JP2011103010A - Method, program and apparatus for synthesizing tone signal - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method for synthesizing a tone signal, a program, and a tone signal generating system for generating quasi musical instrument sounds expressing property possessed by sounds generated from a musical instrument of three-dimensional structure having wires and a body in reality. <P>SOLUTION: In this method for synthesizing the tone signal, a first bending vibration, a second bending vibration and a vertical vibration of wires are calculated based on three equations of motion having an interaction through a wire support end, that is, a first equation of motion representing the first bending vibration of the wires in a first direction with the usage of force acting on the wires calculated in accordance with performance information, a second equation of motion representing the second bending vibration of the wires in a direction different from the first direction, and a third equation of motion representing the vertical vibration of the wires, and the tone signal is calculated based on the information. <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to a technique for synthesizing a musical sound signal by performing a simulation in accordance with a predetermined physical model based on a sound generation mechanism (sound generation mechanism) in a natural musical instrument. In particular, a musical tone suitable for generating a pseudo-instrument sound that realistically represents the characteristics of a sound emitted from a three-dimensional instrument having a string and a main body (a component that supports the string and emits sound in the air) The present invention relates to a signal synthesis method, a program, and a musical tone signal synthesis apparatus.

  Conventionally, a dedicated hardware device configured to include a general-purpose computer, a digital signal processor such as a DSP (Digital Signal Processor), an integrated circuit, a large-scale integrated circuit, etc. A method is known in which a musical instrument sound is synthesized in a pseudo (virtual) manner by performing a simulation in accordance with the predetermined physical model. For example, to generate a pseudo piano sound, a string model unit that simulates vibration of a string, a soundboard model unit that simulates vibration of a piece or soundboard that vibrates with a string in accordance with the vibration of the string, and In accordance with a predetermined physical model provided with a synthesizer, a musical sound signal is synthesized by causing a physical model sound source composed of a DSP or the like to execute a simulation operation. A musical tone synthesizer using such a method is described in Patent Document 1, for example. Patent Document 2 describes an example in which a “non-linear vibration mechanism of a string” is tried when synthesizing a pseudo piano sound.

JP-A-10-63270 Japanese Patent Laid-Open No. 06-83363

  One end of a piano string is supported by a bearing on a frame that is a part of the main body, and the other end is supported by a piece on a soundboard that is a part of the main body. When the key is pressed, the corresponding string is released from the damper, and at the same time, kinetic energy is given to the hammer. The energy of the wave excited in the string by the hammer hitting the string is partially transmitted to the main body through these support ends, and the rest is reflected in the support ends and remains in the string. The vibration is born by the wave generated in the string repeating the reciprocation between the string support ends. The vibration in the direction orthogonal to the axial direction of the string, that is, the bending vibration is initially generated in the direction struck by the hammer, but is also generated in the direction orthogonal thereto due to the influence of the three-dimensional motion piece. In addition to the bending vibrations in the two directions, the string also performs vibrations in the string axis direction, that is, longitudinal vibration.

  When all the strings are released from the damper by depressing the damper pedal, even the strings not hit by the hammer start to vibrate by receiving energy from the main body. In this way, the pronunciation of the piano is based on a three-dimensional coupled vibration mechanism in which not only the strings vibrate the main body but also the main body vibrates the strings. Sound is radiated into the air from the entire surface of the main body, which has a complicated three-dimensional shape composed of soundboards, frames, columns, side panels, etc., creating rich and three-dimensional piano-specific musical sounds. The

  By the way, in the range below about the 40th key of a standard 88-key piano, “Rinling”, “Hinhin”, “Hein”, “Kein”, bell-like, or metallic anharmonicity Sound (hereinafter referred to as ringing sound) may occur. This characteristic ringing sound is prominently generated when hitting hard, and if the level is too high, it may feel uncomfortable, but if you can not hear it at all, the piano sound will be monotonous and boring. End up. Non-linear (finite amplitude) vibration of the strings contributes to the generation of the ringing sound peculiar to this piano.

