[0001] The present invention relates to a technique for synthesizing tones of wind instruments
that generate tones in response to vibration of a reed.
[0002] Heretofore, there have been proposed tone synthesis apparatus of a physical model
type (i.e., physical model tone generators) for synthesizing tones by simulating the
tone generating principles of musical instruments. Among such tone synthesis apparatus
are techniques disclosed in:
R.T. Schumacher "Ab Initio Calculations of the Oscillations of a Clarinet", ACUSTICA,
1981, Volume 48 No. 2, p.75 - p.85 (hereinafter referred to as Non-patent Literature 1); and
S.D. Sommerfeldt, W.J. Strong, "Simulation of a player-clarinet system", Acoustical
Society of America, 1988, 83(5), p.1908 - p.1918 (hereinafter referred to as Non-patent Literature 2). More specifically, Non-patent
Literature 1 discloses a technique for simulating behavior of a clarinet by modeling
a reed as a rigid air valve freely movable in its entirety, and Non-patent Literature
2 discloses a technique for simulating behavior of a clarinet by modeling a reed using
a vibrating member in the form of an elongate plate fixed at one end (i.e., cantilevered
vibrating beam).
[0003] However, although the reed of an actual wind instrument behaves complicatedly in
response to actions of a human player's lip, the techniques disclosed in Non-patent
Literatures 1 and 2 only simulate simple external actions on the reed. Thus, with
these techniques, behavior of the reed of an actual wind instrument can not be reproduced
faithfully, so that it has been difficult to synthesize tones sufficiently approximate
to tones of an actual wind instrument.
[0004] In view of the foregoing, it is an object of the present invention to synthesize
a tone faithfully reflecting therein action of a human player's lip.
[0005] In order to accomplish the above-mentioned object, the present invention provides
an improved apparatus for synthesizing a tone of a wind instrument that is generated
in response to vibration of a reed contacting a lip during blowing or performance
of the wind instrument, which comprises: a first arithmetic operation section that
solves a first motion equation representative of behavior of the reed in an equilibrium
state with external force acting on the lip and a second motion equation representative
of behavior of the lip in the equilibrium state, to thereby calculate displacement
of the lip and displacement of the reed in the equilibrium state; a second arithmetic
operation section that solves a motion equation of coupled vibration of the lip and
the reed with calculation results of the first arithmetic operation section used as
initial values of the displacement of the lip and the displacement of the reed, to
thereby calculate the displacement of the reed; and a tone synthesis section that
synthesizes a tone on the basis of the displacement calculated by the second arithmetic
operation section.
[0006] Because the displacement of the reed is calculated on the basis of the motion equation
of coupled vibration, the present invention can accurately simulate behavior of the
reed as compared to the conventional construction where behavior of the reed is calculated
on the basis of a motion equation that does not reflect therein. As a result, the
present invention can faithfully reproduce tones of an actual wind instrument.
[0007] In a preferred embodiment, each time intensity of the external force acting on the
lip changes, the first arithmetic operation section calculates displacement of the
lip corresponding to the changed intensity of the external force acting on the basis
of the first motion equation and the second motion equation, and the second arithmetic
operation section calculates displacement of the reed by substituting the displacement
of the lip, calculated by the first arithmetic operation section, into the motion
equation of coupled vibration. Because such an arrangement allows any change of the
external force acting on the lip to be reflected in the displacement of the reed,
the present invention can synthesize a variety of tones corresponding to a performance
or rendition style that varies pressing force on the lip.
[0008] In a preferred embodiment, the first motion equation and the second motion equation
include a spring constant of the lip that changes in accordance with a position in
the lip and intensity of pressing force acting on the lip. Such an arrangement can
faithfully simulate the characteristic of an actual lip that a spring constant of
the lip changes in accordance with the intensity of the pressing force and the position
in the lip. As a result, the present invention can accurately synthesize tones of
a wind instrument.
[0009] In a preferred embodiment, the first motion equation includes bending rigidity that
changes in accordance with a position of the reed. Such an arrangement can faithfully
simulate the characteristic of an actual reed that bending rigidity of the reed (product
between a second moment of area and a Young's modulus of the reed MR) changes in accordance
with the position of the reed. As a result, the present invention can accurately synthesize
tones of a wind instrument as compared to the conventional construction where the
reed is simulated with a mere elongated plate-shaped vibrating member that does not
change in sectional shape.
[0010] In a preferred embodiment, the second arithmetic operation section limits the displacement
of the reed to within a predetermined range. Because the displacement of the reed
calculated on the basis of the motion equation of coupled vibration is limited to
within the predetermined range, it is possible to prevent simulation of a situation
where the reed is displaced to outside a displacement range of an actual reed, so
that tones of an actual wind instrument can be reproduced accurately. The range within
which the displacement of the reed is limited is preferably set to a range from the
bottom surface of the lip and a surface of the mouthpiece opposed to the bottom surface.
[0011] In a preferred embodiment, the motion equation of coupled vibration includes at least
one of internal resistance of the lip that changes in accordance with a position in
the lip and internal resistance of the reed that changes in accordance with a position
in the reed. Such an arrangement can simulate a situation where the internal resistance
of the lip and internal resistance of the reed change in accordance with the positions,
and thus, the present invention can faithfully reproduce tones of an actual wind instrument
as compared to the conventional construction where the internal resistance of the
lip and the internal resistance of the reed are set at fixed values.
[0012] In a case where deformation of the lip and reed is relatively small, i.e. where the
deformation is within an elasticity limit), influences imparted from pressing force,
acting on the lip and reed, to the internal resistance of the lip and reed can be
ignored. However, in a case where deformation of the lip and reed is great, i.e. where
the deformation is outside the elasticity limit), the internal resistance of the lip
and reed would also change in accordance with the intensity of the pressing force
as well as positions in the lip and reed. Thus, in a preferred embodiment of the present
invention, the motion equation of coupled vibration includes at least one of internal
resistance of the lip that changes in accordance with a position in the lip and pressing
force acting on the lip and internal resistance of the reed that changes in accordance
with a position in the reed and pressing force acting on the reed. Such an arrangement
can simulate a situation where the internal resistance of the lip and the internal
resistance of the reed change in accordance with the intensity of the pressing force,
and thus, the present invention can faithfully reproduce tones of an actual wind instrument
as compared to the conventional construction where the internal resistance of the
lip and the internal resistance of the reed are set at fixed values.
[0013] The tone synthesis apparatus of the present invention can be implemented not only
by hardware electronic circuitry, such as DSPs (Digital Signal Processors) dedicated
to individual processes, but also by a cooperation between a general-purpose arithmetic
operation processing apparatus and a program. The program of the present invention
is a program for synthesizing a tone of a wind instrument that is generated in response
to vibration of a reed contacting a lip during blowing or performance of the wind
instrument, which causes a computer to perform: a first arithmetic operation step
of solving a first motion equation representative of behavior of the reed in an equilibrium
state with external force acting on the lip and a motion equation representative of
behavior of the lip in the equilibrium state, to thereby calculate displacement of
the lip and displacement of the reed in the equilibrium state; a second arithmetic
operation step of solving a motion equation of coupled vibration of the lip and the
reed with calculation results of the first arithmetic operation step used as initial
values of the displacement of the lip and the displacement of the reed, to thereby
calculate the displacement of the reed; and a tone synthesis step of synthesizing
a tone on the basis of the displacement calculated by the second arithmetic operation
step. Such a program can achieve the same advantageous benefits as the tone synthesis
apparatus of the present invention. Typically, the program of the present invention
is provided to a user in a computer-readable storage medium and then installed into
a computer, or delivered to a user via a communication network and then installed
into a computer.
[0014] The following will describe embodiments of the present invention, but it should be
appreciated that the present invention is not limited to the described embodiments
and various modifications of the invention are possible without departing from the
basic principles. The scope of the present invention is therefore to be determined
solely by the appended claims.
[0015] For better understanding of the object and other features of the present invention,
its preferred embodiments will be described hereinbelow in greater detail with reference
to the accompanying drawings, in which:
Fig. 1 is a block diagram showing an example setup of a first embodiment of a tone
synthesis apparatus of the present invention;
Fig. 2 is a conceptual diagram showing a reed and a neighborhood of the reed in a
wind instrument which are to be simulated by a reed simulating section in the first
embodiment;
Fig. 3 is a schematic view showing contact between a lip and the reed during a performance
of the wind instrument;
Fig. 4 is a block diagram showing functions of the reed simulating section;
Fig. 5 is a conceptual diagram explanatory of discretization in position in an X direction
performed in the first embodiment;
Fig. 6 is a schematic representation of a tubular body portion of the wind instrument;
Fig. 7 is a block diagram showing an example construction of a tubular body model
employed in the first embodiment;
Fig. 8 is a block diagram of a bell section in the tubular body model;
Fig. 9 is a block diagram of a connecting section in the tubular body model;
Fig. 10 is a block diagram of a tone hole portion in the tubular body model;
Fig. 11 is a block diagram of a transmission simulating section;
Fig. 12 is a block diagram of a characteristic parameter conversion section;
Fig. 13 is a block diagram of a shape characteristic parameter conversion section;
Fig. 14 is a diagram explanatory of how a spring constant is measured;
Fig. 15 is a graph showing relationship between pressing force acting on a lip (test
piece) and a spring constant;
Fig. 16 is a block diagram of a characteristic parameter conversion section employed
in a third embodiment of the present invention;
Fig. 17 is graph showing relationship between pressing force acting on the reed and
a displacement amount of the reed; and
Fig. 18 is a block diagram of a characteristic parameter conversion section employed
in a fourth embodiment of the present invention.
