This is the story of how Honda engineers screwed up a big expensive project with a simple arithmetic mistake, tried to fudge their result with sound editing software, and congratulated themselves for being totally awesome.
When I was a kid, my family used to drive up to The Pinery in Ontario, a beautiful park by Lake Huron. Very scenic. My favorite part, though, was a stretch of road a half-hour outside of the park. To discourage reckless Canadians from barreling past the houses and barns, the local government carved five sets of grooves in the road before every stop sign. Drive over them, and the car would vibrate: “vbvbvbvb... vbvbvbvb... vbvbvbvb... vbvbvbvb... vbvbvbvb.” The faster you drive, the higher the pitch.
My Dad is a musicologist, with a particular interest in tuning. So there was no way he was going to pass up the chance to experiment with this instrument. Every time we approached some grooves, he'd start fast over the first set, and try to slow down by the last set, to play a descending scale: G-F-E-D-C. If there was no oncoming traffic after the stop sign, he'd swing over to the other side of the road and play an ascending scale as we sped up.
Ratios of speeds correspond to ratios of vibration frequencies, which correspond to intervals between notes. To play an ascending scale C-D-E-F-G, you need to drive at these ratios to your starting speed: 1 - 9/8 - 5/4 - 4/3 - 3/2 (for example, 24 - 27 - 30 - 32 - 36 mph). 1
Playing a scale with a '95 Toyota Previa is not easy. The notes tend to come out a little wonky — we'd get the half-step between E and F too wide, and with not enough space between F and G. It usually sounded kinda modal... but still awesome.
So imagine my delight when I heard about this musical road [CNET] that Honda built in Lancaster, CA.. A team of engineers carved some grooves into a highway that were carefully spaced to play the William Tell Overture as you drive over them at a constant speed. Awesome, right? The problem is, it's spectacularly out of tune.
Here's the original melody:
And here's the Honda road again:
The Honda version isn't simply out of tune... the notes are just wrong. The original starts with a rising 4th, F-B♭,2 and eventually reaches an octave above the starting note before descending to the tonic F-E♭-D-B♭.3 But Honda's version starts with a rising major 3rd, and its top note is a major 6th above the starting note. Some might have noticed that the last few notes in Honda's commercial sound OK. That's because they edited over them! I can prove it.
The CNET article above speculates that Honda designed the road specifically for the Honda civic driving at the speed limit, and other cars might need to drive at a different speed to make it sound better. But if you're going at a constant speed, all that matters is the spacing between grooves. Speeding up or slowing down just transposes everything. It would be theoretically possible to "correct" the melody by driving at different speeds (like on the road to the Pinery). But the notes on the musical road are too closely spaced for all but consummate musician Mario Andretti.
It also doesn't matter what car you drive.4 The vibration frequency is f = v/d, where v is the car's speed, and d is the distance over which the road pattern repeats. There's no place in the equation for wheel spacing, tire size, side-impact airbags, etc. All of these things affect the quality of the sound, but not the pitch.
So why is the musical road so unmusical?
Honda posted a series of 5 ridiculous videos: [Part 1][Part 2][Part 3][Part 4][Part 5], in which they talk about all the hard work they did and congratulate themselves for being so awesome. There are lots of complicated sounding numbers, there's a "Mathematician/Musician," and plenty of experts. I'm sure some people behind the project understood what was going on. But I think they failed to anticipate a basic misunderstanding on the part of the groove-designers.
In the fourth "making of" video, they mention that the initial note, a low F, has a spacing of 4 inches (4in) between grooves (1:47):
From the video, it looks like the grooves themselves are about 1in wide. Now, suppose you want to make the B♭ a 4th above F. A perfect 4th is a fequency ratio of 4/3, so you should multiply the width by a factor of 3/4... But the width of what?
The width that really matters is the total width of the spacing plus groove (s+g). That's the distance over which the road pattern repeats, so that's the distance over which the car completes one vibration.5 Suppose you didn't know this, and only changed the spacing, from s = 4in to s' = 3/4 × 4in = 3in. Then the frequency ratio is (s+g)/(s'+g) = (4+1)/(3+1) = 5/4, a major 3rd, not a perfect 4th. What about the octave above the starting note? An octave is a frequency ratio of 2/1, but if you only changed the spacing to s' = 1/2 × 4in = 2in, you'd get an actual ratio of (s+g)/(s'+g) = (4+1)/(2+1) = 5/3, a major 6th, not an octave.
There are two ways you could correct this problem:
- Adjust the groove width g as well as the spacing s. For instance, to make an octave, use a spacing s' = 2in and a groove g' = .5in, giving a fequency ratio (s+g)/(s'+g') = 5/2.5 = 2/1. This is probably hard with typical cutting tools. Also, the engineers may have found that they need to make the grooves bigger than some minimum width to get a good sound. So on to method 2...
