For several years I have used the frequencies of sounds to give students practice using exponents. This year I did things backwards: we’ve been studying exponents in various abstract ways first, and then as a sort of conclusion / practice / experiment / “hands-on” kind of thing brought out the tuner.

For this 1-hour lesson you will need:

- A computer per group of students,
- A microphone per computer,
- G-tune,
- Either the money to buy g-tune, the gall to use the demo version in class, or an alternative piece of software, and
- A worksheet much better than this one (docx).
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Suggested Lesson Activity (50 minutes)

**5 minutes**: have a student read the first part of the first question aloud. There is a lot to read here – too much to give to each group to read, I think. After “the chromatic scale,” have groups turn on g-tune and attempt a chromatic scale. This will be fun and hilarious for them and perhaps more so for you. Make sure each group has g-tune working and can identify the notes that it is displaying. Surprisingly, many groups could do this simultaneously – my groups were less than 10 feet apart and did not suffer too badly from interference from nearby singing.

**10 minutes**: have a student read the next part of the first question. Stop after “number to the note,” or whatever you have rewritten it to be (please do rewrite it). Explain that your goal for the class period is that the students will be able to find a mathematical pattern in the musical scale that will allow them to predict the frequencies of different notes (and the notes at different frequencies if you’re studying logs too). Then let each group read the discussion questions aloud and decide on their numbering system. Today I had groups start with 0 at C4 and C3. Another group started with 0 at G#5 and go backwards, and another group started with 1 at A2 and go up by 0.1 (so C3 was 1.3). Of course the scale does not matter as long as it’s linear, so whatever will be easiest for them to work with will be best. Probably the easiest scale would go up by twelfths, but of course you will not tell them that.

**10 minutes:** students will find the specific frequency of ten different notes. After they have done this and recorded their data in duplicate or triplicate, you (the teacher) can take their sheets and swap them between groups to increase measurement speed. You are now done with my worksheet, and luckily have added on to it the last half of this lesson, which I did not have time to do before class.

**10 minutes**: Ask groups, “do you see any patterns in the data?” Your rewritten worksheet says something like “look for patterns in the data.” If they don’t see any, ask how they can look for patterns in numbers. They should be thinking of strategies like “make a table,” or “make a graph.” You might be extra-direct and prompt them to look for a relationship between C3, C4, and C5, or F3, F4, and F5, etc. Today in my class 100% of students noticed that the frequencies approximately double between notes one octave apart.

**10 minutes:** Ultimately you want to be able to bring this back to exponents and logs. I had students graph their data on geogebra, and I asked questions like “what would the frequency for C6 be?” and “what note would be at 1000 Hz?” Inevitably their answers included the exponents and logs (though only one group *called* their logarithms logarithms).

If you are not a small-group kind of teacher, the closure of this lesson seems weak. But just wait until you hear the kind of discussion that happens with 3 or 4 kids trying to figure this out, with such a fun activity (singing) and fun tool (cool waves and stuff that respond to your voice) and high skill levels (my kids can already work with exponents and logs relatively comfortably). One of my four groups actually found the equation that best matched their data, AND its inverse – it was really neat to see it in geogebra. Every group used exponents to talk about the frequencies, and all but one group started to talk about logarithms too. This is practice using math in a casual way and feeling the fun and power of it.

This is a fun class full of joyous noise, and the kids were really into measuring precisely and graphing precisely. Every group eventually made a graph and noticed it looked exponential or logarithmic (depending on what they assigned to which axis). They got to think about what kind of scale makes sense in the first question, and got to see exponents at work in nature. I used the last five minutes of class to indulge in a monologue about the philosophy of sound: “is ‘sound’ the only way to interpret vibrations?” and “the computer registers 180 and we hear a certain pitch – which is more useful?” etc.

- Sometimes when I am self-deprecatory I am actually trying to emphasize how awesome I am. In the case of this worksheet, however, please be advised that you probably actually want a much better one. For one thing, the worksheet provided only covers half of the lesson plan (ran out of time!). ↩

This sounds fabulous! But I don’t think I could do it unless I got help with the technology…

There is a lot of tech in this lesson, to be sure! Luckily g-tune is a small (500kb) program that can install without administrative rights, and it’s relatively simple to use. Weird quirk: you have to click the “Power” button to get the program to start working.

I have done something similar to this before, but using a tone generator (such as the one available in Audacity) to try and find relationships between pairs of numbers which sound good when played together. It leads into discussions of ratio, and ways to build up the ‘standard’ scale using ratios of 2:1 and 3:2 (and their inverses).

Interesting. Do students reliably agree about what sounds are harmonious? What does the lesson look like? I’d love to see a writeup!

Très cool.

Ambitious extension: you’ve noticed that the RATIO of frequencies for notes with the same name is 2:1 or 4:1 or something like that.

And we figured out that the ratio of consecutive notes is a number that, when multiplied by itself 12 times, is 2. (oooh!)

What about other important ratios, like 3:2? Any pairs of notes have that ratio? Answer: almost, and the musicians in the class recognize them as being perfect fifths apart. You can use your tone-generation software (I use Sketchpad, but I know you use Geogebra, so…) to create a reallio-trulio 3:2 fifth and an even-tempered 2^(7/12) fifth, and if your ear is as bad as mine, you can’t tell the difference. But play them together and you’ll hear the beats.

(You can continue to study other Pythagorean intervals such as the Major third — 5:4 and minor seventh — 7:4, comparing them to the orthodox 12-tone values)

Anyhow, the LONG project is to wonder why we have 12 tones. Is it perhaps because 7 equal semitones so well-approximates 3:2? If so, what OTHER numbers of tones accomplish this? (19 is good.)

You know, I think that the relationship between harmonies and ratios is so fascinating, and I’ve tried to talk about it like 6 times in various physics or math classes, but I have never gotten a big response from a single student. This is flabbergasting to me. How can they not think that it is such an amazing insight into our instinctual aesthetic system?

Have you found a way to discuss these things with students in an engaging way?