Critical ThinkingCultureHigher Education

Wikipedia 1, U Chicago News Office 0.

I was browsing my news feed today when I came across the following video, posted by the University of Chicago’s News Office.


Since I’m about to get very critical, let me first talk briefly about the things Cheng does well in this video:

1) She gives a mostly understandable layperson’s description of the circle of fifths, and the problem of the Pythagorean comma (the difference between 7 perfectly-tuned octaves and 12 perfectly-tuned fifths).

2) She gives a very clear idea of what an overtone is.

3) She shows very clearly how the problem of equally dividing an octave into 12 notes has to be solved, if that’s what you want to do.

Okay. With that out of the way, it’s time to vent.

I am shocked and embarrassed that a video containing as many fundamental errors this one does was released by the press office of a major university. Apparently no one involved in the process (including Cheng herself) thought to fact-check its major premise–that Bach’s Well-tempered Clavier has anything at all to do with the creation of an Equal-tempered octave constructed by multiplying frequencies by the twelfth-root of two–by talking to a musicologist or checking Wikipedia.

You see, there’s a reason Bach’s two collections of preludes and fugues in all the keys are called the “Well-tempered Clavier” and not the “Equal-tempered Clavier.” This is because they were not written for equal temperament. In fact, true equal temperament (or rather, the closest possible approximation to it we can muster since any tuning based on irrational numbers is physically impossible) didn’t really catch on until the very late 19th and early 20th centuries–perhaps not coincidentally around the time tonality itself started breaking down.

But let’s back up a minute before we get into all the details, since there’s a lot to unpack here.

Cheng starts off the video with the following statement:

“One of the most exciting things about mathematics is when it turns out to solve a problem that it wasn’t even exactly trying to solve.”

This implies two things that are essentially untrue: first, that the study of numerical ratios was somehow independent of the study of musical tuning (it wasn’t–they were intimately related for centuries), and second, that musicians were just waiting around for some brilliant mathematician to solve this problem for them (they weren’t).

Actually, musicians recognised the possibility of equal temperament for a long time (like, hundreds of years). In fact, the use of a kind of equal temperament is even implied in the writings of the Greek music theorist Aristoxenus in the 4th century BC.

But even without counting that example, the possibility of equal temperament was discussed by lots of music theorists in the decades before 1600. Despite knowing about equal temperament, however, they chose not to use it. This is because “equal temperament” doesn’t actually mean that all the keys are equally in tune. Rather, it means all the keys are equally out of tune. Unless your music is rapidly moving from one key to another, or otherwise frequently borrowing notes and chords from harmonically distant places, equal temperament actually makes the music sound less in tune than it would if one were using the common tuning systems of the day.

That’s the whole problem that makes tuning interesting in the first place: on an instrument like a piano, there is actually no perfect solution to the problem of tuning octaves, fifths, and thirds all at the same time since their ratios have different bases. There is simply no way to make 2:1 octaves line up with 3:2 fifths or 5:4 major thirds.

Equal temperament is not a perfect solution to the problem, despite Cheng’s teleological take on the history of musical temperaments: thirds are significantly out of tune in ET, with a major third being 13.69 cents (hundredths of a semitone) wide of the perfect 5:4 ratio. The difference between ET and pure or “just” intonation is easily audible even to a person with no musical training:


So why don’t we just tune our instruments with pure intervals, if they’re so much better? Unfortunately that’s the central problem: physics don’t allow it, at least not on instruments with fixed pitches. Singers and string players, with their ability to subtly bend pitches to sound more in tune, aren’t quite so limited. This is also a big reason choirs tend to go flat: constant adjusting to keep the group in tune can drive the overall pitch downward over time.

Therefore the choice you make with regard to your tuning system has to be driven by what you want to accomplish musically rather than the other way around. Most music before around 1700 only employs a few tonal centres, and most pieces did not wander too far harmonically from where they started. For this reason it was simply not necessary to employ temperaments that allowed for lots of modulations or distant tonal borrowing: if you needed to play a new piece in a different key, you just re-tuned your instrument. It was therefore simply more practical to use a temperament that kept the keys you were playing in sounding good. Here’s a good example of this effect, using a contemporary piano piece in C major. Since it doesn’t stray too far from the home key, it benefits greatly from the purer intervals offered by the unequal temperament:


