The Final Product

So I made the guitar. The guitarbuilding group make it hard to make a bad guitar, with the beautiful materials they provide, and their expert instruction, however, I’m inordinately proud of myself as well.

Indeed, as more and more of the elements fell into place over the course of the week, it really brought home the affective power of a) building something with your own hands, and b) the iconography of the electric guitar.

Now I have to figure out the logistics of doing this at Fulton. But as the workshop instructors pointed out, even if you don’t have students build one, just bringing the electric guitar into the classroom and saying, “Today we’re going to study sound,” really catches the attention.

Necks, Fretboards, and Scale Length

Pluck a string on a guitar and the sound you hear depends on how fast it vibrates. The frequency is how many times it vibrates back and forth in each second. An A4 note has a frequency of 440 vibrations per second (one vibration per second is one Hertz).

The vibration frequency of a guitar string depends on three things:

the mass of the string
the tension on the string (how tight it’s pulled)
and, the length of the string.

Guitar string sets come with wires of different masses. The guitar has little knobs on the end for adjusting the tension. For building the guitar, you have the most control over the last last parameter, the length of the string, which is called the scale length. Since the guitar string masses are pretty much set, and the strings can only hold so much tension, there are limits to the scale length you can choose for your guitar.

In a guitar, the scale length only refers to the length of the string that’s actually vibrating when you pluck the string, so it’s the distance between the nut and the bridge. For many guitars this turns out to be about 24.75 inches.

For a guitar, the scale length is the length of the strings that are free to vibrate.

Frets

To play different notes, you shorten the vibrating length of the string by using your finger to hold down the string somewhere along the neck of the instrument. The fret board (which is attached to the neck) has a set of marks to help locate the fingering for the different notes. How do you determine where the fret marks are located?

Well, the music of math post showed how the frequency of different notes are related by a common ratio (r). With:

$r = \sqrt[12]{2}$

So given the notes:

Note Number (n)	Note
0	C
1	C#
2	D
3	D#
4	E
5	F
6	F#
7	G
8	G#
9	A
10	A#
11	B
12	C

Since the equation for the frequency of a note is:

$f_n = f_0 \; r^n$

we can find the length the string needs to be to play each note if we know the relationship between the frequency of the string (f) and the length of the string (l).

It turns out that the length is inversely proportional to the frequency.

$l = \frac{1}{f}$

So we can calculate the length of string for each note (l_n) as a fraction of the scale length (L_s).

$l_n = \frac{1}{f_n}$

substituting for f_n gives:

$l_n = \frac{1}{f_0 \; r^n}$

but since we know the length for f₀ is the scale length (L_s) (that inverse relationship again):

$l_n = \frac{1}{\frac{1}{L_s} \; r^n}$

giving:

$l_n = \frac{L_s}{r^n}$

When we play the different notes on the guitar, we move our fingers along the neck to shorten the vibrating parts of the string, so the base of the string stays at the same place–at the bridge. So, to mark where we need to place our fingers for each note, we put in marks at the right distance from the bridge. These marks are called frets, and we’ll call the distance from the bridge to each mark the fret distance (D_n). So we reformulate our formula to subtract the length of the vibrating string from the scale length of the guitar:

$D_n = L_s - \frac{L_s}{r^n}$

The fret marks are cut into a fret board that was supplied by the guitarbuilding team, which we glued onto the necks of our guitars. We did, however, have to add our own fret wire.

Placing the fret wire into the fret cuts. The wire still needs to be fully pressed in.

The team also has an activity for students to use a formula (a different one that’s recursive) to calculate the fret distance, but the Excel spreadsheet fret-spacing.xls can be used for reference (though it’s a good exercise for students to make their own).

The Math of Music

Mark French has an excellent YouTube channel on Mechanical Engineering, including the above video on Math and Music. The video describes the mathematical relationships between musical notes.

Given the sequence of notes: C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C.

Let the frequency of the C note be f₀, the frequency of C# be f₁ etc.

The ratio of any two successive frequencies is constant (r). For example:

$\frac{f_1}{f_0} = r$

so:

$\frac{f_1}{f_0} = \frac{f_2}{f_1} = \frac{f_4}{f_3} = \frac{f_{12}}{f_{11}} = r$

We can find the ratio of the first and third notes by combining the first two ratios. First solve for f₁ in the first equation:

$\frac{f_1}{f_0} = r$

solving for f₁,

$f_1 = f_0 \; r$

now take the second ratio:

$\frac{f_2}{f_1} = r$

and substitute for f₁,

$\frac{f_2}{f_0 \; r} = r$

which gives:

$\frac{f_2}{f_0} = r^2$

We can now generalize to get the formula:

$\frac{f_n}{f_0} = r^n$

or

$f_n = f_0 \; r^n$

where,

n – is the number of the note

From this we can see that comparing the ratio of the first and last notes (f₁₂/f₀) is:

$\frac{f_{12}}{f_0} = r^{12}$

Now, as we’ve seen before, when we talked about octaves, the frequency of the same note in two different octaves is a factor of two times the lower octave note.

