Theoretically Speaking: S1E2

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Music Theory For Non-Musicians

This occasional series is about how music is made, and it’s for people who don’t already make music. It’s part music appreciation and part music theory.
I hope to cover rhythm, melody, intervals, chords, inversions, and more. Maybe we’ll get into extended chords and modes. Let’s see!


#S1E2: Rubberband Man

Go get a rubber band. I’ll wait right here..

Rubber band

Got it? OK, put both your index fingers through it and slowly pull your hands apart so you stretch the rubber band. As you stretch it, pluck it with one of your thumbs. Hear how the pitch changes? It gets higher as your hands get farther apart and lower as they get closer together.

What you have there, my friends, is a string instrument. If you practice, you can play melodies with it. Simple melodies, sure… but you can do it.

Strings vibrate. That vibration vibrates the air around the string, and our ears pick up the vibrating air. The air is crucial. In space, no one can hear you scream, or play banjo.

The pitch of a vibrating string depends on three things: the tension the string is under, the length of the string, and the thickness of the string. Let’s talk about them in that order.

This is a picture of the back of the headstock on one of my basses. The headstock is the part at the end of the neck, and it’s where the tuners are.

Fender Precision Bass Headstock Rear View

Notice how each of the tuners a cog gear meshed with an auger gear. When you twist the flat piece of metal shaped like a clover at the top, the auger spins. As the auger turns the cog, the post on the other side of the headstock tightens the string wound around it.

Music Theory Fender Precision Bass Headstock Front View

The tighter the string, the higher its pitch, just like we heard with the rubber band.

Tuners like these are called machine heads. I suppose that’s because they have gears like a machine. Some machine heads have the gears exposed, like on this bass, and some have them enclosed in a little box. That might be to keep the gears cleaner, but I’ve never heard of anyone having a problem with dirty gears. Maybe it’s just for looks.

Violins, cellos, and various other string instruments use tuning pegs instead of machine heads. Pegs are carved pieces of wood (or molded plastic on cheaper instruments) that have a slightly conical shape. They fit into holes in the headstock, which is known as a scroll box on violins. Once the string is tight enough to play the correct pitch, you push the peg snug into the holes to hold the string at that tension.

Pegs are much harder to tune than the more modern machine heads, but they’re traditional, lightweight, and low tech.

Music Theory Violin Tuning Pegs Headstock

Unlike the rubber band, musical instrument strings have no elasticity, so we can take that out of the equation. When you stretched the rubber band, the pitch went up. That’s a function of its tension, not of its length.

The strings fit into slots near the tuners, and the bar with the slots is called the nut. There are more slots at the other end of the strings, and that’s called the bridge.

The longer a string from the nut to the bridge, the lower its pitch. That’s why violins play higher pitches than the larger cello. 

It’s also why string instruments can produce so many pitches. When the musician holds the string against the fingerboard, the vibrating part of the string is the length from the musician’s finger to the bridge. The closer the finger is to the bridge, the shorter the vibrating part of the string, and therefore the higher the pitch.

You may ask what this has to do with music theory. We’re getting there, but there are a couple more mechanical things to talk about.

Stringed instruments can be categorized as fretted or fretless. (Also acoustic and electric, but that’s another story.)  In the picture below, the bass on the left has frets, which are the metal wires going across the width of the neck. The bass on the right doesn’t have those wires. It’s fretless. 

Music Theory -Fretted vs. Fretless

On a fretted instrument, the distance from each fret to the bridge is exactly the distance needed for each particular pitch.
So when a musician puts her finger on the fingerboard, the string is suspended from the next closest fret to the bridge. And that’s the length of the vibrating string, thereby determining the pitch. It’s very precise.

On a fretless instrument, the finger must hold the string at the exact spot for the pitch to be correct. If the musician isn’t accurate, the note will be out of tune. It takes a lot more practice than a fretted instrument.

The orchestral string instruments – violin, viola, cello, and double bass – are all fretless. The guitar family – lute, ukulele, banjo, mandolin, dulcimer, etc. – are fretted. Some instruments, like the bass guitar, can be either.