<First Embodiment>
[0016] Fig. 1 is a block diagram showing an example setup of a first embodiment of a tone
synthesis apparatus of the present invention. This tone synthesis apparatus 100 is
constructed to synthesize tones by simulating, through arithmetic operations, the
tone generating principles of a single-reed wind instrument, such as a saxophone or
clarinet. As shown in Fig. 1, the tone synthesis apparatus 100 is implemented by a
computer system that comprises an arithmetic operation processing device 10, a storage
device 42 and a sounding device 46.
[0017] The arithmetic operation processing device, such as a CPU (Central Processing Unit)
10, executes programs, stored in the storage device 42, to generate and output tone
data representative of a time-varying waveform of a wind instrument (i.e., temporal
variation of sound pressure). The storage device 42 stores therein programs for execution
by the arithmetic operation processing device 10 and data for use by the arithmetic
operation processing device 10. Magnetic storage device, semiconductor storage device
or other conventionally-known storage device may be employed as the storage device
42.
[0018] The input device 44 includes a plurality of operating members operable by a user
or human player. Via the input device 44, the human player can input, to the arithmetic
operation processing device 10, various parameters to be used for tone synthesis.
Input equipment, such as a keyboard and mouse, and musical-instrument type input equipment,
such as MIDI (Musical Instrument Digital Interface) controller, for inputting information
pertaining to a performance of a wind instrument is employable as the input device
44.
[0019] The sounding device 46 radiates a sound wave corresponding to tone data output by
the arithmetic operation processing device 10. Although not particularly shown in
Fig. 1, the tone synthesis apparatus in practice further includes a D/A converter
for converting tone data into an analog tone signal, and an amplifier for amplifying
and outputting such a tone signal.
[0020] The arithmetic operation processing device 10 functions also as a setting section
12 and a synthesis section 14. In a modification, various functions of the arithmetic
operation processing device 10 may be implemented distributively by a plurality of
integrated circuits. Further, part of the functions of the processing device 10 may
be implemented by dedicated circuitry (DSP) for tone synthesis.
[0021] The setting section 12 sets parameters necessary for tone synthesis. The synthesis
section 14 generates tone data on the basis of the parameters set by the setting section
12, and it includes a reed simulating section 31, a tubular body simulating section
33 and a transmission simulating section 35. The reed simulating section 31 simulates
coupled vibration of the player's lip and the reed. The tubular body simulating section
33 simulates behavior of a tubular portion of the wind instrument from the mouthpiece
to the bell (namely, tubular body portion other than the reed). The transmission simulating
section 35 simulates impartment of transmission characteristics to radiated sounds
from the bell and individual tone holes.
[0022] Fig. 2 is a conceptual diagram showing the reed and neighborhood thereof of the wind
instrument which are to be simulated by the reed simulating section 31. The reed MR
is a vibrating member of an elongated plate shape having one end fixed to the mouthpiece
MP. Let it be assumed here that X, Y and Z axes intersect with one another at an original
point coinciding with a middle point, in a width direction, of a distal end of the
reed MR. The Z axis extends in a width direction of the reed MR. The X axis intersects
with the Z axis in the upper surface (i.e., surface opposed to the mouthpiece MP)
of the reed MR when no external force is acting on the reed MR. Further, the Y axis
extends in a vertical (thickness) direction of the reed MR to intersect with the X
and Z axes.
[0023] Fig. 3 is a schematic exaggerated view of the reed MR and neighborhood thereof, which
are to be simulated by the reed simulating section 31, taken in the Z direction, which
is explanatory of how a human player's lip ML contacts the reed MR at the time of
a performance of the wind instrument. As shown in Fig. 3, the reed simulating section
31 simulates a state where the human player presses the lip ML against the reed MR
with teeth MT during the performance of the wind instrument. The lip ML contacts a
portion of the reed MR from a position xlip1 (adjacent to the distal end of the reed
MR) to a position xlip2 (adjacent to the base of the reed MR) in the X direction.
Further, the teeth MT of the human player contact a portion of the lip ML from a position
xteeth1 (adjacent to the distal end of the reed MR) to a position xteeth2 (adjacent
to the base of the reed MR) in the X direction, to thereby cause pressing force flip(x)
to act uniformly on the reed MR.
[0024] Fig. 4 is a block diagram showing functions of the reed simulating section 31. In
a left area of Fig. 4 are shown parameters set by the setting section 12 and then
stored in the storage device 42. The following lines describe meanings of the parameters.
[0025] First, parameters Stiff(x), Breed(x), A(x), µ reed(x) and ρ reed(x) pertaining to
the reed MR will be described. Stiff(x) represents bending rigidity (N·m
2) of the reed MR at a position x in the X direction. Namely, the bending rigidity
Stiff(x) corresponds to a product between a Young's modulus of the reed MR and a second
moment of area I(x) [m
4] of the reed MR at the position x. As shown in Fig. 2, Breed(x) represents a horizontal
width [m] (i.e., dimension in the Z direction) at the position x, and A(x) is a sectional
area (i.e., area in a Y-Z plane passing the position x) [m
2] of the reed MR at the position x. In the illustrated example, the sectional shape
of the reed MR varies depending on where the position x in the X direction is. Thus,
the second moment of area I(x), horizontal width Breed(x) and sectional area A(x)
of the reed MR to be used in calculation of the bending rigidity Stiff(x) are functions
of the position x. Further, µ reed(x) represents a distribution of internal resistance
[(kg/sec)/m] of the reed MR, and ρ reed(x) represents a density [kg/m
3] of the reed MR.
[0026] Next, parameters klip(x), dlip(x), A(x), µ lip(x) and mlip(x) pertaining to the lip
ML will be described. klip(x) represents a distribution of spring constant [N/m
2], in the X direction, of the lip ML (e.g., spring constant for a unit length, in
the X direction, of the lip ML). dlip(x) represents a dimension in the Y direction
(i.e., thickness) [m] of the lip ML at the position x when no external force acts
on the lip ML. µlip(x) represents a distribution of internal resistance [kg/sec)/m]
of the lip ML at the position x. mlip(x) represents a distribution of mass [kg/m],
in the X direction, of the lip ML. The distribution of spring constant klip(x), thickness
dlip(x), distribution of internal resistance µ lip(x) and distribution of mass mlip(x)
vary depending on where the position x in the X direction is.
[0027] Further, in Fig. 4, P represents pressure (Pa) within the mouth cavity of the human
player, and ρ air represents a density of air (kg/m
3) at normal temperature (e.g., 25°C). H(x) represents a position, in the Y direction,
on the surface of the mouthpiece MP opposed to the reed MR, as seen in Fig. 2; such
a position H(x) will hereinafter be referred to as "facing position". Once displacement
y(x,t), in the Y direction, of the reed MR reaches the facing position H(x), the upper
surface of the reed MR contacts the mouthpiece MP; thus, the facing position H(x)
corresponds to a limit value (i.e., lower limit value) of the displacement of the
reed MR. Further, Zc represents characteristic impedance to an air flow at a starting
point of a portion of the mouthpiece MP that can be regarded as a tubular body (i.e.,
the base of the reed MR).
[0028] As shown in Fig. 4, the reed simulating section 31 comprises first, second, third
and fourth arithmetic operation sections 311, 312, 313 and 314. The first arithmetic
operation section 311 calculates displacement y
0(xf) of the reed MR and displacement yb(xf) of the bottom surface of the lip ML when
the lip ML is in an equilibrium state with pressing force flip(xf) caused to statically
act on a position xf, in the Y direction, of the lip ML. The second arithmetic operation
section 312 calculates displacement y(x,t) in the Y direction at a time t at each
position x, in the X direction, of the reed MR by solving a motion equation of coupled
vibration between the lip ML and the reed MR using the displacement y0(xf) and displacement
yb(xf), calculated by the first arithmetic operation section 311, as initial displacement
values (i.e., values when t = 0) of the reed MR and lip ML. The third and fourth arithmetic
operation sections 313 and 314 calculate pressure POUT of a sound wave to be output
from the reed MR to the tubular body portion (adjacent to the mouthpiece MP) on the
basis of the displacement y(x,t) of the reed MR. Details of processing performed by
the reed simulating section 31 will be discussed below.
[0029] Let's now consider an equilibrium state achieved by causing pressing force flip(xf)
to act from the teeth MT on a position xf (xteeth≦xf ≦xteeth2) of the human player's
lip ML, as shown in Fig. 3. Assuming that the reed MR has been deformed in the Y direction
is deformed in the Y direction by a distance d1 and the lip ML by a distance d2 due
to pressing force flip(xf), resilient force R1 acting from the reed MR on the lip
ML and resilient force R2 acting from the lip ML on the reed MR can be expressed by
the following mathematical expressions. Note that, although in reality the upper surface
of the lip ML contacts the lower surface of the reed MR, Fig. 3 shows in a schematically
simplified manner the upper surface of the lip ML as positioned on the upper surface
of the reed MR.