Harmonic_octaves — Click the waves to hear the different octaves. The wavelengths of the sounds are shown (in meters).

So, the frequency ratio between the first C (f₀) and the second C (f₁₂) is 2:

$\frac{f_{12}}{f_0} = 2$

therefore:

$\frac{f_{12}}{f_0} = 2 = r^{12}$

so we can now find r:

$r^{12} = 2$

$r = \sqrt[12]{2}$

Finally, we can now find the frequency of all the notes if we know that the international standard for the note A4 is 440 Hz.

Mark French has details on the math in his two books: Engineering the Guitar which is algebra based, and Technology of the Guitar, which is calculus based.

The Rolling Shutter Effect

When you pluck a guitar string, the string moves up and down really fast. However, if you take a video of it with a digital camera with a rolling shutter (which most cameras have at the moment) it captures the motion of the string in a wavelike pattern that is proportional to the frequency of the motion of the string; the smaller strings move faster, create a higher pitched sound, and shows up as shorter-wavelength waves. Note: this is not the way the strings actually move, it’s an interesting, and potentially useful optical effect.

Because the optical effect really makes it look like there are a lot of internal waves rolling along the string — which there are not — I’d be quite cautious about using this in physics class. However, if a student wanted to go into the detail to understand how it works — and then explain it to the class, they can start with the math about standing waves in instrument strings and the relationship between sound pitch and wave motion, and a visual explanation of the rolling shutter effect:

The Sounds of Space

Charged electromagnetic particles from the solar wind interact to create sound in space.

You can also listen at radio-astronomy.net.

DarwinTunes: Watching Music Evolve

Take randomly generated sound waves (using sine curves for example), mix them together to get beats, and then let people decide which ones sound best. Let the best ones mate — add in small mutations — and wait a few thousand generations for the sound patterns to evolve into music.

That’s what DarwinTunes does, and they let you participate in the artificial selection process (artificial as opposed to natural selection).

The details are included in their article: Evolution of music by public choice by MacCallum et al. (2012).

Generating (and Saving) Tones with SoX

I’ve been using the command line program SoX to generate tones for my physics demonstrations on sound waves.

Single frequency tones can be used for talking about frequency and wavelength, as well as discussing octaves.

Combine two tones allows you to talk about interference and beats.

SoX can do a lot more than this, so I though I’d compile what I’m using it for in a single, reference post. For the record: I’m using SoX in Terminal on a Mac.

Using SoX

To play a single note (frequency 173.5 Hz) for 5 seconds, use:

> play -n  synth 5 sin 347

To save the note to a mp3 file (called note.mp3) use:

> sox -n note.mp3 synth 5 sin 347

The SoX command to play two notes with frequencies of 347 and 357 Hz is:

> play -n synth 15 sin 347 sin 357

and to make an mp3 file use:

> sox -n beat_10.mp3 synth 15 sin 347 sin 357

Listen for the Beat

beats — Two sound waves with slightly different frequencies sometimes cancel each other out (destructive interference) and sometimes add together (constructive interference) to create a sound that gets loud and quiter with a beat. The two lower sound waves (green and blue) are out of phase, and their combination (superposition) creates the third (red) wave.

Play two sound tones that are close together in frequency and the sound waves will overlap to create a kind of oscillating sound called a beat.

beats-2-anno — When you hear the beat (see below), you're hearing the alternating of the high amplitude region and the low amplitude region.

Below are two tones: separated and then mixed — listen for the beat.

	Frequency	Sound File (mp3)
Tone 1	347 Hz	1m.mpg
Tone 2	357 Hz	1m-357.mp3
Mixed Tones (with beat)	347 Hz + 357 Hz	beat_10.mp3

Interestingly, you can sometimes hear the beat as a third tone if the frequency difference is just right. The frequency of the beat is the difference between the frequency of the two tones.

Notes

The SoX command to play two notes with frequencies of 347 and 357 Hz is:

> play -n synth 15 sin 347 sin 357

to make an mp3 file use:

> sox -n beat_10.mp3 synth 15 sin 347 sin 357