Flute acoustics: measurement, modelling and design - School of ...

CHAPTER 8. SOFTWARE IMPLEMENTATION 109
Head
UnitCell 1 UnitCell 2
embouchure hole
headjoint
body
tone holes
Figure 8.2: A schematic diagram of a ‘flute’ with two finger holes, showing the conceptual division
of the instrument into a ‘head’ and two ‘unit cells’. Note that tone holes are modelled as
lumped elements and that the various elements are divided at the midpoint of each tone hole.
Keefe (1990) (Figure 8.2). The first segment is the headjoint, which in this context includes the
embouchure hole, upstream air column and downstream air column between the embouchure
hole and first tone hole. The remaining segments are L-shaped unit cells which, as explained
by Keefe, include a single hole and any bore section between the hole and the next hole downstream.
This arrangement confers the distinct advantage that for each frequency the transfer
matrices need only be calculated once. The impedance of the flute fingered differently may
then be obtained by multiplication of saved matrices, thereby reducing computation time dramatically.
A single transfer matrix relates the acoustic pressure and flow at the input of the
instrument to those immediately upstream of the first tone hole. Two transfer matrices are
calculated for each unit cell: one for the open hole and the other for the closed hole.
The Woodwind structure is defined in Woodwind.h as
typedef struct woodwind_str {
Head head;
Vector cells;
double flange;
} *Woodwind;
The member head contains all of the flute dimensions upstream of the first finger hole, and
cells is a linked list of ‘unit cells’, each describing one finger hole and any bore segments
downstream before the next hole. The member flange is a dimensionless number related to
the termination condition at the end of the instrument. It is equal to the wall thickness divided
by bore radius for an open end. For a closed end (such as at the cork of a flute) flange is equal
to -1.
The Head structure is defined as
typedef struct head_str {
EmbouchureHole embouchureHole;
Vector upstreamBore;
double upstreamFlange;
Vector downstreamBore;
Map matrixMap;
} *Head;
and the UnitCell structure as