Keep this in mind when listening to a classical virtuoso like Hilary Hahn or Yo-Yo Ma, or a jazz player like Jaco Pastorius whose fretless instrument was known as The Bass Of Doom. They’re playing with nuance, speed, and dynamics, all the while getting their fingers on the exact millimeter of string for each note, without the help of frets. It’s astounding. 

In the photo above, notice the fretted bass has dots on the neck every so often. These are called fret markers and they help the musician know where on the neck they’re placing their fretting hand. Also notice that one of the frets has two fret markers. There’s a good reason for this:

This part is more physics than music theory. Vibrations are waves. Sound is vibrating air, which is why we call it “sound waves.” Those waves are much easier to see on a string instrument than on a wind instrument. We can actually see the waves on a string.

When a string is first plucked, it’s a single arch from the bridge to the nut. But as it decays, it divides in half and becomes a wave with one nadir and one peak. This diagram is obviously exaggerated but it shows the basic idea.

Music Theory- Sound wave bridge nut

The straight dotted line indicates where the string is at rest, and the blue line is the vibrating string. See that point in the middle where the two lines cross? It’s exactly halfway between the bridge and the nut. If the musician were to put her finger at that precise point and pluck the string again, she would get the same note but an octave higher. 

Let’s say it’s the A string. Plucking it open, meaning to not fret it anywhere, would produce an A. Plucking it while holding it down at that halfway point would produce an A an octave higher. That spot, on a fretted instrument, is indicated with the two fret markers.

As the string’s vibration decays some more, the wave subdivides into thirds, then quarters, and so on. Each of these wave-within-waves produce their own pitches, though they’re much quieter than the original note. The original is called the fundamental, and the others are called harmonics. This is why the sound of a note changes slightly over time. You’ll hear the fundamental throughout, and the harmonics as they come and go.

That’s part of what makes an orchestra’s string section sound so full. When you get twenty violins, ten violas, eight cellos and six double basses, and each of those forty-four instruments has its own physical characteristics, the multitude of harmonics will overlap each other and form a huge, full-spectrum sound.

Composers know this, and they’ll use it to evoke the emotion they want to portray.

Listen to Symphony No. 7 in C major by Sibelius. It’s only 22 minutes long but it captivates as it changes from one emotion to another. There’s a section around the 12 minute mark where the strings go from anxiety to relief. Perhaps it should be subtitled “Jean’s Mood Swings.”

This is only possible because Sibelius understood how the strings and their harmonics would interact with each other and with the rest of the orchestra, particularly the brass instruments in this case. Composers understand music theory, and do more than just come up with melodies. They need to know every instrument thoroughly. 

At the very least, they need to know every instrument’s lowest note and highest note so they don’t write a part out of the instrument’s range. However, the greatest composers understand and take advantage of every instrument’s particular qualities and strengths, their tone and timbre, their volume potential, and their limitations of all of the above. Like a painter who mixes his own paints, a great composer thoroughly knows every instrument and can use the right combination of their abilities to express any emotion.

With electric string instruments, there’s no end to the possibilities.

Musicians can run them through all sorts of electronic effects to change their timbre, volume, and even pitch. Electric guitars can sound like church organs, if you have the right equipment. It’s really just a matter of taste.

We didn’t get to how a string’s thickness helps to determine its pitch. This is partly because I don’t quite understand the physics of it. All I know for sure is the thicker the string, the lower its pitch, but I can’t explain why. If you can, please tell me in the comments. 

There’s much more to cover when it comes to string instruments. Vibrato! Bends! Whammy bars!
We’ll get to these all in a future article.

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Bill Bois

Bill Bois - bassist, pie fan, aging gentleman punk, keeper of the TNOCS spreadsheet:
https://docs.google.com/spreadsheets/d/138BvuV84ZH7ugcwR1HVtH6HmOHiZIDAGMIegPPAXc-I/edit#gid=0

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ArchieLeech
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May 6, 2022 1:39 pm

OK, so every class has That Student, the one who tells himself he’s helping the teacher when, really, he or she is just showing off. Please to introduce myself, I’ve always been That Student. (You know how to reach me if I say too much, BIll.)

