Musical Mathematics

on the art and science of acoustic instruments

 

Table of Contents

 

 

Buy now at Amazon.com: click New or Used.

 

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

 

© 2013–2017 Cristiano M.L. Forster
All rights reserved.

 

www.chrysalis-foundation.org

 

 

 

*     *     *

 

Chrysalis I

 

 

Chrysalis II

 

 

*     *     *

 

 

 

 

Building New Chrysalis II

 

October 2013–March 2015

 

 

 

 

I

 

 

 

 

 

II

 

 

 

III

 

 

IV

 

 

V

 

 

VI

 

 

 

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.

 

 

 

 

Stringing

 

 

 

 

 

 

 

Chrysalis II