Calculation AlgorithmThe heart of The Whistle Calculator is
the algorithm that calculates the placement of the tone holes.
All the rest is just there to make it easy for the user to access and
control the calculation.
Historically there have been two main free sources of this calculation. Flutomat, by Pete Kosel, and TWCalc by Daniel Bingamon.
Both of these authors were kind enough to allow me to copy their
algorithms for use in this program. Without this generosity The
Whistle Calculator would not exist, because I know almost nothing about
the physics of whistle music making. I do not think that it is
any secret that these algorithms are approximations. That is,
there is sometimes a discrepancy between the hole positions suggested
by the Whistle Calculator and the positions that are actually
needed. For some combinations of whistle size and key they work
very well, but for other sizes, or keys they may not work nearly as
well. One goal of The Whistle Calculator is to allow the user to
select the most appropriate version of the algorithm for the whistle
I have replicated the algorithms of both Flutomat and TWCalc as
separate pieces of code, and these can be selected on the GUI of The
I have found a minor flaw in the 'live' flutomat code on Pete Kosel's website.
The flaw does not lead to significant differences in the hole
placements until the holes get large relative to the tube bore, or the
tube wall thickness gets large. Even then the differences are small.
The correct algorithm is, however, easier to implement so you will see
a 'corrected flutomat' calculator in the list.
Hans Bracker at Soft Flute has
been working on the fine tuning of the alorithm and I have created a
version of the calculator with his modifications to the algorithm.
As a further development I have now created one calculator that
implements all the others as sub-classes. This is because much of
the mechanism is common between them, with the differences between then
expressed in seven functions.
As a specialised subclass of this 'Unified' calculator I have created a
calculator that allows the user to express the contents of these
The user can then alter the behavious of the calculator in an attempt
calculator has defined for it seven functions:
- closedCorrection: Closed hole for tone hole
n. The length of the vibrating air column is effectively
increased by each closed tone hole which exists above the first open
tone hole. This function is called C_c(n) in the Flutomat code.
- cutoffFrequency: Calculates the cutoff
frequency above which the open hole correction is not valid.
Instrument should be designed so that all second register notes are
well below this frequency. This function is called f_c(n) in the
- endCorrection: Calculates the distance from physical open
end of flute to effective end of vibrating air column. The
vibrating air column ends beyond the end of the flute and C_end is
always positive. This function is called C_end() in the Flutomat code.
- embouchureCorrection: Calculates the distance from theoretical start of air column to the center of embouchure hole; the air column effectively extends beyond the blow hole center by this distance. This function is called C_emb() in the Flutomat code.
- effectiveThickness: Effective wall thickness, i.e. height of air
column at open finger holes; air column extends out past end of hole a
fraction of the hole diameter. This function is called te(n) in the
Flutomat code.\n" ;
- firstHoleDistance: Calculates the effective distance from the
first (\"single\") tone hole to the end of the vibrating air column
when only that hole is open. This function is called C_s() in Flutomat
code. It is only used in iterative mode.
The base version of these functions are functionally equivalent to the "Corrected Flutomat" algorithm.
- subsequentHoleDistance: Calculates the effective distance from
the second and subsequent tone holes to the end of the vibrating air
column when all holes below are open. NOTE: the value of this
correction is invalid if the frequency of the note played is above the
cutoff frequency. This function is called C_o(n) in Flutomat code. It
is only used in iterative mode.;
There are two other sets of functions, Those defined by Hans Bracker,
and those that were part of TWCalc. These extra two sets each
only replace part of the set of functions. Where a function
replaces one in the set, the replacement function is used wherever
needed. So TWCalc redefines effectiveThickness for example.
So wherever effectiveThickness is called when using TWCalc set of
functions, the version supplied by the TWCalc set is used. We say
that the TWCalc version of this function 'overrides' the base version.