[0030] From force balance at the contact point (position xf) between the reed MR and the
lip ML, R1 - R2 = 0 is established, and
[0031] From force balance at the contact point (position xf) between the lip ML and the
teeth MT, Flip(xf) = 0 is established.
[0032] Further, from relationship between deformation and displacement of the reed MR, d1
= y0(xf) is established, and
[0033] Further, from relationship between deformation and displacement of the lip ML, d2
= {yb(xf) - dlip(xf) - y0(xf)} is established.
[0034] From the individual mathematical expressions above, Motion Equations A1 and A2 can
be derived.
[0035] The first arithmetic operation section 311 shown in Fig. 4 calculates displacement
yb(xf) of the bottom surface of the lip ML and displacement y0(xf) of the reed MR
by solving Motion Equations A1 and A2 by substituting thereinto the bending rigidity
Stiff(xf), pressing force flip(xf), spring constant klip(xf) and thickness dlip(xf).
More specifically, the first arithmetic operation section 311 calculates displacement
y0(xf) of the reed MR from Motion Equation A1 using difference equation conversion,
Gaussian elimination method or the like and then calculates displacement yb(xf) of
the lip ML by substituting the calculated displacement y0(xf) into Motion Equation
A2. How to solve Motion Equation A1 will be described later.
[0036] Dynamic characteristics when the lip ML and reed MR vibrate in a coupled manner can
be expressed by Motion Equation B below.
[0037] The second arithmetic operation section 312 calculates displacement y(x, t) of the
reed MR by setting the displacement y0(xf), calculated by the first arithmetic operation
section 311, as an initial value of the displacement y(xt) of the reed MR and substituting
the displacement yb(xf), calculated by the first arithmetic operation section 311,
into the displacement yb(x) of the lip ML in Motion Equation B. The right side of
Equation B represents external force fex(x) acting on the position x, in the X direction,
of the reed MR. First, the second arithmetic operation section 312 calculates external
force fex(x) by not only substituting into the right side of Motion Equation B the
parameters breed(x), P, klip(x) and dlip(x) set by the setting section 12 and pressure
p(t) calculated by the fourth arithmetic operation section 314 but also substituting
the displacement y0(xf) and displacement yb(xf), calculated by the first arithmetic
operation section 311, into the right side of Motion Equation B as initial values
of the displacement y(x, t) and displacement yb(x). The pressure p(t) is pressure
in a portion of a gap between the reed MR and the mouthpiece MP close to the distal
end of the reed MR (hereinafter referred to as "immediately-above-reed portion").
Calculation, by the fourth arithmetic operation section 314, of the pressure p(t)
will be described later.
[0038] Second, the second arithmetic operation section 312 calculates displacement y(x,
t) of the reed MR by substituting the parameters mlip(x), A(x), µ reed(x), Stiff(x)
and ρ reed, set by the setting section 12, into the left side of Motion Equation B
and setting the external force fex(x) calculated earlier into the right side of Motion
Equation B. How to solve Motion Equation B will be described later.
[0039] The second term in the left side of Motion Equation B can be transformed as follows:
[0040] Therefore, Motion Equation B can be transformed into Equation B1 below.
[0041] Next, the time t is discretized as a product between an integer i and a predetermined
value Δt (i.e., t = i · Δt), and then the time derivatives are substituted by the
following differences.
[0042] Further, as shown in Fig. 5, the position x in the X direction is discretized in
such a manner that the discretized positions are distributed at equal intervals Δx.
Namely, the position x is discretized as a product between an integer n and a predetermined
value Δx (i.e., x = n · Δx), and then the position derivatives are substituted by
the following differences.
Note that "y(n, i)" above is an abbreviation of y(n · Δx, i · Δt). Thus, Mathematical
Expression B1 above can be rewritten as Equation B2 below.
Note, however, that, in Equation B2 above, the individual terms are results of the
following substitutions:
Note that "(n, i)" added to some letters in Equation B2 above is an abbreviation
of y(n · Δx, i · At).
[0043] Next, Equation B3 approximately expressing Equation B2 above is derived by adding
together (1) an equation obtained by multiplying the second term through to the fourth
term in the left side of Equation B2 by 1/2 and (2) an equation obtained by substituting
"i" in Equation B2 by (i + 1) and then multiplying the second term through to the
fourth term in the left side of Equation B2 by 1/2.
[0045] Assuming that the reed MR is fixed to the mouthpiece MP at a position N as shown
in Fig. 5, y(N, i) and y(N+1, i) become zero at a given time point i. Further, because
acceleration (∂
2y(0, i)/ ∂ x
2) and sheer force (∂
3y(0, i)/∂ x
3) become zero at the distal end of the reed MR where no external force acts (n = 0),
the following Equation B4_1 and Equation B4_2 are established:
[0046] Further, the following Equation B4_3 is derived by adding together Equation B4_1
and Equation B4_2, and the following Equation B4_4 is derived by subtracting Equation
B4_2 from three times of Equation B4_3.
[0047] Further, the following Equation B4_5 is derived by substituting 2 into n in Equation
B4 above.
[0048] Further, the following Equation B5 is derived from an equation derived by substituting
n = 3 to N - 1 into Equation B4 and from Equation B4_3 and Equation B4_4.
[0049] The Gaussian elimination method is suitable as a solution method for Equation B5
above. Because two rows and two columns in a left upper portion of Equation B5 above
constitute a diagonal matrix by Equation B4_3 and Equation B4_4 being derived from
Equation B4_1 and Equation B4_2, there can be achieved the benefit that the necessary
quantity of arithmetic operations to be performed in the Gaussian elimination method
can be reduced.
[0050] The second arithmetic operation section 312 calculates displacement y(x, t) of the
reed MR by solving Equation B5 using the displacement (y0(xf), yb(xf)), calculated
by the first arithmetic operation section 311, as initial values of the displacement
y(x, y) and yb(x). More specifically, the second arithmetic operation section 312
first calculates variables y(0, i+1) to y(N-1, i+1), representing future displacement,
in the left side of Equation B5, by not only substituting variables y0(0) - y0(N-1)
and y0(2) to y0(N-1), calculated by the first arithmetic operation section 311, into
both of the variables y0(0, i) to y0(N-1, i), representing current displacement, in
the right side of Equation B5 and variables y(2, i-1) to y(N-1, 1-1), representing
previous displacement, in the right side of Equation B5 but also substituting the
displacement yb(xf), calculated by the first arithmetic operation section 311, into
yb(2) to yb(N-1) of Equation B5. Second, in order to advance the time by Δt, the second
arithmetic operation section 312 calculates variables y(0, i+1) to y(N-1, i+1), representing
future displacement in the left side of Equation B5, by solving Equation B5 by not
only substituting variables y(2, i) to y(N-1, i), representing current displacement,
into variables y(2, i-1) to y(N-1, i-1), representing previous displacement, in the
right side of Equation B5, but also substituting variables y(0, i+1) to y(N-1, i+1),
representing last-calculated future displacement, into variables y(0, i) to y(N-1,
i), representing current displacement, in the right side of Equation B5. By repeating
the above-mentioned arithmetic operations for calculating the displacement y(0, i+1)
to y(N-1, i+1) at the time point (i+1) by solving Equation B5 by substituting thereinto
the displacement y(0, i) to y(N-1, i) at the time point i, the second arithmetic operation
section 312 calculates a change over time of the displacement y(x, t) at each position
x of the reed MR.
[0051] Further, each time the pressing force flip(x) set by the setting section 12 changes,
the first arithmetic operation section 311 calculates new y0(xf) and yb(xf) by substituting
the changed pressing force flip(x) into the pressing force flip(xf) in Motion Equations
A1 and A2. Each time the first arithmetic operation section 311 calculates new displacement
yb(xf), the second arithmetic operation section 312 updates the numerical value to
be substituted into yb(2) to yb(N-1) with the new displacement yb(xf). With the aforementioned
arrangements, it is possible to synthesize tones faithfully reproducing a style of
performance or rendition where the pressing force flip(xf) is changed as desired.
However, even when the first arithmetic operation section 311 has calculated new displacement
y0(xf) in response to a change of the pressing force flip(xf), the second arithmetic
operation section 312 does not reflect the calculated new displacement y0(xf) for
the displacement y(0, i) to y(N-1, i) of Equation 5. Thus, with the aforementioned
arrangements, it is possible to avoid any discontinuous change of the displacement
y(x, t), so that auditorily natural tones can be generated.