  In short, to realistically express the characteristics of musical sounds emitted from a piano, which is a three-dimensional structure having a string that is a part that generates a scale and a main body that is a part that supports the string and emits sound in the air. The three-dimensional coupled vibration mechanism of the string and the main body, the three-dimensional acoustic radiation mechanism of the main body, and the non-linear (finite amplitude) vibration mechanism of the string are taken into account for high-precision simulation. However, a method (calculation algorithm) for realizing this has not been proposed yet.

  In the piano, various musical nuances can be expressed by controlling the key press depth or the damper pedal press depth on the time axis. For example, after a key is pressed, the nuance of how the sound stops is completely different depending on whether the pressed key is released slowly or suddenly. Also, in a piano, it is possible to “resonate moderately” instead of “maximally resonate” a string by using a technique called “half pedal” in which the damper pedal is depressed halfway through the entire stroke. A conventional electronic sound source cannot generate the various sound-stopping and string resonance effects described above.

  A general grand piano has a pedal called a shift pedal. By depressing this pedal, the position of all the hammers shifts to the high-pitched sound side, and the relatively soft part of the hammer that normally does not touch the strings comes into contact with the strings. Furthermore, when fully depressing it, for example, in a range with three strings corresponding to one key, the hammer will only strike two of the three strings, and the remaining one Works as a resonant string. Conventional electronic sound sources cannot express the fine musical nuances obtained by stepping on the shift pedal.

  The present invention has been made in view of the above circumstances, and is a musical tone signal synthesis capable of generating a pseudo-instrument sound that realistically represents the characteristics of a sound emitted from a three-dimensional musical instrument having a string and a main body. It is an object to provide a method, a program, and a musical tone signal synthesizer.

  In order to solve the above-described problems, the present invention inputs a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by applying a force from at least a first direction and a string support end that supports the string. A musical sound signal synthesis method generated according to performance information to be obtained, wherein first information representing a force acting on the string calculated according to the performance information is acquired, and interaction via the string support end is obtained. A first equation of motion representing a first bending vibration of the string in the first direction using the first information, and the string in a direction different from the first direction. Based on the second equation of motion representing the second bending vibration of the second string and the third equation of motion representing the longitudinal vibration of the string, the second bending vibration, the second bending vibration, and the second vibration representing the longitudinal vibration of the string A string model calculation process for calculating information, and the second information Based on, to provide a tone signal synthesis method characterized by comprising a tone signal calculation process of calculating the tone signal.

  In another preferred embodiment, the third equation of motion is expressed using the first bending vibration and the second bending vibration.

  According to the present invention, a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted with a force from at least the first direction and a string support end that supports the string is determined according to input performance information. A first tone information representing a force acting on the string calculated in accordance with the performance information, and two equations of motion having an interaction via the string support end. The string based on a first equation of motion representing a first bending vibration of the string in the first direction using the first information and a third equation of motion representing a longitudinal vibration of the string. A string model calculation process for calculating second information indicating the first bending vibration and the longitudinal vibration, and a tone signal calculation process for calculating the tone signal based on the second information. A method for synthesizing a musical sound signal is provided.

  According to the present invention, a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted with a force from at least the first direction and a string support end that supports the string is determined according to input performance information. A first tone information representing a force acting on the string calculated in accordance with the performance information, and two equations of motion having an interaction via the string support end. A first equation of motion representing a first bending vibration of the string in the first direction using the first information, and a second bending vibration of the string in a direction different from the first direction. A string model calculation process for calculating second information indicating the first bending vibration and the second bending vibration of the string based on the second equation of motion, and the musical tone signal based on the second information And a musical sound signal calculation process for calculating Providing a tone signal synthesis method of the symptoms.

  In addition, the present invention provides a performance in which a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted a force from at least a first direction and a string support end that supports the string is input to a computer. A program generated according to information, wherein the computer acquires first information representing a force exerted on the string calculated according to the performance information, and has an interaction via the string support end A first equation of motion representing a first bending vibration of the string in the first direction using the first information; a first equation of motion of the string in a direction different from the first direction; Based on a second equation of motion representing two bending vibrations and a third equation of motion representing longitudinal vibration of the string, second information indicating the first bending vibration, the second bending vibration and the longitudinal vibration of the string is obtained. A string model calculating means for calculating; Based on the serial second information providing program for functioning as a tone signal calculating means for calculating said tone signal.

