Chords and Scales

Sound Waves and Overtones

I think most people have at least a rough understanding that what we think of as "sound" is just a membrane in our ear that vibrates and sends that signal to our brains.  Those vibrations are caused by waves that propagate through the air.  That's probably enough detail for our purposes here.

Have a look at a piano string.  How does it make sound?  A wooden hammer from inside the piano hits it, and it vibrates.  This causes the air around it to vibrate, and that wave propagates to our ears.  How does a clarinet make sound?  By making a thin piece of wood called a "reed" vibrate.  How does a trumpet make sound?  The player actually vibrates his lips.  How does a voice make sound?  Through vibration of vocal cords.  And so forth.

Now, draw a wave on a chalkboard.  Suppose this is the wave that we hear as a "C".

  

Now play the C one octave up.  Does it sound like the same note?  I mean, yeah, it's higher, but something about that note sounds the same as the lower C.  The reason we think this is because the wave of the higher C is exactly half as long as the lower C.

As it turns out, piano strings and pretty much any other musical instrument have some small imperfections that cause little sub-vibrations for that higher C.  This is called an "overtone."  Try the following experiment and listen for yourself.  Push the key for the higher C down, not so hard that it plays the note, the goal is just to release the little pads (called "dampers") that normally hold the string still.  We want that higher C to vibrate freely any time it comes in contact with another wave that it matches.  OK, now play the lower C, very loud and very short so that the damper holds it still.  You should hear a faint vibration of that higher C.

You might be thinking "yeah of course it's vibrating, the damper isn't holding it steady."  Ah, well, try this experiment again, except use the B right below the higher C.  Strike the lower C key again.  You should hear nothing.  Now strike the lower B key.  Hear it now?


Go back to the C keys.  Now shift everything up an octave.  Hold down the key 2 octaves above middle C and strike the key 1 octave above middle C.  Now ask the mathematicians in the room:  if the first C is 1/2 the wavelength of middle C, and this higher C is 1/2 the wavelength of that, what is the high C in relation to middle C?  What's half of a half?  A quarter.  1/4.  Turns out, that middle C is also generating those 1/4 waves.  Try causing the 2-octave C to vibrate by striking middle C.  Still there, but not quite as loud, is it?

Now wait a minute.  We've got 1/2, and we've got 1/4.  What happened to 1/3?

Yup, it's there too.  So what note does it sound like?  Well, it's got to be something in between 1/2 (the one-octave C) and 1/4 (the two-octave C) right?  Because 1/3 is larger than 1/4, but not as large as 1/2.  Well there aren't any C's in between them, it must be another note.  It's the G.  Try causing the G to vibrate by striking middle C.  (Not the G immediately above middle C, the one that's an octave up from that.)  Hear it?

So now, we've got 1/2, 1/3, and 1/4.  For the sake of completeness let's draw its wave:

By now, everyone's got the idea and are probably asking about 1/5.  That's the E above the 2-octave C.  If you try the experiment, you may hear it but it's getting really faint.  As the overtones get higher, they also get fainter until you can't really distinguish them any more.

Let's just do one more step, shall we?  1/6.

What note is that?  Well, 1/6 is half of 1/3.  1/3 is a G, so 1/6 is the next G above that.  What about 1/7?  This one is unfortunately not on the piano, but we're already getting pretty faint so let's just not worry about that one.  1/8 of course is half of 1/4, so it's the C three octaves up, and so forth.

Music Theory for Ward Choir

Sarah got called to be the ward choir director around the end of January.  Of course, her due date was soon after, so she immediately had to ask for subs.  I filled in for one of the weeks, and I've always thought that ward choir shouldn't just be for people who already know music.  I mean, we always say "everyone is welcome, no experience required" but then we give them music for which they have to learn parts that aren't just the melody that everyone knows.  So I spent part of the time going over some music theory.  Nothing high-level, mind you, just some stuff I thought they might find interesting about how music works.  When Sarah found out, she rolled her eyes at me and mumbled something that sounded like "nerd" but then asked them next week if they thought that was useful.  She got mostly positive answers.  So she's continuing the music lessons.  I thought it might be worth writing down the stuff we've been teaching here.  This is still a rough draft and we may change things around as we go.

Chords and Scales

1. Sound waves and overtones
2. The major scale
3. Circle of 5ths
4. Key signatures and accidentals
5. Tonic, dominant, and sub-dominant

Rhythm

1. 4/4, 3/4, 2/4, and downbeats
2. Whole notes, half notes, quarter notes, eighth notes, sixteenth notes (also triplets?)
3. Dotted notes and ties
4. Upbeats
5. Basic conducting