[0052] As shown in Fig. 4, the second arithmetic operation section 312 includes a range
limiting section 32 that limits the displacement y(x, t) of the reed MR to within
a predetermined range. The range limiting section 32 limits the displacement y(xt)
of the reed MR, calculated from Equation B5, to a range from the displacement yb(xf)
of the lip ML (i.e., position of the bottom surface of the lip ML which the teeth
MT contacts), calculated by the first arithmetic operation section 311, to the facing
position H(x) set by the setting section 12. Namely, when the displacement y(x, t)
of the reed MR exceeds the displacement yb(xf) in the case where the value of downward
displacement, in the Y direction, of the reed MR exceeds that of the lip ML is considered
to be positive), the range limiting section 32 changes the displacement y(x, t) to
the displacement yb(xf), but when the displacement y(x, t) of the reed MR exceeds
(falls below) the facing position H(x), the range limiting section 32 changes the
displacement y(x, t) to the facing position H(x). With the aforementioned arrangements,
it is possible to avoid simulation of an absurd situation where the reed MR is located
beneath the bottom surface of the lip ML or above the mouthpiece MP. The displacement
yb(x) of the bottom surface of the lip ML has been described above as the upper limit
value of the displacement y(x, t) of the reed MR, but, because the lip ML has a thickness,
a given position closer to the facing position H(x) than the displacement yb(x) by
a predetermined value corresponding to the thickness of the lip ML (e.g., a fixed
value corresponding to a minimum value of the thickness of the lip ML, or a variable
value corresponding to a minimum value of the thickness of the lip ML and variable
in accordance with the pressing force flip(x)).
[0053] Note that the same method as used for the calculation, by the second arithmetic operation
section 312, of the displacement y(x, t) is used for the calculation, by the first
arithmetic operation section 311, of the displacement y0(x) (i.e., solution for Motion
Equation A1), as outlined below. Motion Equation A1 is transformed into the following
Difference Equation A1_A1 in a similar manner to the above-mentioned transformation
from Motion Equation B1 to Equation B2.
If the individual terms in Equation A1_A1 are rearranged per type of the variable
y, the following Equation A1_2 can be derived:
Note, however, that the individual terms in Equation A1_2 are ones previously substituted
as follows:
[0054] Equation A1_2 is transformed into the following Difference Equation A1_3 in a similar
manner to the above-mentioned transformation from Equation B4 to Equation B5.
[0055] The first arithmetic operation section 311 calculates displacement y0(x) (y(0) to
y(N-1) in Equation A1_3) using the Gaussian elimination method or the like. The foregoing
has been a specific example of the solution for Motion Equation A1.
[0056] The third arithmetic operation section 313 of Fig. 4 calculates a volume flow rate
f(t) in the immediately-above-reed portion on the basis of the parameters H(x), ρ
air, breed(x) and Zc set by the setting section 12 and the displacement y(x, t) calculated
by the second arithmetic operation section 312. The third arithmetic operation section
313 in the instant embodiment calculates, as the volume flow rate f(t) of the immediately-above-reed
portion, a difference value between a volume flow rate U(t) resulting from a pressure
difference between the upper and lower surfaces of the reed MR, and a volume flow
rate u(t) resulting from displacement (y(x, t)) of various portions of the reed MR
(namely, f(t) = U(t) - u(t)).
[0057] The volume flow rate u(t) can be expressed by the following Equation C1, where "leff
represents a distance from the distal end to the supporting point of the reed MR (i.e.,
effective length of the reed MR).
[0058] The third arithmetic operation section 313 calculates the volume flow rate u(t) by
substituting into Equation C1 the width Breed(x) of the reed MR set by the setting
section 12 and a time derivative of the displacement y(x, t) (i.e., velocity of the
reed MR) calculated by the second arithmetic operation section 312 to perform numeric
integration, such as the Simpson's method.
[0059] Further, the volume flow rate U(t) can be calculated in accordance with the following
arithmetic operational sequence. First, the third arithmetic operation section 313
calculates a gap ξ (t) [m] between the mouthpiece MP and the reed MR at the distal
end of the reed MR. More specifically, the gap ξ (t) calculates, as the gap ξ (t),
a difference between displacement y(0, t) of the distal end (x = 0) of the reed MR
of the displacement y(x, t) of the reed MR, calculated by the second arithmetic operation
section 312, and a facing position H(0) at the distal end (x = 0) (i.e., gap ξ (t)
= y(0, t) - H(0)).
[0060] Then, the third arithmetic operation section 313 calculates effective mass M(t) [Kg]
of air passing through the gap between the mouthpiece MP and the reed MR. The effective
mass M(t) can be expressed by the following equation C2:
, where R(t) represents a relative ratio between the horizontal width Breed(0) and
the gap ξ (t) at the distal end of the reed MR (i.e., ration R(t) = Breed(0)/ ξ(t)).
Namely, the third arithmetic operation section 313 calculates effective mass M(t)
by substituting into Equation C2 the horizontal width Breed(0) and air density ρ air
of the reed MR, set by the setting section 12, and the relative ratio R(t).
[0061] For the effective mass M(t) and volume flow rate U(t), the following Equation C3
is established:
, where A represents a predetermined coefficient (e.g., A = 0.0797). The following
method is used in the calculation of the volume flow rate U(t) using Equation C3 above.
[0062] Equation C3 can be transformed into the following Equation C4 using Equation D1 and
Equation D2 to be described later:
[0064] The third arithmetic operation section 313 calculates, as a volume flow rate f(t),
a difference between the volume flow rate U(t) and the volume flow rate (t) calculated
in accordance with the above-described arithmetic operation sequence.
[0065] The fourth arithmetic operation section 314 of Fig. 4 calculates output wave pressure
POUT(t) and sound pressure p(t) of the immediately-above-reed portion p(t). The output
wave pressure POUT(t) is pressure of a sound wave traveling forward from the reed
MR through the interior of the tubular body portion (hereinafter referred to as "output
wave"). Portion of the sound wave traveling from the reed MR through the interior
of the tubular body reflects off an open end (bell) of the wind instrument, and then
that portion having traveled through the interior of the tubular body (hereinafter
referred to as a "reflected wave") travels backward through the interior of the tubular
body to reach the interior of the mouthpiece MP. Thus, the output wave pressure POUT(t)
corresponds to a sum of pressure produced by the volume flow rate f(t) and pressure
PIN of the reflected wave traveling from the interior of the tubular body to the mouthpiece
MP (this pressure will be referred to as "reflected wave pressure PIN"). The reflected
wave pressure PIN is calculated or arithmetically determined by the tubular body simulating
section 33.
[0066] Because the pressure produced by the volume flow rate f(t) is a product between the
volume flow rate f(t) and the characteristic impedance Zc, the output wave pressure
POUT(t) can be expressed by the following equation D1:
[0067] The fourth arithmetic operation section 314 calculates the output wave pressure POUT(t)
by substituting into Equation D1 above the characteristic impedance Zc set by the
setting section 12, volume flow rate f(t) calculated by the third arithmetic operation
section 313 and reflected wave pressure PIN calculated by the tubular body simulating
section 33.
[0068] Because the output wave pressure POUT(t) and reflected wave pressure PIN act on the
immediately-above-reed portion, the sound pressure p(t) of the immediately-above-reed
portion p(t) can be expressed by the following Equation D2:
[0069] The fourth arithmetic operation section 314 calculates the pressure P(t) by substituting
into Equation D2 above the output wave pressure POUT(t) calculated on the basis of
Equation D1 and reflected wave pressure PIN(t) calculated by the tubular body simulating
section 33. The pressure P(t) calculated by the fourth arithmetic operation section
314 is fed back to the calculation (Equation B) of the external force fex(x) by the
second arithmetic operation section 312 and calculation (Equation C) of the volume
flow rate U(t) by the third arithmetic operation section 313.
[0070] Next, a description will be given about the functions of the tubular body simulating
section 33. As shown in Fig. 6, a tubular body section (extending from the mouthpiece
to the bell) of an actual wind instrument can be approximated by a structure comprising
k (k is a natural number) tubular unit portions U (U[1] - U[k]) connected together
in series. Diameters and overall lengths of the individual tubular unit portions (namely,
shape of each of the tubular body portions) are variably set. The tubular body simulating
section 33 realizes behavior of a sound wave inside the tubular body portion by use
of a physical model (hereinafter referred to as "tubular body model") simulating the
structure of Fig. 6.
[0071] Fig. 7 is a block diagram showing an example construction of the tubular body model
used by the tubular body simulating section 33. As shown in Fig. 7, the tubular body
model includes: delay elements DA (DA[1] - DA[k]) provided on a path r1 in corresponding
relation to the unit portions U; delay elements DB (DB[1] - DB[k]) provided on a path
r2 in corresponding relation to the unit portions U, junctions or connecting sections
J (J[1] - J[k-1]) provided between adjacent ones of the delay elements DA and between
adjacent ones of delay elements DB; hole portions TH (TH[1] - TH[k-1]) connected to
some of the connecting sections J which are located at positions corresponding to
tone holes of the wind instrument; and a bell section BL corresponding to the bell
of the wind instrument. The path r1 simulates behavior of an output wave traveling
through the interior of the tubular body portion from the mouthpiece MP to the bell
(i.e., output wave pressure POUT(k, t)), while the path r2 simulates behavior of an
output wave traveling through the interior of the tubular body portion from the bell
to the mouthpiece MP (i.e., reflected wave pressure PIN(k, t)).