  In addition, the present invention provides a performance in which a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted a force from at least a first direction and a string support end that supports the string is input to a computer. A program generated according to information, wherein the computer acquires first information representing a force exerted on the string calculated according to the performance information, and has an interaction via the string support end Two equations of motion based on a first equation of motion representing a first bending vibration of the string in the first direction using the first information and a third equation of motion representing a longitudinal vibration of the string. And functioning as a string model calculation means for calculating second information indicating the first bending vibration and the longitudinal vibration of the string, and as a tone signal calculation means for calculating the tone signal based on the second information. Pro for To provide the ram.

  In addition, the present invention provides a performance in which a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted a force from at least a first direction and a string support end that supports the string is input to a computer. A program generated according to information, wherein the computer acquires first information representing a force exerted on the string calculated according to the performance information, and has an interaction via the string support end Two equations of motion, the first equation of motion representing the first bending vibration of the string in the first direction using the first information, and the string in a direction different from the first direction Based on a second equation of motion representing a second bending vibration, a string model calculation means for calculating second information indicating the first bending vibration and the second bending vibration of the string, and based on the second information , Calculate the tone signal It provides a program to function as a sound signal calculating means.

  According to the present invention, a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted with a force from at least the first direction and a string support end that supports the string is determined according to input performance information. 3 is a musical tone signal generating device for generating first information representing a force acting on the string calculated in accordance with the performance information and having an interaction via the string support end. A first equation of motion representing a first bending vibration of the string in the first direction using the first information, and a second bending vibration of the string in a direction different from the first direction. A string model for calculating second information indicating the first bending vibration, the second bending vibration, and the longitudinal vibration of the string based on the second motion equation representing the third motion equation representing the longitudinal vibration of the string Based on the calculation means and the second information, Providing a musical tone signal generating apparatus characterized by comprising a tone signal calculating means for calculating a signal.

  According to the present invention, a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted with a force from at least the first direction and a string support end that supports the string is determined according to input performance information. A musical tone signal generating device for generating first information representing a force acting on the string calculated in accordance with the performance information, and two equations of motion having an interaction via the string support end The string based on a first equation of motion representing a first bending vibration of the string in the first direction using the first information and a third equation of motion representing a longitudinal vibration of the string. A string model calculating means for calculating second information indicating the first bending vibration and the longitudinal vibration, and a tone signal calculating means for calculating the tone signal based on the second information. A musical sound signal generating apparatus is provided.

  According to the present invention, a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted with a force from at least the first direction and a string support end that supports the string is determined according to input performance information. A musical tone signal generating device for generating first information representing a force acting on the string calculated in accordance with the performance information, and two equations of motion having an interaction via the string support end A first equation of motion representing a first bending vibration of the string in the first direction using the first information, and a second bending vibration of the string in a direction different from the first direction. A string model calculating means for calculating second information indicating the first bending vibration and the second bending vibration of the string based on the second equation of motion, and the musical tone signal based on the second information. A musical sound signal calculating means for calculating Providing a musical tone signal generating apparatus according to claim.

  ADVANTAGE OF THE INVENTION According to this invention, the musical tone signal synthesis method, program, and musical tone signal synthesis | combination apparatus which can produce | generate the pseudo musical instrument sound which realistically expressed the characteristic which the sound emitted from the musical instrument emitted from the three-dimensional structure which has a string and a main body have are provided can do.

