Today’s question

Last weekend, Youngest Son participated in the UNI Honor Band [1]. At the multi-day event, the students learned and rehearsed pieces, developed their musical skills in small groups, attended concerts, and performed the works in a very nice concert. Somewhere along the way, probably in one of the small group sessions, one of the UNI music faculty said something like, Does anyone have any questions about music?

Youngest reports that he then asked a question that he’s been wondering,

What is the harmonic series of a conical-bore instrument that is open at both ends?

The UNI professor did not know. But I expect he understood the question. It took me a day or two of thinking about it to understand what Youngest was asking.

From what I understand, long hollow instruments (like recorders, flutes, saxophones) can have a cylindrical bore, in which the diameter remains consistent, or a conical bore, in which the diameter gradually increases or decreases. Clarinets and flutes have cylindrical bores; saxophones and cornetti have conical bores. Such instruments can be closed at one end or open at both ends.

Different choices have different effects on the number of overtones you have. For example, it turns out that an instrument with a cylindrical bore that is closed at one lacks [2] the even overtones. A physicist can probably explain why. But that’s one of the reasons clarinets sound the way they do.

Did Youngest have a particular instrument in mind when he asked the question? No, it seems not. We could not think of any conical-bore open-ended instrument. But he’d been playing with an instrument design program and was interested in the question of what kinds of sounds such an instrument would produce.

I’m pretty sure that he looked online. However, most online sources, such as this page on pipes and harmonics at the University of New South Wales [3] or The Wikipedia page on bores in wind instruments don’t show instruments with open conical bores. Perhaps that’s because there’s an assumption that the cone ends in a point?

None of the music folks we’ve asked have had an answer. The first applied mathematician also did not know.

Perhaps we should ask a physicist. Or maybe we could ask one of Grinnell’s world-music [4] faculty, both of whom seem to know about a wide range of instruments. If only I had a son who was taking a Physics class and regularly talked to our world-music faculty.

Or maybe I should read things again. Let’s see …. Wikipedia says that

The bore of a modern recorder has a reversed taper, being wider at the head and narrower at the foot of the instrument.

Doesn’t that make it a kind of conical bore? Is a recorder open at both ends? I think so. I’m pretty sure that recorders have the full harmonic series. So there you go. I may have an answer for my son!

But I’ll still ask a physicist [5].

[1] I am not bragging. I am just stating a fact.

[2] Or at least has comparatively quiet.

[3] The page is from UNSW; the pipes and harmonics may not be.

[4] Or should I say world-class world-music faculty?

[5] Or one of the Grinnell music faculty.

Version 1.0 of 2018-02-14.