- Over-adjust the groove spacing so that the total g+s is correct. For instance, to make an octave, adjust the groove spacing to s' = 1.5in, so you get a frequency ratio of (s+g)/(s'+g) = 5/2.5 = 2/1.
Armed with this theory for why the musical road sounds so bad, I crunched some numbers in Mathematica, and was able to reproduce Honda's result, sort of...
Here's Mathematica playing the correct William Tell Overture:
And here's Mathematica programmed to make the mistake I think Honda's engineers made:
And here's honda's commercial version again:
Notice that a few notes in the commercial sound different from Mathematica's version. Particularly at the end. Honda's last few notes are sort of... in tune! Turns out that's a bit of Hollywood magic. Here's a recording I stole from a different video of someone driving down the Musical Road:6
What happened to the ending? It's all funky again. Go back and listen to the Mathematica version that mimics Honda's mistake. Same funky ending.7 Whoever put together the Honda commercial must have edited over the ending, assuming that as long as the last few notes were correct, no one would notice anything wrong.8
What I don't understand is: if they were going to doctor the sound, why didn't they just correct the whole thing? It's not that hard. My dad did this version in about 20 minutes:
I learned something else kind of ridiculous from this analysis: if Honda didn't doctor the overall pitch of the melody in their commercial, then they were speeding. The opening frequency is about 238Hz, which corresponds to a speed of about 67mph if the road pattern repeats over 5in. But they mention in one of the videos that the speed limit is 55! Crap.
In fact, in this youtube video, where they explicitly state they're going 55mph, the melody starts a minor third below the Honda commercial. A minor third is a frequency ratio of 6/5, so this is consistent with Honda's driver doing 6/5 × 55mph = more than 10mph over the speed limit...
Another funny point is that some of the intervals you get from Honda's miscalculation are pretty bizarre. The D, a major 6th above the starting F, should have a frequency ratio of 5/3 above the starting frequency. Instead, it has a ratio 5/(4 × 3/5+1) = 25/17. This isn't really in the western scale. It's about 2/3rds of the way between an augmented 4th and a pure 5th. Microtonal composers like Easley Blackwood might have found a use for it, but I don't think it's what Honda was after.
If I were them, I'd seriously consider paving over the road. In fact, it seems like some local residents might do it for them. There is another option, though. If they bring in the bulldozers, and shuffle around a few chunks of asphalt at the end of the road, they might get a decent rendition of "When The Saints Go Marching In."
Update [12/30/08]: Added picture comparing grooves to Civic wheelbase
Update [5/2/11]: I am both sorry and delighted to hear that they rebuilt the musical road (see, e.g., here), and they fixed nothing. Here it is on April 28, 2011:
Update [4/15/18]: This post was recently featured on Tom Scott's Youtube Channel "Amazing Places." As of today, the video currently has about 5 million views. It was also mentioned in The New York Times.
- If anyone's wondering what happened to the 1/12th powers of 2 in this whole tuning discussion, I'm using what's called Just Intonation, which is an (often better-sounding) approximation to the Equal Temperament system most people know. Actually, it's really the other way around: the reason we use 12 equal semitones is that it lets us approximate nice integer ratios like 3/2, 4/3, 5/4, etc.. This is a long story that I'm not going to get into here.
- Actually, the starting note in the recording is around a B♭. I'm going to pretend like everything is in the key of B♭ (so the starting note is F), since that's the key they talk about in the making-of videos. Sorry to the perfect-pitch people.
- The original melody actually has a run down to the B♭: F-E♭-D-C-B♭. Honda apparently decided this was too complicated and used a simplified version. That's what I'll stick to here.
- With one exception that can't fix the tuning. See my comment, below.
- More precisely, once you know the force driving the vibrations is periodic with period T=d/v, it follows that the vibrations themselves have that periodicity, so the Fourier transform of any resultant sound is only nonzero at integer multiples of f=1/T. For more explanation, see the second comment, below.
- I've actually transposed it up to be in approximately the same key as the other recordings in this article. By the way, there are hundreds of such videos on Youtube.
- Aside from a single passing note. If I change the closing notes from F-E♭-D-B♭-D-B♭ to F-E♮-D-B♭-D-B♭, and apply the Honda miscalculation, it sounds almost exactly like the undoctored recording of the musical road. So it appears that there are two errors at work here: the groove spacing miscalculation, and replacing an E♭ with an E♮.
- It seems like Honda fixed up some of the other notes, too, to get a more pleasant sound. Some might object that it's easy to make the notes sound bad by speeding up or slowing down as you drive down the road. However, I don't hear anything like that in the random person's recording. The melody returns to previous notes with reasonable accuracy, which it wouldn't do if the speed were varying.