It is a block diagram which shows the structure of the electronic musical instrument which concerns on 1st Embodiment of this invention. It is a block diagram which shows the structure of the musical tone signal synthetic | combination part which concerns on 1st Embodiment of this invention. It is a figure explaining the structure of a standard grand piano. It is a block diagram which shows the structure of the electronic musical instrument which concerns on 2nd Embodiment of this invention. It is a block diagram which shows the structure of the musical tone signal synthetic | combination part which concerns on 2nd Embodiment of this invention. It is a block diagram which shows the structure of the electronic musical instrument which concerns on 3rd Embodiment of this invention. It is a block diagram which shows the structure of the musical tone signal synthetic | combination part which concerns on 3rd Embodiment of this invention. It is a block diagram which shows the structure of the electronic musical instrument which concerns on 4th Embodiment of this invention. It is a block diagram which shows the structure of the musical tone signal synthetic | combination part which concerns on 4th Embodiment of this invention.

  Hereinafter, an embodiment of the present invention will be described.

<First Embodiment>
FIG. 1 is a block diagram showing a configuration of an electronic musical instrument 1 according to the first embodiment of the present invention. The electronic musical instrument 1 is, for example, an electronic piano, and includes a control unit 11, a storage unit 12, a user operation unit 13, a performance operation unit 15, and a sound emission unit 17. These units are connected to each other via a bus 18.

  The control unit 11 includes a CPU (Central Processing Unit) 11a, a DSP 11b, other peripheral circuits (not shown), a ROM (Read Only Memory) 11c, a RAM (Random Access Memory) 11d, a signal interface 11e, and an internal bus 11f. A DMA (Direct Memory Access) controller and a video processor may be included as other peripheral circuits. The CPU 11a reads out a control program stored in the ROM 11c, loads it into the RAM 11d, and executes it, thereby controlling each part of the electronic musical instrument 1 via the bus 18 and performing a tone signal synthesis process to be described later. The synthesis unit 100 and the like are realized. The RAM 11d functions as a work area when the CPU 11a processes each data.

  The storage unit 12 is, for example, a large-capacity storage unit such as a hard disk, and stores musical tone control data such as MIDI (Musical Instrument Digital Interface) data, musical tone signals generated by musical tone signal synthesis processing described later, and the like. In this example, the musical tone control data includes data indicating changes in key depression amount, hammer speed, damper pedal depression amount, and shift pedal depression amount according to time progress. These data may be loaded from an information storage medium DP (for example, a compact disk) or downloaded from a server via a network.

  The user operation unit 13 includes an operation panel 13 a and a display unit 14. The operation panel 13a includes, for example, a mouse 13b, an operation switch 13c, a keyboard 13d, and the like. When the user operates the mouse 13b, the operation switch 13c, and the keyboard 13d, data representing the operation content is output to the control unit 11. As a result, the user gives an instruction to the electronic musical instrument 1. The display unit 14 is a display device such as a liquid crystal display that displays video on the screen, and is controlled by the control unit 11 to display various screens such as a menu screen. The menu screen may be automatically displayed when power is supplied to the electronic musical instrument 1.

  The performance operation unit 15 includes a keyboard unit 15 a and a pedal unit 16. The keyboard portion 15a corresponds to a keyboard portion such as an electronic piano, and has a keyboard on which a plurality of keys (black key 15b, white key 15c) are arranged. Further, when each key is pressed into each key 15b, 15c in the keyboard portion 15a, a key position sensor 15d that outputs information indicating the amount of pressing the key, and a key speed sensor 15e that outputs information indicating the pressing speed. Is provided. The keyboard unit 15a outputs information KS obtained by converting the information indicating the key pressing amount from the analog format to the digital format, and information KV obtained by converting the information indicating the pressing speed from the analog format to the digital format via the bus 18. And periodically output to the signal interface 11e of the control unit 11. The keyboard unit 15a outputs these pieces of information KS and KV together with information KC (for example, key number) representing the depressed key. At this time, the hammer speed is calculated by the control unit 11 based on information output from the keyboard unit 15a. The pressing speed may be calculated from the key pressing amount output from the key position sensor 15d, and the key speed sensor 15e may not be provided. In this case, the keyboard unit 15a may be provided with a calculating unit that calculates the pressing speed from the key pressing amount. Further, the CPU 11a of the control unit 11 may calculate the pushing speed from the information KS.

