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)[ ].

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.

Professionals?

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:



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And here’s the Hon­da road again:



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The Hon­da ver­sion isn’t sim­ply “out of tune”… the notes are just wrong. The orig­i­nal starts with a ris­ing 4th, F-B♭[ ], and even­tu­al­ly reach­es an octave above the start­ing note before descend­ing to the ton­ic F-E♭-D-B♭.[ ] 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[ ]. 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?

The Error

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.[ ] 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.

Oops.

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 .

The Coverup

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:



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And here’s Math­e­mat­i­ca pro­grammed to make the mis­take I think Honda’s engi­neers made:



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And here’s honda’s com­mer­cial ver­sion again:



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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[ ]:



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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[ ]. 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.[ ]

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:



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Aftermath

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.”



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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:

Just… wow.

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.