Eurekarina

In 1987, I moved 1,600 miles from home to attend graduate school in Tucson. During a conversation about home towns, my friends there didn't believe me when I said that back in Urbana, we had experimental cows with windows in their stomachs. Home on a break, I rode my bike over to the South Farms and took this photograph to show everyone:

I was reminded of that picture recently in the geek lab, where I have apparently been industriously re-inventing the wheel.

Perhaps you recall me mentioning that the 7-hole chickarina produces a 12-note chromatic octave and then some, with several built-in pitch redundancies. I speculated then that with fewer holes, I could still get a diatonic octave scale. You might also recall that the number of holes in the ocarina is one of the variables that affects tone quality: increasing the number of holes usually necessitates fipple adjustments. Perhaps, I thought yesterday, I can attain better tone quality by building a diatonic scale using fewer holes.

How many holes would be necessary?

Two holes of different sizes yield four pitches total:
(1) 0 0 (both holes closed)
(2) 0 1 (H1 open)
(3) 1 0 (H2 open)
(4) 1 1 (both holes open)
Four notes out of eight, right there! Add a third hole:
(1) 0 0 0
(2) 0 0 1
(3) 0 1 0
(4) 0 1 1
(5) 1 0 0
(6) 1 0 1
(7) 1 1 0
(8) 1 1 1

"But what has this to do with reinventing wheels, or with cows with windows?," you ask. I'm getting there.

The fingering inventory above doesn't automatically produce a diatonic scale on a chickarina: I need holes of appropriately tuned sizes so that, e.g., 0 0 1 doesn't sound the same as 0 1 0 and 1 0 0. Moreover, according to that most honorable and meticulously edited of sources, Wikipedia, "the [pitch on an ocarina] is dependent on the ratio of the total surface area of opened holes to the total cubic volume enclosed by the instrument. This means that, unlike a flute or recorder, sound is created by resonance of the entire cavity and the placement of the holes on an ocarina is largely irrelevant." This suggests that two holes with a total area A would affect pitch and tone quality in the same way that a single larger hole with area A would, although in practice, as I've pointed out above, this isn't always the case--thus the motivation for fewer holes. But it does mean that if I want eight distinct pitches with three holes, 1 0 0 not only can't equal 0 0 1 or 0 1 0, it also can't equal 0 1 1. (Incidentally, I love that the Wiki author[s] wrote "cubic volume." I'll add that combo to the Redundancy List right after truly unique and labium lip. Perhaps if I want to learn about Helmholtz resonators, I should consult more rigorous texts than Wikipedia.)

Further complicating things, for any given resonating chicken-shaped cavity, the higher the pitch, the larger a newly opened hole has to be to produce a given interval. That is, if I want, say, a major second, I need a smaller hole at lower frequencies (e.g. do-re), and a larger hole at higher frequencies (e.g. la-ti). Consequently, with three holes, the bottom of a scale might be in tune, but as you approach the top of the scale, the pitches become woefully flat: you need a fourth hole. Adding a fourth hole adds eight new fingering combinations (1 0 0 0, 1 0 0 1, 1 0 1 0, etc.), making room for pitch redundancy.

So: four holes.

For my birds, I based hole-size decisions on the intervals produced between 0 0 0 0 and each singly open hole (0 0 0 1, 0 0 1 0, 0 1 0 0, and 1 0 0 0):
H1 = major second
H2 = major third
H3 = perfect fourth
H4 = perfect fourth. Yes, H3 and H4 are the same size. There's room for redundancy.

These holes yield a diatonic scale thus:
Do: 0 0 0 0 (no holes open)
Re: 0 0 0 1 (H1 open)
Mi: 0 0 1 0 (H2 open)
Fa: 0 1 0 0 (H3 open)
Sol: 0 1 1 0 (H3+H2 open. You'd think it should be H4+H1, but that tends to be a little flat: 0 1 0 1 isn't equivalent to a major second plus a perfect fourth--in theory a perfect fifth--because, remember, the higher you go up the scale, the larger the holes need to be to produce a given interval. H3+H2 is a little sharp, but under-blowing lowers the pitch just fine.)
La: 1 1 0 0 (H3+H4 open. On most instruments, a perfect fourth plus a perfect fourth yields a minor seventh. On an ocarina, the interval is flat--a major sixth.)
Ti: 1 1 1 0 (H3+H4+H2 open. A would-be octave is flat--a slightly flat major seventh)
Do': 1 1 1 1 (all holes open. A flat tenth becomes a perfect octave).

You can get a chromatic scale (and then some) out of this, although you have to half-cover H1 for Ra. Pitches can also be tweaked by changing wind speed. You can't really overblow an ocarina, but underblowing bends pitches quite well.

Thus did I reinvent the wheel: the internets tell me a British mathematician named John Taylor invented the four-hole chromatic octave ocarina in 1964. I suppose I could have started my ocarina adventures by looking up fingering charts online, but in exchange for taking the empirical route--thinking for months like the pianist that I am (7-hole ocarinas) before thinking like a mathematician (4 holes)--yours truly got to enjoy a pretty exciting oceureka moment.

As for the cows, the resonating cavities of my chickens vary in size and shape, so the acoustics of the individual ocarinas vary. Whistle wall thickness also affects pitch and tone quality. Thus, while most of the 4-hole birds I made looked just fine, a particularly thin-walled ocarina required humongous H3 and H4 holes and ended up looking like this:

So you can see why I was thinking about cows with windows in their stomachs.

In the midst of all this research, a friend dropped by and observed that my chickens were starting to look a little uniform. Personally, I think they all look wildly different, as far as these guys go, but I will concede that most of the current birds lean to the left. Given North Carolina's recent shameful vote to add a bigoted amendment to our state constitution, I'm not keen on having right-leaning chickens in my flock right now. Here, two lefties show a stunned right-winger that they haven't disappeared even though he voted their civil rights away.