[0072] The delay element DA[i] of an i (i = 1 - k)-th stage is an element for delaying output
wave pressure POUT(i, t), supplied from a preceding stage, by a predetermined delay
amount dA[i]; for example, it is a shift register that differs in the number of stages
in accordance with the delay amount dA[i]. Output wave pressure POUT(t) calculated
by the reed simulating section 31 (fourth arithmetic operation section 314) is supplied,
as an initial value POUT(1, t), to the delay element DA[1] of the first stage to be
sequentially delayed by the delay elements DA[1] - DA[k] of the individual stages,
and then reaches the bell section BL. Namely, the delay element DA[i] simulates a
propagation delay of the output wave pressure POUT(i, t) in the 1-th unit portion
U[i].
[0073] The bell section BL simulates radiation of a sound wave from the bell of the wind
instrument and reflection of the sound wave at the distal end of the bell. A shown
in Fig. 8, the bell section BL includes a filter section 62 and a multiplication section
64. Output wave pressure POUT(k, t) output from the delay element DA[k] of the k-th
stage (i.e., last stage) on the path r1 is supplied to the bell section BL. The filter
section 62 includes a low-pass filter portion 621 and a subtraction portion 622. The
low-pass filter portion 621 filters out components of a time waveform of the output
wave pressure POUT(k, t), output from the k-th stage delay element DA[k], which exceed
a cutoff frequency fCB. Multiplied value CB of a multiplier in the low-pass filter
portion 621 is a value that satisfies CB = 2π·fCB·Δt. The subtraction portion 622
calculates radiated sound pressure PB(t) by subtracting the output of the low-pass
filter portion 621 from the output wave pressure POUT(k, t) of the k-th stage delay
element DA[k]. Namely, the subtraction portion 622 functions as a high-pass filter
that filters out components of the output wave pressure POUT(k, t) which fall below
the cutoff frequency fCB. The radiated sound pressure PB(t) is equivalent to pressure
of the sound wave radiated from the bell.
[0074] The multiplication section 64 simulates reflection of a sound wave at a boundary
between inner and outer sides of the bell of the wind instrument. Namely, the multiplication
section 64 calculates reflected wave pressure PIN(k, t) by multiplying the output
from the low-pass filter portion 621 by a coefficient rB and then outputs the calculated
reflected wave pressure PIN(k, t) to the path r2 (more specifically, to the delay
element DB[k] of Fig. 7). Because the sound wave reverses its phase and causes some
loss at the time of the reflection, the coefficient rB is set at a negative number
whose absolute value is, for example, smaller than one.
[0075] Similarly to the delay element DA[i], the delay element DB[i] of Fig. 7 delays reflected
wave pressure PIN(i, t), input from a preceding stage (closer to the bell section
BL), by a predetermined delay amount dB[i]. Namely, the delay element DB[i] simulates
a propagation delay of the reflected wave pressure PIN(k, t) in the i-th unit portion
U[i]. The reflected wave pressure PIN(k, t) calculated by the bell section BL are
sequentially delayed by the delay elements DB[k] - DB[1], and the reflected wave pressure
PIN(1, t) output from the first-stage delay element DB[1] is used, as reflected wave
pressure PIN(t), in arithmetic operations by the reed simulating section 31 (fourth
arithmetic operation section 314).
[0076] The connecting section (or junction) J simulates output wave diffusion and energy
loss arising from inner diameter variation of the tubular body portion. The connecting
section (or junction) J may be of either a two-port type as shown in (A) of Fig. 9
or a three-port type as shown in (B) of Fig. 9. The two-port type connecting section
J[i] includes: a multiplication section 71 for multiplying output wave pressure POUT(i,
t), supplied via the path r1, by a coefficient αi; a multiplication section 72 for
multiplying reflected wave pressure PIN(i+1, t), supplied via the path r2, by a coefficient
βi; an addition section 73 for adding together an output (αi·POUT(i, t)) from the
multiplication section 71 and an output (βi · PIN(i+, t)) from the multiplication
section 72; a subtraction section 74 for outputting a difference between the output
from the addition section 73 and the output wave pressure POUT(i, t) to the path r2
as new reflected wave pressure PIN(i, t); and a subtraction section 75 for outputting
a difference between the output from the addition section 73 and the reflected wave
pressure PIN(i+1, t) to the path r1 as new output wave pressure POUT(i+1, t). Such
a two-port type connecting section J[i] is employed where no tone hole portion TH
is connected, such as the connecting sections J[1] and J[2] shown in Fig. 7.
[0077] The three-port type connecting section J[i] shown in (B) of Fig. 9 is employed where
a tone hole portion TH is connected, such as the connecting sections J[3] and J[4]
shown in Fig. 7. The three-port type connecting section J[i] includes, in addition
to the aforementioned components of the two-port type connecting section J[i], a subtraction
section 76 for outputting a difference between the output from the addition section
73 and sound pressure Ri(t) output from the i-th tone hole portion TH[i] to the tone
hole portion TH[i] as sound pressure Qi(t), and a multiplication section 77 for multiplying
the sound pressure Ri(t) by a coefficient γi.
[0078] The tone hole portion TH[i] simulates radiation of a sound wave from an i-th tone
hole and reflection of the sound wave at the tone hole. As shown in Fig. 10, the tone
hole portion TH[i] includes delay elements DE1 and DE2, a filter section 66 and a
multiplication section 68, similarly to the bell section BL of Fig. 8. The delay element
DE1 delays sound pressure Qi(t), supplied from the three-port connecting second J[i],
by a delay amount dE1. The filter section 66 includes a low-pass filter section 661
for filtering out components of the delayed sound pressure Qi(t) which exceed a cutoff
frequency fCTH, and a subtraction section (high-pass filter) 662 for calculating radiated
sound pressure PHi(t) by subtracting the output of the low-pass filter section 661
from the sound pressure Qi(t). Multiplicities value CTH of a multiplier in the low-pass
filter portion 661 is a value that satisfies CTH = 2π · fCTH · Δt. The radiated sound
pressure PHi(t) is equivalent to pressure of the sound wave radiated from the i-th
tone hole. The multiplication section 68 calculates sound pressure Ri(t) by multiplying
the output of the low-pass filter section 661 by a coefficient rHi (e.g., positive
or negative number whose absolute value is, for example, below one), in order to simulate
a situation where phase inversion does not occur when the i-th tone hole is closed
or where sound wave loss and phase inversion occur when the tone hole is opened. Namely,
the multiplication section 68 simulates reflection of a sound wave at a boundary between
inside and outside of the tone hole. The sound pressure Ri(t) is delayed by the delay
element DE2 by a delay amount dE2 and then output to the three-port connecting section
J[i] (multiplication section 77). The foregoing has been a discussion of the functions
of the tubular body simulating section 33.
[0079] The transmission simulating section 35 of Fig. 1 simulates impartment of transmission
characteristics to radiated sounds from the bell and individual tone holes of the
wind instrument. As shown in Fig.11, the transmission simulating section 35 includes
a multiplication section 351 corresponding to the bell, k multiplication sections
353 corresponding to the unit portions U[1] - U[k], and an addition section 355 for
adding together the outputs of the multiplication section 351 and k multiplication
sections 353. The multiplication section 351 multiplies sound pressure PB(t), calculated
by the bell section BL, by a coefficient MB. The i-th multiplication section 353 multiplies
radiated sound pressure PHi(t), calculated by the tone hole portion TH[i], by a coefficient
MHi. The coefficient MHi is set at 0 when the i-th tone hole is closed or not provided
in the wind instrument, but set at a predetermined value greater than 0, such as 1,
when the i-th tone hole is opened. Thus, listening sound pressure Pmix(t) calculated
by the addition section 355 represents sound pressure of a sound wave (listening sound)
comprising a mixture of the radiated sound from the bell and radiated sound from a
tone hole that is opened by a human player. The listening sound pressure Pmix(t) is
output, as tone data, from the arithmetic operation processing device 10 to the sounding
device 46.
[0080] Next, a description will be given about the setting section 12. As shown in Fig.
1, the setting section 12 includes a characteristic parameter conversion section 21
and a shape characteristic parameter conversion section 23. The characteristic parameter
conversion section 21 converts various parameters, pertaining to characteristics of
the reed MR and lip ML, to parameters necessary for tone synthesis. The shape characteristic
parameter conversion section 23 converts various parameters, pertaining to the shape
and dimensions of the wind instrument, to parameters necessary for tone synthesis.
[0081] Fig. 12 is a block diagram showing specific functions of the characteristic parameter
conversion section 21. The user operates the input device 44 to input or designate
various parameters, listed in a left region of Fig. 12, to the arithmetic operation
processing device 10. Among such parameters designated by the user are physical property
values pertaining to air (i.e., Cair and ρair), physical property values pertaining
to the lip ML (
ρ lip, Elip and tan
δ lip), a dimension pertaining to a particular sample of the lip (hereinafter referred
to as "lip sample") (blip_sample), physical property values pertaining to the reed
MR (ρreed, Ereed and tan δ reed), dimensions pertaining to a particular sample of
the reed (hereinafter referred to as "reed sample") (breed_sample, lreed_sample and
dreed_sample), breath pressure Po, and tone pitch fn.
