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.


© 2013–2020 Cristiano M.L. Forster
All rights reserved.




*     *     *


Chrysalis I



Chrysalis II



*     *     *





Building New Chrysalis II


October 2013–March 2015




























Butt jointing seven quarter-sawn Sitka spruce boards to make one of two soundboards











Routing eight rib channels and the circumference of the right soundboard











Undercutting the right soundboard and gluing the first birch plywood support ring

















Machining and dry-fitting the ribs of the right soundboard








Drilling and tapping four multi-screw rib clamps











Gluing the first set of four ribs to the right soundboard














Setting up and cutting the octagonal Delrin hub

















Drilling eight spoke holes in the hub











Machining the soundboard spacers























Comparing the spider assembly dimensions to a to-scale drawing











Drilling, routing, and dry-fitting the second birch plywood support rings




















Dry-fitting the spider assembly between the second birch plywood support rings











Machining and installing the buttress support screws






































Machining and installing two axle clamps
























































Dry-fitting and centering the right soundboard with a Delrin axle sleeve








Gluing the second birch plywood support ring to the right soundboard








Turning and installing Honduras rosewood spokes to replace the Delrin spokes














Testing radial and lateral runout








Marking and drilling a template designed for locating and drilling

82 identically spaced tuning gear post holes

and 82 identically spaced nut pin holes

of the left and right soundboards

















Using the drilling template and two transfer punches to mark the centers

of 82 post holes and 82 nut pin holes of the right soundboard








Counterboring the tuning gear post holes for press-fitting the tuning gear

ferrules of the left and right soundboard/support-ring assemblies

















Drilling eight soundboard and two bridge support screw holes in the hub











Epoxying the base of the bridge support screw flange to the right soundboard











Installing the bridge and soundboard support screws of the right soundboard


























Drilling six bridge screw holes and installing the bridge of the right soundboard























Turning the axle to length, then drilling/reaming two holes and

milling four slots to accommodate the three longest

strings of the left and right soundboards




















Installing the machined axle and the three longest strings of the right soundboard

















Machining the ash molding


























Soaking the ash molding in boiling water and clamping it around a form


























Gluing a semicircular ash molding to the right soundboard


Phase I










Phase II








Machining full-length cylindrical tuning gear ferrules

to replace standard partial-length conical ferrules





























Spray finishing the soundboards








Machining the cherry buttons that cover the machine screw holes

in the soundboards of Chrysalis I and Chrysalis II














Color-coding the buttons for Chrysalis I and Chrysalis II











Machining the Delrin nuts or the downbearing supports of the strings at the tuning gears



































Machining the Lexan soundboard shields























Sawing slots into the Delrin nuts


An earlier sequence of photos shows that the circumferences of two Sitka spruce soundboards are encircled by two birch plywood support rings. This construction provides maximum anchorage for the tuning gears and gives maximum structural resistance against bending the soundboards under the tension of the strings.

Since the two circular aluminum bridges are attached directly to the soundboards, they function as true bridges by transmitting mechanical energy from the vibrating strings directly to the soundboards. However, because this stringing design requires that 164 individual Delrin nuts sit on top of the soundboard/plywood assemblies, numerous tests indicate that these nuts have bridge-like functions; they also transmit energy from the strings to the soundboards. Consequently, the nuts significantly affect the quality of sound, or the amplitude, ring-time, and timbre of the vibrating strings.

Instead of the strings passing across the nuts and touching them over an area of contact, I discovered that the strings sound best if they pass through the nuts and only touch them at a point of contact. To achieve a bridge-like contact point, I sawed slots into the nuts with a circular saw blade. The following sequence of drawings and photos shows that by clamping the nuts in the milling machine vice at an angle of 53.5° and passing them into a revolving saw, the saw cuts a tangent line or a straight channel into the nuts. So, although the nuts have a wide base to counteract the downbearing force of the strings at the tuning gears, they also have a point at the top to maximize the transmission of mechanical energy from the vibrating strings to the soundboards.





















Widening the slots to eliminate motion of the nuts while tuning the strings











Assembling the wheel of Chrysalis II: two Sitka spruce soundboards,

two sets of 82 tuning gears, and two circular aluminum bridges




















Based on the dimensions of the original Chrysalis I stand, designing a proportionately

 larger stand for Chrysalis II without changing the critical  distance

 between the floor and the bottom of the new wheel











Two sides of the new birch plywood stand clamped together





Drilling eighteen dowel holes into two sides of the stand clamped together





Drilling three holes for the Delrin ball bearing pillow block supports





Drilling four flat bottom holes for the load bearing maple dowels

on the inner stand surfaces covered with beech veneer





Drilling off-center lock screw holes that prevent the dowels, which hold

the sides of the stand together, from turning on their center screws


Two dowels, like the kind shown below, each have two Honduras rosewood

blocks for mounting the instruments four casters.








Assembling the stand with eighteen maple dowels

and two Delrin ball bearing pillow block supports











Gluing the beech veneer to the outer stand surface











The 82 strings of the right soundboard and a newly powder coated bridge

with a 14° string relief angle on the rim of Ring B

















Installation of 16 brackets made of powder coated brass designed

to counteract movement of the soundboards at the tuning gears

due to the tension of the strings;


and 16 attached pads made of felt covered Delrin designed

to stop the wheel against the top of the knee

while playing with both hands.













Chrysalis II