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Diary of an OCW Music Student, Week 9: A 12-Tone Pythagorean Set

Education Insider talks a lot about OpenCourseWare (OCW), so maybe it's time we put some to the test. So it is that over ten weeks this fall we'll be taking a course from the University of California - Irvine ourselves. What does a semi-professional musician stand to learn from a primer in music theory? This week we look at how to construct a 12-tone set using mathematics.

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By Eric Garneau

music brain

Math is the Answer

After this lecture, we've only got one class to go in UC-Irvine's Introduction to Pitch Systems in Tonal Music OCW series - can you believe it's almost over? Before we finally tackle the issue of how to tune acoustic instruments in a way that's both mathematically and harmonically satisfying, lecturer John Crooks asks us to consider one more possibility born out of simple equations that could give all us DIY rockstars an easy handle on instrument tonality.

In this ninth installment, Crooks tries to find a tuning system that works by building a Pythagorean set (pitches constructed using a harmonious 3:2 ratio starting with the root note's frequency) out of 12 tones, not seven like we did a few weeks ago. If you think about it, that makes a lot of sense - on a piano each octave takes up 12 keys, after all, and the reason we couldn't build a mathematically sound Pythagorean set out of seven pitches might have been because we didn't go all the way around the circle of fifths when calculating our frequencies, so we had to take some shortcuts to complete our set.

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A Comma Problem

And yet after Professor Crooks does all the math (which again involves multiplying your starting tone's frequency and all those that result by 3/2, then dividing any that surpass your original octave range by a power of two), we still don't have a tuning system that works mathematically. Our process builds a satisfying 12-tone half-step scale, but when we come back around to the octave of our initial pitch, we end up with a tone slightly more than double our starting frequency, creating another discordant comma like Crooks discussed last week. That seems especially problematic since we know from earlier lessons that our ears naturally pick up on octaves as means of organizing a pitch system. If our octaves are out of tune, what are we left with?

Again, Crooks makes use of the televised serial-like nature of his presentations to tease next week's lesson, the final one in our course. Students who've been following along with us may feel that Crooks spends an inordinate amount of time showing us methods of tuning that don't work, and with the upcoming tenth lesson we'll finally figure out a way that does... at least that's the idea. Crooks offers up lots of interesting tidbits about just what it is we'll be learning - notably that the tuning system we currently use is a 'compromise' that makes the best of both mathematical and harmonic discrepancies. Given Crooks' introductory video to this course, in which he expresses hopes that students will perhaps be inspired to create their own tuning systems, I can't help but wonder what he's been building to. Even if some of these more math-heavy lessons have seemed a bit esoteric, then, at least this learner finds the ultimate endpoint of these videos truly compelling.

How in-tune were these Duke University students who crafted a musical tribute to Richard Nixon?

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