Microwave Theory of a Crumhorn

By Charlie Jackson

From the Spring 1999 IEEE Microwave Theory and Techniques Society Newsletter. Original Copyright IEEE. Adapted for a web browser.

Applying microwave design techniques to renaissance woodwind musical instruments has been a lifelong passion for me. Ever since I first played a recorder, when I was 8 years old, I have been fascinated with transmission line theory. Years later I dog-eared my copy of "Horns, Strings, and Harmony" by Arthur Benade (now available from Dover). My first calculator was bought so that I could design a simple flute, rather than ace my homework. I started reading Ramo, Whinnery, and Van Duzer's book on electromagnetic theory because of the transmission line theory it contained, rather than its insights into Maxwell's equations.

crumhorn picture

Figure 1. A pair of crumhorns from a renaissance woodcut.

This note describes how microwave design theory can be used to design a musical instrument that flourished in the renaisance called a Crumhorn, Figure 1. The Crumhorn is a capped double reed instrument that is more related to the bagpipe than the oboe.

It has a small diameter bore, with small diameter tone holes, and a reed that allows the pitch to be changed a bit more than some would like; all this makes it ideal to model with a simple transmission line theory. Basically, it sounds like a kazoo. To learn more about the Crumhorn, go to the Crumhorn Home Page .

Microwave design techniques include a bag of tricks and tools. The two most important tools for the design of a Crumhorn are transmission line theory, and a philosophy of modeling. The Microwave design community has employed a brute-force approach to design circuits, that the Acoustics community has not embraced. Microwave designers break a circuit up into little parts and cascade the circuit into one big circuit that models the complexity needed for an accurate model. Programs such as LIBRA are used pervasively throughout the microwave industry. A program called DELTAE is the only acoustical program that follows the same philosophy. There are no programs such as SONNET, or HFSS which solve the feild equations for arbitrary geometries. This is not surprising; I will be the first to point out that the market for designing renaissance musical instruments will not support the development of such tools.

Acoustical models of woodwinds using the method that I will present here were first described 20 years ago. Acoustical models used custom coded programs at about the time when a SPICE and COMPACT were in general use with electrical engineers. The Acoustical community is now more concerned with economically viable programs using time domain models that approximate woodwinds, and capture the transients that make up the essence of a woodwinds sound. To design a Crumhorn we need to have equations that model the:

We will ignore effects due to the:

Condition for Resonance

The condition for resonance is that the input impedance Z of the tube be maximum, or that the admittance Y=1/Z be a minimum (zero). To model the circuit (Crumhorn) all we need to do is find the peaks in the impedance as a function of frequency. To model a flute, the admittance should be maximum.

Impedance of The tube

The impedance of the tube is given by

crumhorn picture

where ro is the density of the gas, c is the speed of sound, and S is the cross sectional area. This equation is used the same way that the impedance of a microstrip line is used. For contrast, we can express the equation for the impedance of a wide microstrip line as:

crumhorn picture

where mo is the permeability of free space, c is the speed of light, W is the width of the line, d is the substrate thickness, and er is the effective dielectric constant.

Very small diameter tubes exhibit effects due to the viscosity of the gas, so for frequencies of musical interest, a 1 mm hole is the smallest that we can use, without significant modifications to the equation for the impedance.

Transmission Line

A key equation relates the input impedance Zt and the impedance of a tube Zo, the length of the tube l, the speed of light and the frequency k=w/c, and the terminating impedance Zl by

cascaded impedance

equation 1

Another key element of our model is that when the termination impedance is pure imaginary, then the total impedance is pure imaginary, and only one term of a complex pair needs to be tracked. To recap: Zt is pure imaginary if Zl is pure imaginary.

Losses in Propagation

The transmission line equation shown above has already implicitly assumed that the losses are negligible.

cross section

Figure 2 Cross section of a woodwind showing a closed and open hole.

circuit

Figure 3 Circuit of the cross section of Figure 2.

Open Circuit Condition (Closed Hole)

Figure 2 shows the cross section of a crumhorn. An open hole and a closed hole is shown. The open hole short circuits the pressure (hence it is short circuit), and a closed hole corresponds to an open circuit. When the terminating impedance is infinite, then the transmission line equation (1) reduces to

open hole

Short Circuit Condition (Open Hole)

When the terminating impedance is zero, then the transmission line equation (1) reduces to

closed

parallel

Figure 4 Parallel impedances

Parallel Impedances

The hole impedance is in parallel with the transmission line impedance. Both impedances are imaginary (no real part), so they are added with the standard equation

parallel equation

By starting at the far end of the crumhorn, where the open hole impedance is zero, and using the transmission line equation, the short circuit and open circuit equations and the parallel addition equations, it is possible to calculate the impedance of the instrument. Each fingering needs to be analyzed, and summarized.

Reed Input System

How will we model the reed? Actually, we won't. It turns out that differences in reeds can be accounted for, to first order, by pushing the reed in, or pulling it out until the effective length of the reed equals the design length. Open Hole Lattice

It turns out that there are some holes at the end of a crumhorn that serve two purposes. First, they allow the lowest pitch to be tuned, and second, they provide a low pass filter that suppresses higher order modes. This low pass filter is called an open hole lattice by A. Benade.

Radiation Impedance

The radiation impedance is ignored for the small diameter holes of the crumhorn. For a flute, the effects are large, and cannot be ignored. This makes the crumhorn a great starting point for woodwind modeling.

impedance plot

Figure 5 Plot of Impedance vs Frequency for "g" fingering.

Brief Description of the Program

The program to model a crumhorn consists of many excel sheets in a workbook. First, there is a version page, with some cryptic comments to help the imagined user. Second is a shareware page to prevent rampant plagiarism. Next there are a lot of pages that calculate the impedance and resonance frequency of each fingering. Each note page has a plot of the impedance as a function of frequency (Fig. 5), and the peak frequency is calculated and passed to the summary sheet. Finally, there is a top level summary page containing a table of dimensions and physical constants, a table to enter pitches, a fingering chart, a calculation of each hole combinations resonant frequency (Fig. 6), along with the total deviation (Fig. 7), and a plot of the design and calculated frequencies. A fingering chart is used to check the individual pages of the excel program.

calc vs design

Figure 6 Plot of design and calculated frequency for each fingering.

To reduce the size of the program so it is about the size of a floppy disk, the number of frequency points per note page is minimal. More points per note page will allow a more accurate determination of the resonance frequency.

Dimensions of the hole diameters, the wall thickness, the lengths of tubes, and the tube diameter can be changed to fine tune the instrument. Lucite tubing is commonly available in a number of different sizes, so now it is possible to design a crumhorn using these materials.

error plot

Figure 7 Plot of deviation from design for each fingering in cents.

The dimensions of an alto crumhorn are from a book by Trevor Robinson called "The Amateur Wind Instrument Maker" published by the University of Massachusetts Press. The name of the finger holes follows his convention.

The program is posted in a zip file on my web page.

Conclusion

Microwave design techniques can be applied to the design of renaissance woodwind instruments. The program described here shows that the dimension of a crumhorn provided in Trevor Robinson's book are pretty good, and certainly provide an excellent starting point for the fabrication of a crumhorn. The next step is to extend the program so that it can model other renaisance instruments including a cornetto, a lyzard, a serpent, and a shawm.