Why correct musical pitch depends on spoke length and not on thickness

The fundamental frequency of vibration of a stretched string or wire varies according to the following formula, which is derived from basic physical laws.¹

F1 =[(1/(2·L)]·sqrt(T/m)

where

F ₁	= the fundamental frequency in cycles per second
L	= the length of the string
T	= tension of the string in pounds or dynes
m	= mass per unit of length in lb/in or gm/cm.

The cross-sectional area of the spoke and the mass per unit length m are exactly proportional to each other. Therefore, for two different strings or wires of equal length, one thick and another thin, the frequency is the same if the tension per unit of cross-sectional area is the same. One way to think of this is to imagine two identical spokes side by side, both of the same gauge and at the same tension. They vibrate at the same frequency. Now imagine lightly connecting them together all along their length. They still vibrate at the same frequency. Finally, imagine merging them into one, thicker spoke. It still vibrates at the same frequency.

These facts greatly simplify the measuring of spoke tension for wheelbuilders. To determine whether a spoke is optimally tensioned, we don't have to measure the thickness or, what is more difficult, the tension, since the musical pitch translates directly into the tension per unit of cross-sectional area.

Note that the fundamental frequency of a spoke increases only as the square root of tension. Therefore, every doubling of frequency - one musical octave - raises the tension by a factor of 4. A spoke whose fundamental frequency is only 1.2 times as high as the value given in the table - a musical minor third higher - is already under more than 1.4 times as much tension, and is likely to fail quickly.

Bicycle spokes rarely break due to excessive tension; but the rim may not withstand it, and when the rim relaxes around the spoke holes, the wheel fails. Weight loading on a wheel decreases tension of the few spokes at the bottom of the wheel greatly, and raises tension of the remaining spokes only very slightly, but lateral loading while leaning the bicycle from side to side while pedaling out of the saddle causes significant increases in spoke tension and can lead to rapid failure of an overtensioned wheel.² I have seen newly-built wheels that failed in this way the first time they were ridden. They had been tensioned to about 1.5 times the amount I recommend and were built using high-quality box-section aluminum-alloy rims.

A spoke whose fundamental frequency is .83 of the nominal value I recommend -- a musical minor third lower - is carrying only 70 percent of the optimum tension, and a wheel built to this tension level can carry only about 70 percent of the weight it could if properly tensioned. A loose wheel is likely to break spokes and go out of true, because the spokes can go completely slack under heavy loading. The excess motion in a slack wheel is what breaks the spokes and allows the nipples to unscrew.

As the thickness of a wire increases, our formula for musical pitch becomes slightly inaccurate. The bridge of a steel-stringed guitar provides an everyday example of this phenomenon. The guitar's lower-pitched strings are made of thicker wire, whose greater bending stiffness adds its contritution to the stiffness generated by the string's tension.The vibrating part of the string (the part between the bridge and fret, or bridge and nut if the string is not fretted) is, to a first approximation, clamped, and is nominally slightly shorter than the actual span For this reason, the guitar bridge does not go straight across at a right angle to the strings, but is slanted to make the lower-pitched strings longer, so they will play in tune on all frets of the guitar's neck.

The discrepancy for which the guitar bridge compensates is not large. Even in bicycle spokes, which are much thicker than guitar strings, the adjustment only amounts to a few percent. The table in the accompanying article includes a correction which I determined empirically by measuring the musical pitch of tensioned spokes clamped off at different lengths.

Part of a spoke at the outer end is inside the spoke nipple, and part at the inner end is in contact with the hub. These parts can not contribute to the vibrating length. The table also accounts for this. The ends of a butted spoke are enough thicker than the shaft of the spoke that the ends contribute only slightly to its effective vibrating length. This, and the greater strength at the threads and head of a butted spoke, account for the higher musical pitch I recommend for butted spokes.

The yield strength of good steel is about 150,000 pounds per square inch or 1.04 x 10¹⁰ dyn/cm², and the tension recommended in the table is 1/3 this, about 50,000 pounds per square inch or 3.45 x 10⁹ dyn/cm².

The tension of spokes measured using a commercial tensiometer and using musical pitch has been compared, and the results have been published. They show good correlation².

1. The derivation of this formula is given, for example, in Alonso and Finn, Fundamental University Physics vol. 2, Fields and Waves (1967, Addison Wesley), section 18-7.

2. It follows from the analysis here that the changes in tension of spokes in a bicycle wheel under load may simply and elegantly be demonstrated by measuring the changes in their musical pitch. This is easiest in a radially-spoked or unlaced wheel.

3. John S. Allen, Comparison of measurements of bicycle spoke tension using a mechanical tensiometer and musical pitch, Human Power #53, page 3.