I just want to point out that perhaps the closest direct relative to the rubber band as a musical instrument was the old washtub bass, also know as a gutbucket. A washtub as a resonator, a broom handle, and a string which could be stretched or slackened to change pitch – the player could also manipulate the length of the string by squeezing it in different places, like more sophisticated string instruments. Check out this guy!

https://www.youtube.com/watch?v=0HS922HEhIo

I probably shouldn’t mention that slide guitar players who use a bottleneck or a steel tube on a finger on their left hand are, in effect, using a movable fret. Think about it, you’ll get it…

One of the great mysteries of the world is how all music can be quantified as numbers. It’s not just an egghead abstraction – the way all these things work together can make you smile, cry, or dance. It all fits together. Bill’s doing this exactly right – I’m looking for to each installment.

mt58
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mt58
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May 6, 2022 4:29 pm
Reply to  ArchieLeech

Co-opting one of my old bits from myself:

It’s The tnocs.com Vocabulary Challenge!

We all have 72 hours to shoehorn the word “gutbucket” in a conversation. Context is not required, and the honor system is in effect. Report back on Monday, and good luck.

Phylum of Alexandria
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May 9, 2022 2:24 pm
Reply to  mt58

Gonna be drinking moonshine tonight. Time to bring out the gutbucket…

mt58
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mt58
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May 6, 2022 6:33 pm
Reply to  Virgindog

Absolutely.
I’m on a plane right now. I have to figure out if I should casually drop it in conversation to my seat mate, or go with the flight attendant.

If I’m detained upon landing, well, let’s just say I took one for the team.

Pauly Steyreen
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May 7, 2022 11:46 am

Here’s an explainer about the physics behind thicker strings creating lower pitch.

https://physics.stackexchange.com/questions/454866/how-does-the-thickness-of-a-string-affect-the-frequency

Phylum of Alexandria
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May 10, 2022 12:09 pm
Reply to  Pauly Steyreen

Nice, thanks.

I also found this video helpful in thinking about the physics behind timbre.

https://www.youtube.com/watch?v=JsVqb6PAA1k

dutchg8r
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dutchg8r
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May 7, 2022 12:47 pm

Nom-nom-nom…

I feel like Audrey II in Little Shop of Horrors –

FEED ME, VDOG!!! Must learn more!!!!

Excellent write up.

mumchance
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mumchance
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May 8, 2022 12:39 am

Thanks for the explainer, VDog. I had never really understood how the effect of having large numbers of strings in an orchestra worked; that was really fascinating. I took guitar for an entire semester(!) in high school (spoiler alert: I was not very good, and it made my fingers sore), and one thing I’ve always wondered is why chiming a string produces a higher sound (one octave up, IIRC?). Do you know? 

mumchance
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mumchance
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May 9, 2022 12:06 am
Reply to  Virgindog

Yes, that thing!

mumchance
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mumchance
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May 9, 2022 6:35 pm
Reply to  Virgindog

Very cool, thank you!

Phylum of Alexandria
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May 9, 2022 2:22 pm

Awesome! Very interesting, thanks.

I still don’t quite understand what it is that contributes to timbre. Is it variations in harmonic interactions, fluctuating across time? Is there a special role for intermodulation harmonic frequencies (the sums and differences of two or more fundamentals as well as their integer harmonics) that contribute to those distinct qualities?

Phylum of Alexandria
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May 10, 2022 8:11 pm
Reply to  Virgindog

Thanks. It seems my idea of intermods was wrong; apparently that’s a source of distortion. Here’s a video for guitar players:

https://www.youtube.com/watch?v=jvV_ga3X0Cs

Dance Fever
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May 11, 2022 10:35 pm

Sorry to be posting so late, Bill Bois, but as I explained on another post, these are hectic days at our school. I showed your post to our resident music teacher(35 years in the business of teaching) and he gave you an A. Don’t know if that will help your post grad grades any but it shows you know your stuff!

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