Now, some of you may actually prefer the slight buzziness of the equal-tempered version: we have been trained through the pervasive use of equal temperament in modern music to expect thirds to sound slightly out of tune, and consequently we sometimes find pure thirds to sound a bit empty. This is neither good nor bad (these kinds of preferences are just culturally determined), but it is good to be aware of it in discussions like this. After all, just because we accept the out-of-tune thirds of ET as natural does not mean that everyone does. In fact, they are precisely the reason people like Vincenzo Galilei (father to that other guy) rejected equal temperament as basically unusable for “modern music” about a hundred years before Bach was even born.*

It is true, of course, that when you start to play highly chromatic music in temperaments designed for only a few keys things can start to get quite out of tune:


This isn’t necessarily a bad thing, though. Whenever you have a feature like this built into a system, you can bet that there will be people who take advantage of it for expressive purposes. The out-of-tune-ness of distant tonal areas thus becomes another tool in the composer’s arsenal, rather than in inherent limitation imposed by “inadequate” tuning systems (and remember, they knew about equal temperament, they just didn’t use it). Just as we accept the bad thirds of equal temperament, people hundreds of years ago accepted the out-of-tune-ness of distant keys as simply being part of the sonic landscape.

So what about Bach and the Well-tempered Clavier? Well, the exciting thing about the “well”-tempered systems (as opposed to the mean-tone systems that preceded them) was not so much that they allowed one to play in all the keys, but that they were designed in such a way that made the “home” keys (i.e. those with relatively few accidentals in the key signature like C, G, and F) sound relatively pure while still keeping chords from distant keys from sounding unusably bad. The use of an uneven temperament also meant that the various keys all had slightly different sounds to them, with some more in tune than others. The effect is subtle, but certainly audible:


It was precisely this that was the real challenge: to write music that not only used all the keys, but also took aesthetic advantage of their individual characters. If C major is beautifully in tune, you can write something relatively placid that uses the stillness of pure intervals to good effect:


And if C-sharp major is much less in tune, you adapt the energy from the constant beating of harmonics into something a bit more active or even relentless:


But here’s the thing. None of this is new information. I teach this concept to my non-major music appreciation students every year. The history of tuning and temperament is explained in great detail on Wikipedia. So can someone please explain to me what the University of Chicago was thinking? How could a video like this ever have been posted without anyone involved even checking the basic premises? I can see the press office not knowing the difference, but Cheng is actually a professor!

I suppose the two cultures strike again.


Someone fix me a drink.



*There was actually a whole lot of fascinating stuff going on back then where tuning is concerned, including octaves split into many more than twelve steps to compensate for the problems outlined above. I think it’s especially powerful that some people thought that a 17-note octave with split chromatic keys on the keyboard was a much more reasonable solution to the temperament problem than 12-tone equal temperament. But that’s a post for another time.

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Dan has a PhD in historical musicology and has taught music history and theory at a major Canadian university. He mainly studies music from the Italian Renaissance when he's not busy performing stand-up comedy or playing JRPGs with his cat, Roy. He occasionally tweets as @incontrariomotu and blogs about geeky stuff at The Otaku Skeptic. He is also the glorious editor-in-chief of School of Doubt.


  1. September 12, 2014 at 6:08 pm —

    Wow, what a ludicrous overreaction to a perfectly benign video.

    Its “major premise” is not “that Bach’s Well-tempered Clavier has anything at all to do with the creation of an Equal-tempered octave”; it is “Maths is interesting and crops up in real life”. I guess this is the “Two Cultures” thing you’re referring to — you, as a musicologist, take a video by a mathematician, about mathematics, and treat it as if it’s about musicology.

    Also “any tuning based on irrational numbers is physically impossible”? Orly?

    • September 13, 2014 at 1:39 am —

      Uh, if it has ‘nothing at all’ to do with it, then why does she say the opposite? Did you actually watch the video?

      There are plenty of ways to make a video about how math intersects with tuning and temperament without using specious and historically ignorant examples. What are you arguing exactly? That it shouldn’t be pointed out when these kinds of mistakes are made? This isn’t some random YouTuber, here. A fairly prestigious university has endorsed something that reinforces a flawed and easily disproven historical narrative.

      And yes, true equal temperament based on the twelfth-root of two (as opposed to one close enough for human hearing) is not possible to achieve in real life. Take a string tuned to A=440. Now tune string to Bb=440 * 2^(1/12) and /prove it/.

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