no subdominant chord.  

There are plenty of other important chords, but those are the three biggies to know if you want to stand there with a guitar on American Idol and con people into thinking you know what you're doing: The tonic, the dominant, and the subdominant.  There are lots of other chords with their own fancy names you may want to learn, especially if you take up an instrument.  But I'm not trying to make you a musician.  I'm laying the groundwork for understanding music, not just music in general, but specific pieces of music that we are going to hear in the future.  There were questions left unanswered in the OPUS 1 lesson about Mozart's Magic Flute that we can address better next time.

So why these three chords?  What makes them so special that they pop up everywhere with such frequency, that they need special names?  

Here we go back to ratios.  The two notes with the simplest ratios to C were G (3/2) and F (4/3).  There is something rather magnetic about those notes that draw our attention to them.  They want to steal the spotlight away from C, the tonic, the master of the scale, the alpha and omega of the scale!  In a song, the music oftentimes temporarily just abandons C for another key, something more exciting like F and G.  In this case, it doesn't just change chords, but changes scale, with whole Do-re-mi schema being shifted several notes to the left or right on the keyboard.  

Let's look at the G major KEY, the dominant of C, for a moment.  If you were to play in G, this would now be your Do-Re-Mi.

Do is now G, Re is now A...  What happened to Ti?  Well, it had to become one of those evil black keys.  Likewise, the scale for the F Major key has its own black key, with the B key reduced to B flat.  Any other major key other than C that we try to play in is going to require that we leave the lazy comfort of the white keys in some way.  

But still, it's pretty close to the same, isn't it?  You only have to learn one black key.  It could have been worse.  It could have been B Major, in which case Do-Re-Mi would require we play five black keys and three white keys.  Dreadful.  And I'm running short of time so I can't make a picture of B Major for you.  Trust me on this.  B Major is the darkest, most evil circle of Hell.

But let's see.  G and F are both harmonically close to C, as shown by their ratios, and they have a lot of other notes in common.  It makes sense, does it not, that they would have a seductive quality to them that leads us astray from safe, sound, C Major?  Indeed they do,  First they lure us with their beautiful three-note chords, , then one thing leads to another, and we wake up in bed next to them with a hangover!  How did I get here? you ask yourself, as you try to slide out of bed without waking up this strange key.  Nope, no coffee, gotta get to work, I'm late as it is, besides, didn't you tell me you were just a subdominant?  You're nice, you deserve better than me, and I've got a beautiful C major waiting for me at home, I'd leave you my card, but I'm fresh out, so, oops, look at the time!

And that, my friends, [rewinding us all the way back to OPUS 1] is why the Magic Flute Overture starts in E-flat Major (the tonic), migrates to B-Flat Major (the dominant of E-flat) during the exposition, then gets lost in a host of strange keys before finally returning to E-flat major again in the recapitulation, properly chastened and bearing flowers and a shit-eating grin.  

Originally posted to Dumbo on Thu Jul 15, 2010 at 05:08 PM PDT.

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