Musical Mathematics

on the art and science of acoustic instruments


Table of Contents



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Also available from the publisher at Chronicle Books, San Francisco.


© 2000–2018 Cristiano M.L. Forster
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9 May 2004


Hi Jessie,


In the last email you wrote, (1) How are the glasses mounted? I'm curious about this, just as far as how smoothly they rotate, how they are kept stable, how they are put there in the first place. I imagine that the actual construction of the instrument was difficult and not without problems... Are there any specific problems you encountered with the construction?


And (2) How are the glasses set up? Are they in the form of an ascending scale? I want to know this in order to know how the player orients themselves to the instrument.”


I hope that the following three sections from my book Musical Mathematics will answer these questions. The most complex problem in building this instrument was to keep the drive and transmission components quiet.








Section 12.14


          Plate 9 (see Instruments and Music > Glassdance) shows the Glassdance, an instrument that consists of 48 revolving crystal glasses. With respect to parts, materials, and tooling, this is by far the most complex instrument I have built to date. Initially, I planned to build a traditional glass armonica based on the design by Benjamin Franklin (1706–1790). Franklin’s invention requires a series of graduated glass bowls that fit closely inside each other so that only the rims of the bowls are exposed. The bowls are mounted on a horizontal metal axle that passes through holes in the center of the bowls. Musicians rotate the axle by activating a foot pedal, and play the bowls by simply touching the revolving rims. Because the bowls revolve, one may easily play two or three bowls with one hand.


          Franklin’s design poses three significant problems. (1) An instrument builder not trained in glass making must depend on a specialist for the graduated bowls. (2) Since the glassmaker is also the tuner of the bowls, it would be very costly to order a nonstandard series of bowls tuned to a new tuning system. Given such a tuning, and given that the graduated bowls must fit closely inside each other, the dimensions of all the bowls would have to be recalculated. (3) In the event that an old bowl breaks, or that the instrument requires a new bowl with a different tuning, the axle design is exceedingly impractical. If the old bowl is near the center of the axle, then all the other bowls from either the left or the right end of the axle must first be removed to access the old bowl.


          The design of the Glassdance resolves all of these difficulties. First, all the crystal glasses on this instrument were produced by commercial manufacturers. Second, the need for a series of graduated bowls does not exist because each glass has its own center of revolution. Third, because all the glasses revolve independently of each other, it is very easy to install any number of new glasses.


          Plate 10 (see Instruments and Music > Glassdance > Detail) is a detail shot of the inside of the case that houses the drive components of the glasses. Note three strips of black neoprene that divide the front of the case into four separate panels. The panels diminish the transmission of sound from the drive components because they prevent the front from vibrating like a large soundboard. Each panel includes 12 large red chain sprockets, 12 ball bearings (or one ball bearing behind each sprocket), and one red drive chain. These chains do not have individual links like conventional machine or bicycle chains. Instead, they consist of 1/32 in. diameter stainless steel cables covered with polyurethane. Since the chains bend by virtue of being flexible, they are completely silent!


          In addition, each panel includes one large red drive sprocket, one small red idler sprocket, and 2 more ball bearings. These sprockets are partially hidden from view by four large and four small red transmission sprockets mounted on the bottom of the case. A heavier yellow polyurethane transmission chain that drives these sprockets wraps around a raised ninth sprocket in the lower left corner of the case. This sprocket sits at the end of a drive shaft that connects to a variable speed DC motor housed inside a soundproof enclosure. A brown felt-covered table — that I bolted to the back of the Glassdance stand — supports the motor and enclosure. The blue liner on the bottom side of the case is a material called E.A.R., which is an acronym for energy absorbing resin. (See Section 12.3.) This material eliminates the structure-borne sound produced by the ball bearings that support the eight transmission sprockets. A black box near the upper left corner of the case contains the DC motor controls. The left side of the box has an on/off toggle switch and an infinitely adjustable speed control switch. The performer accesses these switches through an opening in the left side of the case.


Section 12.15


          To understand the coupling between the crystal glasses and the chain sprockets, turn to Figure 12.7. Because commercially produced glasses have glass stems that are extremely fragile, I removed all the stems of the crystal glasses on the Glassdance. First, I drilled a hole dead center through the bottom of each glass with a diamond core drill bit attached to a water pump. Next, I cut off the stems with a diamond wire saw. Figure 12.7(A) shows that I passed a round head machine screw (a) through two standoff washers (b), the hole in the bottom of the crystal glass (c), two more standoff washers (b), and a new round aluminum stem (d). A nut (e) at the far end of the machine screw holds these seven parts of the stem assembly together. Figure 12.7(B) shows that I epoxied a neoprene ball bearing sleeve (f), which holds ball bearing (g), into a hole in the back of the Glassdance panel (h). The neoprene sleeve acts as a resilient mounting collar that significantly reduces the transmission of sound from the revolving bearing. I also epoxied an aluminum tube (i) into the inner race of the ball bearing. The hub of a chain sprocket (j) fastens into the inner opening of the tube, and a natural rubber liner (k) slips into the outer opening of the tube. The rubber liner provides a resilient seat for the outer portion of the aluminum stem, and for the large standoff washer outside the crystal glass. Finally, the aluminum stem passes through the rubber liner, the sprocket hub, and a neoprene retainer (l). A hose clamp (m) tightens around the retainer, which holds the crystal glass securely in the panel assembly.





Section 12.16




          . . . Figure 12.9 gives the frequency ratios of the Glassdance. The lowest tone is G4 (or G above middle C) at 392.0 cps. The first “octave” up to G5 at 784.0 cps contains 24 glasses, and the second “octave” up to G6 at 1568.0 cps, 23 glasses. For all rows, the frequencies of the glasses increase from left to right . . .


          (To translate some of these just intoned ratios into conventional Western note names, see M.M. Pages > Just Intonation.)