[0082] The parameter Cair represents the sound speed [m/sec] in air, and the parameter ρ
air represents the density [kg/m
3] of air. The breath pressure P0 represents air pressure within the mouth cavity of
the user or human player during a performance of the wind instrument. The tone pitch
fn is a numerical value indicative of a pitch of a tone to be synthesized by the arithmetic
operation processing device 10. Desired performance tone can be synthesized by appropriately
changing the tone pitch fn.
[0083] The physical property values pertaining to the lip ML includes density ρ lip [kg/m
3] of the lip ML, Young's modulus Elip [Pa] of the lip ML, and loss coefficient tanδlip
of the lip ML. The physical property values pertaining to the lip sample include a
width (i.e., dimension in the Z direction) blip_sample [m]. The lip sample is a structure
made of a material which has generally the same physical characteristics as an actual
human lip but is different from the actual human lip in that it is simplified in shape
into a plain three-dimensional shape (rectangular parallelepiped in the illustrated
example). Thus, the horizontal width (i.e., dimension in the Z direction) blip_sample
is a fixed value that does not depend on the position in the X direction. In place
of the aforementioned arrangement where the user individually inputs the physical
property values and dimensions pertaining to the lip ML and lip sample, the instant
embodiment may employ an arrangement where values of the individual parameters (ρlip,
Elip, tan δ lip and blip_sample) are stored in advance in the storage device 42 in
association with a plurality of types of lips ML so that the characteristic parameter
conversion section 21 can acquire, from the storage device 42, values of the parameters
pertaining to a particular type of lip ML selected by the user via the input device
42.
[0084] The physical property values pertaining to the reed MR include density
ρ reed [kg/m
3] of the reed MR, Young's modulus Ereed [Pa] of the reed MR, and loss coefficient
tan δ reed of the reed MR. The physical property values pertaining to the reed sample
include a horizontal width (i.e., dimension in the Z direction) breed_sample [m],
a length (i.e., dimension in the X direction) lreed_sample [m], and a thickness (i.e.,
dimension in the Y direction) dreed_sample [m]. The reed sample is a structure made
of a material which has generally the same physical characteristics as an actual reed
but is different from the actual reed in that it is simplified in shape into a plain
three-dimensional shape (rectangular parallelepiped in the illustrated example). Thus,
the physical property values (breed_sample, lreed_sample and dreed_sample) pertaining
to the reed are fixed values. In place of the aforementioned arrangement where the
user individually inputs the physical property values and dimensions pertaining to
the reed MR and reed sample, the instant embodiment may employ an arrangement where
values of the individual parameters (ρ reed, Ereed, tan δ reed, breed_sample and lreed_sample)
are stored in advance in the storage device 42 in association with a plurality of
types of reeds MR so that the characteristic parameter conversion section 21 can acquire,
from the storage device 42, values of the parameters pertaining to a particular type
of reed MR selected by the user via the input device 42.
[0085] The characteristic impedance Zc of the mouthpiece MP of the wind instrument can be
expressed by the following Mathematical Expression (a1):
[0086] As shown in Fig. 12, the characteristic parameter conversion section 21 calculates
the characteristic impedance Zc by performing Mathematical Expression (a1) above with
respect to the sound speed cair, density ρair and diameter φ in. Note that φ in represents
an inner diameter [m] of the mouthpiece MP at the base of the reed MR (i.e., portion
of the reed MR fixed to the mouthpiece MP). For example, the inner diameter φ 1 of
the first unit portion U[1] of the tubular body model is used as the diameter φ in.
[0087] Further, a distribution of spring constant klip(x) [N/m
2] of the lip ML can be expressed by the following Mathematical Expression (a2):
[0088] As shown in Fig. 12, the characteristic parameter conversion section 21 calculates
a distribution of spring constant klip(x) [N/m
2] of the lip ML with respect to the physical property values and dimensions (Elip,
blip(x) and dlip(x)) of the lip ML. In Mathematical Expression (a2) above, the horizontal
width blip(x) and thickness dlip(x) at the position x in the X direction can be determined
from the tone pitch fn, as will be described later.
[0089] Distribution of inner resistance µ lip(x) of the lip ML can be expressed by the following
Mathematical Expression (a3), in which mlip_sample represents a mass [kg] of the lip
sample, llip_sample represents a length, in the X direction, of the lip sample, and
klip_sample represents a distribution of spring constant [N/m] of the lip sample.
[0090] As shown in Fig. 12, the characteristic parameter conversion section 21 calculates
the distribution of inner resistance
µ lip (x) of the lip ML by performing arithmetic operations of Mathematical Expression
(a3) with respect to the physical property values (ρ lip, Elip and tan δ lip) of the
lip ML and dimensions (blip_sample) of the lip ML. Note that, because the distribution
of inner resistance
µ lip(x) is represented by the calculated value of Mathematical Expression (a3) for
the lip sample of a simple parallelepiped shape, the distribution of inner resistance
µ lip(x) takes a fixed value that does not depend on the position x.
[0091] Distribution of inner resistance
µ reed(x) of the reed MR, on the other hand, can be expressed by the following Mathematical
Expression (a4), in which mreed_sample represents a mass [kg] of the reed sample,
Ireed_sample represents a second moment of area of the reed sample [m
4], and kreed_sample represents a distribution of spring constant [N/m] of the reed
sample.
[0092] As shown in Fig. 12, the characteristic parameter conversion section 21 calculates
the distribution of inner resistance µreed(x) of the reed MR by performing arithmetic
operations of Mathematical Expression (a4) with respect to the physical property values
(ρ reed, Ereed and tan δ lip) of the reed MR and dimensions (breed_sample, dreed_sample
and lreed_sample) of the reed sample. Note that, because the distribution of inner
resistance
µ reed(x) is represented by the calculated value of Mathematical Expression (a4) for
the reed sample of a simple parallelepiped shape, the distribution of inner resistance
µ reed(x) takes a fixed value that does not depend on the position x.
[0093] Further, as shown in Fig. 12, the characteristic parameter conversion section 21
determines a plurality of parameters (blip(x), dlip(x), xteeth1, xteeth2, xlip1, xlip2
and Flip(x)) pertaining to an embouchure (i.e., state of the lip ML during a performance),
a coefficient for adjusting the breath pressure P0 and a plurality of parameters (rHl
- rHk, rB, MH1 - MHk and MB) pertaining to fingering of the wind instrument on the
basis of the tone pitch fn through a key scale process ("KSC" in Fig. 12). The key
scale process is a process for determining values of various parameters, corresponding
to an actually designated tone pitch fn, from a table where various numerical values
the tone pitch fn can take and values of the parameters are associated with each other.
[0094] The plurality of parameters pertaining to an embouchure include a horizontal width
(i.e., dimension in the Z direction) blip(x) of the lip ML, a thickness (i.e., dimension
in the Y direction) dlip(x) [m] of the lip ML when no external force acts on the lip
ML, force Flip(x) [N] with which the human player's teeth MT press the lip ML, and
parameters (xlip1, xlip2, xteeth1 and xteeth2) pertaining to positions of the human
player's lip ML and teeth MT relative to the reed MR.
[0095] Further, the characteristic parameter conversion section 21 determines a horizontal
width blip(x) and thickness dlip(x) of the lip ML corresponding to the tone pitch
fn through the key scale process and calculates a distribution of mass mlip(x) [kg/m]
by multiplying a product between the width blip(x) and the thickness dlip(x) by the
density
ρ lip of the lip ML. The horizontal width blip(x) and thickness dlip(x) are also applied
to the aforementioned calculation of the distribution of spring constant klip(x).
[0096] In order to discretize the individual positions x in the X direction as shown in
Fig. 5, the characteristic parameter conversion section 21 arithmetically determines,
as discretized positions (nlip1, nlip2), numerical values obtained by dividing the
positions (xlip1, xlip2) by a distance Δx, and arithmetically determines, as discretized
positions (nteeth1, nteeth2), numerical values obtained by dividing the positions
(xteeth1, xteeth2) by the distance Δx. Further, the characteristic parameter conversion
section 21 determines, as discretized positions (nlip1, nlip2), numerical values obtained
by dividing the positions (xlip1, xlip2) by a distance Δx, and determines, as discretized
positions (nteeth1, nteeth2), numerical values obtained by dividing the positions
(xteeth1, xteeth2) by the distance Δx. Further, the characteristic parameter conversion
section 21 determines, as a length lteeth in the X direction of the teeth MT, a difference
between the positions xteeth1 and xteeth2, and determines, as a length llip in the
X direction of the lip ML, a difference between the positions xlip1 and xlip2. Then,
the characteristic parameter conversion section 21 determines pressing force flip(x)
[N/m] acting from the teeth MT approximately on a unit length flip(x) [N/m] (flip(x)
= Flip(x) / lteeth).