  The pedal portion 16 has a plurality of pedals corresponding to the damper pedal 16a and the shift pedal 16b. In addition, the damper pedal 16a and the shift pedal 16b are provided with a pedal position sensor 16c that outputs information indicating the depression amount of the pedal when the pedal is depressed. The pedal unit 16 periodically outputs information PS obtained by converting information representing the pedal depression amount from an analog format to a digital format to the signal interface 11e of the control unit 11 via the bus 18. The pedal section 16 outputs this information PS together with information PC indicating the pedal that has been depressed. Thus, the keyboard part 15a and the pedal part 16 output each performance information by the operation.

  The sound emitting unit 17 includes a digital / analog converter 17a, an amplifier (not shown), and a speaker 17b. A musical sound signal input under the control of the control unit 11 is converted from a digital format to an analog format in the digital / analog converter 17a, amplified by an amplifier, and output as a sound from the speaker 17b. In this example, this musical tone signal is generated as a result of musical tone signal synthesis processing described later. The above is the description of the configuration of the electronic musical instrument 1. Next, the tone signal synthesis unit 100 realized by the control unit 11 executing the control program will be described with reference to FIG. It should be noted that each of the functions of the tone signal synthesizer 100 described below may be realized by hardware.

  FIG. 2 is a block diagram showing the configuration of the tone signal synthesizer 100. The musical sound signal synthesizing unit 100 synthesizes a musical sound signal indicating a pseudo piano sound by a physical model composed of a plurality of models (damper model, hammer model, string model, main body model, air model) described below. The standard piano has 88 keys on the keyboard. One hammer, one to three strings, and 0 to multiple dampers (contact with the strings at multiple points). It means). The number of strings and the number of dampers are different for each sound range.

  FIG. 3 is a diagram illustrating the configuration of a standard grand piano. The plurality of models are based on a standard grand piano (acoustic piano) 21 shown in FIG. The grand piano 21 includes a keyboard 21b including 88 keys 21a, a hammer 21c connected to the key 21a via an action mechanism 21d, a string 21e, and a damper 21f capable of contacting the string 21e. The string 21e is connected to a piece 21ea at one end and a bearing 21eb at the other end. Most of the key 21a, the hammer 21c, the action mechanism 21d, the string 21e, and the damper 21f are housed in the cabinet 21h. The number of strings 21e and the number of contact points of the damper 21f vary depending on the sound range. The cabinet 21h, the frame, the wooden frame, the piece 21ea, the bearing 21eb, and other vibration parts that emit piano sound constitute the main body 21j. In the following description, a string, a hammer, a damper, and a main body indicate the configuration of the standard grand piano 21 and do not indicate the configuration included in the electronic musical instrument 1.

  The musical tone signal synthesis unit 100 shown in FIG. 2 includes a comparison unit 101, damper model calculation units 102-1 and 102-2 that calculate a damper model for each corresponding string, a hammer model calculation unit 103 that calculates a hammer model, and a string model. String model calculation units 104-1 and 104-2 for calculating a string for each string, a body model calculation unit 105 for calculating a body model, and an air model calculation unit 106 for calculating an air model.

  The damper model calculation units 102-1 and 102-2 calculate the vibration of a certain string 21e using a damper model. The string model calculation units 104-1 and 104-2 calculate the vibration of a certain string 21e based on the string model. The hammer model calculation unit 103, the main body model calculation unit 105, and the air model calculation unit 106 calculate the vibration of a certain string 21e using the hammer model, the main body model, and the air model, respectively.

  The comparison unit 101 is connected to the damper model calculation units 102-1 and 102-2. The damper model calculation units 102-1 and 102-2 are connected to the string model calculation units 104-1 and 104-2, respectively. The hammer model calculation unit 103 is connected to both the string model calculation units 104-1 and 104-2. The string model calculation units 104-1 and 104-2 are connected to the main body model calculation unit 105. The main body model calculation unit 105 is connected to the air model calculation unit 106. The musical sound signal is output from the air model calculation unit 106.