[0097] Further, the characteristic parameter conversion section 21 determines a pressure
P within the mouth cavity of the human player by determining a coefficient pmul, corresponding
to the tone pitch fn, through the key scale process and multiplying the breath pressure
P0 by the coefficient pmul. The coefficient pmul is a coefficient that varies in accordance
with the tone pitch fn. In the case of actual wind instruments, there is a tendency
that a breath pressure range of a human player for sounding the wind instrument differs
depending on the tone pitch; for example, the breath pressure range for a performance
of high-pitch tones is greater than that that for a performance of lower-pitch tones.
Because the coefficient pmul to be multiplied to the breath pressure P0 is a variable
value depending on the tone pitch fn, the instant embodiment can faithfully simulate
the aforementioned characteristics of the wind instrument even where the breath pressure
P0 is selected independently of the tone pitch fn.
[0098] Further, the characteristic parameter conversion section 21 determines, through the
key scale process, coefficients rH1 - rHk to be used in the tone hole portions TH[1]
- TH[k] of the tubular body simulating section 33 and in the bell section BL, and
coefficients MH1 - MHk and coefficient MB to be used in the transmission simulating
section 35. For example, the coefficient MHi is set at zero when the first tone hole
is closed during a performance of the tone pitch fn, but set at a predetermined value
greater than zero, such as one. Similarly, the coefficient rHi is set at a different
value depending on whether the i-th tone hole is closed or opened.
[0099] Fig. 13 is a block diagram showing specific functions of the shape characteristic
parameter conversion section 23. As shown in Fig. 13, the shape characteristic parameter
conversion section 23 is supplied with various parameters pertaining to the shapes
and dimensions of the reed MR and tubular body portion. Such parameters supplied to
the shape characteristic parameter conversion section 23 include parameters (Li, φ
i, ti, ψi) of the shape of each unit portion U[i] constituting the tubular portion,
thickness yd(x, z) of the reed MR, positions (zleft(x), zright(x)) of left and right
end portions, in the Z direction, and position yc(x), in the Y direction, of an axis
line functioning as a basis of the second moment of area I(x).
[0100] For the shape of the i-th unit portion U[i], the length Li and inner diameterφi of
the unit portion U[i] and the depth ti and inner diameterψ i of the tone hole are
designated, as shown in Fig. 6. First, the shape characteristic parameter conversion
section 23 determines coefficients pertaining to the connecting section J[i] (i.e.,
coefficients α1 and β1 for the two-port type connecting section, but coefficients
α1, β1 and γ1 for the three-port type connecting section) from the aforementioned
coefficients. Second, the shape characteristic parameter conversion section 23 determines
a delay amount dA[i] of the delay element DA[i] and delay amount dB[i] of the delay
element DB[i] on the basis of the length Li of the unit portion U[i]. In addition
to the aforementioned parameters, the shape characteristic parameter conversion section
23 may variably set a cut-off frequency fCB of the bell section BL and a cut-off frequency
fCTH and delay amount (dE1, dE2) of the tone hole portion TH[i].
[0101] Third, the shape characteristic parameter conversion section 23 calculates a horizontal
width Breed(x) of the reed MR by substituting the positions (zleft(x), zright(x))
of the left and right end portions of the reed MR into the following equation (b1):
[0102] Fourth, the shape characteristic parameter conversion section 23 calculates a sectional
area A(x) of the reed MR at the position x by integrating the thickness yd(x, z) over
a region from the left end position zleft(x) to the right end position zright(x) of
the reed MR, as represented by the following equation (b2):
[0103] Fifth, the shape characteristic parameter conversion section 23 calculates a second
moment of area I(x) pertaining to the axial line of the position yc(x) by the following
Equation (b3):
[0104] In the instant embodiment, as set forth above, the displacement y(x, t) of the reed
MR is calculated on the basis of Motion Equation B that expresses coupled vibration
of the reed MR and lip ML. Thus, the instant embodiment can faithfully simulate the
behavior of the reed MR as compared to the technique of Non-patent Literature 1 which
models a reed as a rigid air valve freely movable in its entirety and the technique
of Non-patent Literature 2 which models a reed using a vibrating member in the form
of an elongate plate. Further, because, each time the pressing force flip(x) acting
from the lip ML on the reed MR is changed, the displacement yb(x) of the lip ML in
Motion Equation B is updated with a result calculated from the changed pressing force
flip(x) on the basis of Motion Equation A1 and Motion Equation B, the instant embodiment
can faithfully simulate a rendition style which changes the pressing force flip(x).
Because the displacement y(x, t) of the reed MR in Motion Equation B is maintained
even when the pressing force flip(x) is changed, the instant embodiment can effectively
minimize an uncomfortable feeling of a tone arising from a discontinuous change of
the displacement y(x, t).
<Second Embodiment>
[0105] Next, a description will be given about a second embodiment of the present invention.
Whereas the first embodiment has been described above in relation to the case where
the spring constant klip(x) does not depend on the pressing force flip(x) from the
teeth MT, the second embodiment uses a spring constant klip(x) (x, flip(x)) that depends
on the pressing force flip(x). In the following description of the second and other
embodiments, similar elements to those in the first embodiment are indicated by the
same reference numerals and characters as used for the first embodiment and description
of these similar elements are omitted here as necessary to avoid unnecessary duplication.
[0106] Relationship between the spring constant klip(x) (x, flip(x)) of the lip ML and the
pressing force flip(x) is determined through actual measurement. Fig. 14 is a diagram
explanatory of how the spring constant klip(x) (x, flip(x)) is measured. As shown
in Fig. 14, an outer surface of a test piece 82 placed on a working table 80 is pressed
by a pressing member 84. The test piece 82 is an elastic member having substantially
the same elastic characteristic as the lip ML. The pressing member 84 presses only
part of the surface of the test piece 82 in generally the same manner as where the
teeth MT of the human player presses the lip ML. Operation for measuring an amount
of deformation of the test piece 82 to determine a spring constant klip(x) (x, flip(x))
is repeated while varying the intensity of the pressing force flip(x) and changing
the position x to be pressed by the pressing member 84. Through the aforementioned
test, the relationship between the spring constant klip(x) (x, flip(x)) of the lip
ML and the pressing force flip(x) is measured per position x.
[0107] Fig. 15 is a graph showing relationship between the pressing force flip(x) and the
spring constant klip(x) (x, flip(x)) observed when particular positions x of the test
piece 82 were pressed by the pressing member 84. As shown in Fig. 15, the spring constant
klip(x) (x, flip(x)) of the test piece 82 varies according to the intensity of the
pressing force flip(x). Namely, the spring constant klip(x) (x, flip(x)) increases
as the intensity of the pressing force flip(x) increases.
[0108] Upon completion of the aforementioned measurement, a function, such as a spline function,
approximating the relationship between the pressing force flip(x) and the spring constant
klip(x) (x, flip(x)) is determined for each of a plurality of positions x. Further,
a function (hereinafter referred to as "resiliency function") defining relationship
among the position x, on which the pressing force flip(x) acts, the intensity of the
pressing force flip(x) and the spring constant klip(x) (x, flip(x)) is determined
for each of a plurality of types of lips ML by the aforementioned operations being
repeated for a plurality of test pieces 82 differing from one another in physical
property and dimension. Each of the thus-determined resiliency functions is stored
into the storage device 42 of the tone synthesis apparatus 100.
[0109] The user selects any one of the plurality of types of lips ML by operating the input
device 44. The characteristic parameter conversion section 21 of Fig. 1 acquires,
from the storage device 42, the resiliency function corresponding to the user-selected
lip ML and then calculates a spring constant klip(x) (x, flip(x)) by substituting
the pressing force flip(x) into the resiliency function. The spring constant klip(x)
(x, flip(x)) thus calculated by the characteristic parameter conversion section 21
is used in arithmetic operations by the reed simulating section 31 (more specifically,
by the first and second arithmetic operation sections 311 and 312).
[0110] In the instant embodiment, as set forth above, the spring constant klip(x) (x, flip(x))
varies in accordance with not only the position x on which the pressing force flip(x)
acts, but also the intensity of the pressing force flip(x). Namely, the instant embodiment
can faithfully reproduce behavior of an actual wind instrument in which the generated
tone varies in accordance with the intensity of the pressing force flip(x) acting
from the teeth on the lip during a performance and position (x) of the teeth relative
to the lip. In this way, the instant embodiment can faithfully synthesize a variety
of tones corresponding to various rendition styles.
[0111] Whereas, in the above-described measurement, the pressing force flip(x) is caused
to act on part of the test piece 82, there may be employed an alternative method in
which the pressing force flip(x) is caused to act on the entire upper surface of the
test piece 82 so as to measure a spring constant klip(x) (x, flip(x)). In the case
where such an alternative method is employed, a spring constant klip(x) (x, flip(x))
that varies in accordance with the pressing force flip(x) but does not depend on the
position x is defined by the elastic function. In this way, it is possible to reproduce
behavior in which the generated tone varies in accordance with the pressing force
acting from the teeth to the lip.
<Third Embodiment>
[0112] In the above-described first embodiment, the internal resistance µ lip(x) of the
lip ML and the internal resistance µreed(x) of the reed MR take fixed values that
do not depend on the position x. However, in a third embodiment to be described below,
the internal resistance µlip(x) of the lip ML and the internal resistance µreed(x)
of the reed MR are varied in accordance with the position x.
[0113] If the horizontal width blip_sample of the lip sample in Mathematical Expression
(a3) above is substituted by a horizontal width blip(x) corresponding to the position
x, the following Mathematical Expression (a3 - 1) is derived:
[0114] Similarly, for the internal resistance µreed(x) of the reed MR, there can be derived
the following Equation (a4 - 1) where the sectional area A(x) of the reed MR that
varies in accordance with the position x and the spring constant kreed(x) are variables:
[0115] Fig. 16 is a block diagram showing the characteristic parameter conversion section
21 employed in the third embodiment. As shown, the characteristic parameter conversion
section 21 calculates the internal resistance µlip(x) corresponding to the position
x by performing the arithmetic operation of Equation (a3 - 1) with respect to the
physical property values and dimension (tan δ lip, blip(x), ρ lip and Elip(x)) of
the lip ML. The horizontal width blip(x) is calculated from the tone pitch fn through
a key process as in the above-described first embodiment.
[0116] Further, the characteristic parameter conversion section 21 calculates the internal
resistance µreed(x) corresponding to the position x by performing the arithmetic operation
of Equation (a4 - 1) with respect to the physical property values (tan δ reed, ρ reed,
A(x) and kreed(x)). The sectional area A(x) calculated by the shape characteristic
parameter conversion section 23 performing the arithmetic operation of Equation (b2)
is used in the arithmetic operation of Equation (a4 - 1). Numerical value stored in
the storage device 42 or designated via the input device 44, for example, is used
as the spring constant kreed(x) [N/m] of the reed MR in Equation (a4 - 1).
[0117] The internal resistance µlip(x) and internal resistance µreed(x) calculated in the
aforementioned arithmetic operation sequence are used in the arithmetic operation
of Motion Equation B by the second arithmetic operation section 312. With the instant
embodiment, where the internal resistance µlip(x) of the lip ML and internal resistance
µ reed(x) of the reed MR change in accordance with the position x, it is possible
to faithfully reproduce tones of an actual wind instrument as compared to the construction
(e.g., construction of the first embodiment) where the internal resistance µlip(x)
and internal resistance µ reed(x) are set at fixed values.
<Fourth Embodiment>
[0118] In a case where deformation of the lip ML and reed MR is relatively small, i.e. where
the lip ML and reed MR deform within an elasticity limit), even the third embodiment
where the internal resistance µlip(x) and internal resistance µreed(x) depend only
on the position x can faithfully reproduce tones of an actual wind instrument. However,
in a case where deformation of the lip ML and reed MR is great, i.e. where deformation
of the lip ML and reed MR is outside the elasticity limit), the internal resistance
µ lip(x, flip(x)) of the lip ML depends not only on the position x but also on the
pressing force flip(x), and the internal resistance µreed(x, freed(x)) of the reed
MR depends not only on the position x but also on the pressing force freed(x) on the
reed MR.
[0119] Fig. 17 is graph showing relationship between the pressing force freed(x) acting
on the reed MR and the displacement (amount) of the reed MR. As shown, once the pressing
force freed(x) exceeds a predetermined value fTH, i.e. once the pressing force freed(x)
reaches the elasticity limit, the displacement of the reed MR changes non-linearly.
Namely, as the intensity of the pressing force freed(x) increases, the spring constant
klip(x) (x, flip(x)) decreases (i.e., the reed MR becomes easier to deform). Because
the pressing force freed(x) acting from the lip ML on the reed MR is equal to the
pressing force flip(x) acting from the reed MR on the lip ML, the pressing force freed(x)
is written as the pressing force flip(x), for convenience sake, in the following description.
[0120] The internal resistance µlip(x, flip(x)) of the lip ML is defined by Equation (a3
- 2) below. Because the spring constant klip(x) (x, flip(x)) in Equation (a3 - 2)
is a function of the pressing force flip(x), the internal resistance µlip(x, flip(x))
changes in accordance with the position x and pressing force flip(x). Similarly, the
internal resistance µreed(x, flip(x)) of the reed MR changes in accordance with the
position x and pressing force flip(x) (spring constant kreed(x, flip(x)), as defined
by Equation (a4 - 2) below.
[0121] Fig. 18 is a block diagram showing the characteristic parameter conversion section
21 employed in the fourth embodiment. As shown, the characteristic parameter conversion
section 21 has two types of tables (Tlip, Treed). The table Tlip correlates values
of the pressing force flip(x) and the spring constant klip(x) (x, flip(x)) of the
lip ML to each other, and the table Treed correlates values of the pressing force
flip(x) and the spring constant kreed(x) (x, flip(x)) of the reed MR to each other.
Contents of the table Tlip and table Treed are set in accordance with results of experiments
where pressing force was applied to an actual lip and reed. The characteristic parameter
conversion section 21 searches through the table Tlip for a spring constant klip(x)
(x, flip(x)) corresponding to pressing force flip(x) per unit length calculated by
dividing pressing force Flip(x), calculated through a key scale process, by a length
lteeth of the teeth MT, and then searches through the table Treed for a spring constant
kreed(x) (x, flip(x)) corresponding to the pressing force flip(x).
[0122] Then, the characteristic parameter conversion section 21 calculates internal resistance
µlip(x, flip(x)) corresponding to the position x and pressing force flip(x) by performing
the arithmetic operation of Equation (a3 - 2) with respect to the spring constant
klip(x) (x, flip(x)) searched out from the table Tlip and physical property values
(mlip and tan δlip) of the lip ML. As in the above-described first embodiment, the
distribution of mass mlip(x) in Equation (a3 - 2) above is a result of multiplication
between the horizontal width blip(x) and the density ρlip. Further, the characteristic
parameter conversion section 21 calculates internal resistance µreed(x, flip(x)) corresponding
to the position x and pressing force flip(x) by performing the arithmetic operation
of Equation (a4 - 2) with respect to the spring constant kreed(x) (x, flip(x)) searched
out from the table Treed and physical property values and dimension (tan δreed, ρ
reed and A(x)) of the reed MR.
[0123] The internal resistance µlip(x, flip(x)) and internal resistance µ reed(x) (x, flip(x))
calculated in the aforementioned arithmetic operational sequence are used in the arithmetic
operation of Motion Equation B by the second arithmetic operation section 312. With
the instant embodiment, where the internal resistance µlip(x, flip(x)) of the lip
ML and internal resistance µreed(x)(x, flip(x)) of the reed MR change in accordance
with the position x and intensity of the pressing force flip(x), it is possible to
faithfully reproduce tones of an actual wind instrument as compared to the construction
(e.g., construction of the first embodiment) where the internal resistance µlip(x)
and internal resistance µreed(x) are set at fixed values. Whereas the foregoing description
has been made assuming that deformation of the lip ML and reed MR is outside the elasticity
limit, the construction of Fig. 18 is also applicable to the case where deformation
of the lip ML and reed MR is only within the elasticity limit.
<Modification>
[0124] The above-described embodiments may be modified variously as set forth below by way
of example.
(1) Modification 1:
[0125] Whereas the embodiments have been described above in relation to the case where the
characteristic parameter conversion section 21 and shape characteristic parameter
conversion section 23 convert user-input parameters into parameters necessary for
tone synthesis, there may be employed an alternative construction where various parameters
to be used in arithmetic operations by the synthesis section 14 are input directly
by the user. For example, although Fig. 12 illustratively shows the construction where
parameters pertaining to the embouchure and fingering are calculated through the key
scale process, there may be employed an alternative construction where such parameters
pertaining to the embouchure and fingering are input or designated directly to the
arithmetic operation processing device 10 by the user via the input device 44.
(2) Modification 2:
[0126] Whereas the embodiments have been described above in relation to the case where the
product between the Young's modulus and the second moment of area I(x) of the reed
MR is determined as bending rigidity Still(x) of the reed MR, there may be employed
an alternative construction where bending rigidity Still(x) of the reed MR is determined
from results of actual measurements. In one example, bending rigidity Still(x) is
determined from displacement of a test piece, simulating the reed MR, measured with
pressing force applied to various positions x of the test piece, and then a function
(hereinafter "rigidity function") approximating relationship between the position
x and the bending rigidity Still(x) is created. Such rigidity functions of a plurality
of types of reeds MR, differing in physical property value and dimension, are sequentially
created in the aforementioned manner and stored into the storage device 42. The reed
simulating section 31 (more specifically, the first and second arithmetic operation
sections 311 and 312) of the arithmetic operation processing device 10 acquires, from
the storage device 42, rigidity function corresponding to any one of the reeds MR
(e.g., reed MR selected by the user) and uses the acquired rigidity function in subsequent
arithmetic operations. Such arrangements too can achieve substantially the same advantageous
benefits as the first and second embodiments.
(3) Modification 3:
[0127] Tone synthesis based on the displacement y(x, t) calculated by the second arithmetic
operation section 312 may be performed in any desired manner. For example, there may
be employed a construction where simulation of sound wave losses in tone holes and
boundary between inside and outside of the bell is omitted.