  The musical tone signal obtained by the musical tone signal synthesizing process of the musical tone signal synthesizing unit 100 is based on a physical model in the case where there are two corresponding strings in a specific key. That is, the string model calculation units 104-1 and 104-2 for calculating the string model are connected in parallel to the main body model calculation unit 105 for calculating the main body model. Here, if there are three or more strings, the main body model calculation unit 105 is connected to the string model calculation unit 104-iw (iw = 3,4,...) And the string model connected in parallel. The damper model calculation unit 102-iw (iw = 3, 4,...) May be increased. When there are a plurality of keys, the number of the damper model calculation unit 102, the hammer model calculation unit 103, and the string model calculation unit 104 is increased according to the number of keys, and the string model calculation unit corresponding to each key 104 may be connected to the main body model calculation unit 105. Therefore, the tone signal synthesizer 100 shown in FIG. 2 has generality.

The input signal in the musical tone signal synthesis unit 100 is a signal generated by the control unit 11 and generated according to information indicating the key pressing amount output from the keyboard unit 15a (hereinafter referred to as an input signal). 1 (referred to as e _K (nΔt)), a signal indicating a hammer speed generated according to information indicating the key pressing speed and the pressing acceleration (hereinafter referred to as input signal 2 (V _H (nΔt))), pedal unit 16 is a signal generated according to information indicating the depression amount of the damper pedal output from 16 (hereinafter referred to as input signal 3 (e _P (nΔt)), and a signal generated according to information indicating the depression amount of the shift pedal. (Hereinafter referred to as input signal 4 (e _S (nΔt))). Each of these four signals is input to the tone signal synthesis unit 100 as a signal on a discrete time axis (t = nΔt; n = 0, 1, 2,...) Based on the information KS, KV, PS described above. Shall be. These four signals may be generated when the control unit 11 reads out the musical tone control data stored in the storage unit 12.

  On the other hand, the output signal in the musical sound signal synthesis unit 100 is a musical tone signal (hereinafter referred to as a musical tone signal (P (nΔt))) indicating a waveform of “sound pressure at an observation point in the air” output from the air model calculation unit 106. ).

First, in the physical model of the musical tone signal synthesis process in the musical tone signal synthesis unit 100 in this example, the following 28 assumptions are made.
(Assumption 1) Gravity is ignored.
(Assumption 2) It is assumed that a string in a state where it is straightly stopped by receiving an axial force (hereinafter referred to as a static equilibrium state) has an elongated cylindrical shape.
(Assumption 3) The thickness of the string is assumed to be unchanged. That is, the beam theory is adopted.
(Assumption 4) The cross section perpendicular to the central axis of the chord is assumed to be flat after deformation and perpendicular to the central axis. That is, Bernoulli Euler's assumption is adopted.
(Assumption 5) The amplitude of the string is small but not necessarily very small.
(Assumption 6) The strings are assumed to be homogeneous.
(Assumption 7) The string stress is given as the sum of a component proportional to strain and a component proportional to strain rate. That is, it is assumed that internal viscous damping (also referred to as rigidity proportional viscous damping) acts on the string.
(Assumption 8) It is assumed that external viscous damping (also referred to as mass proportional viscous damping) acts in the axial direction of the string.
(Assumption 9) One end of the string is supported by a point on a bearing that is a part of the main body, and the other end is supported by a point on a piece that is a part of the main body. (The string shall not be constrained from rotating at the support end.)
(Assumption 10) The action / reaction between the string and air is ignored.
(Assumption 11) The shape of the portion that contacts the string of the hammer (hereinafter referred to as the tip of the hammer) is assumed to be cylindrical, the bottom radius of the cylinder is infinitesimal, and the height of the cylinder is the other string. To the extent that it does not interfere with.
(Assumption 12) When there are a plurality of strings corresponding to one hammer, the central axes of the strings in the static equilibrium state are on the same plane.
(Assumption 13) When there are a plurality of strings corresponding to one hammer, the one hammer has the same number of hammer tips as the number of the strings.
(Assumption 14) The direction of the central axis of the tip of the hammer (cylinder) is orthogonal to the direction of the central axis of the string (cylinder) in a static equilibrium state.
(Assumption 15) The center of gravity of the hammer moves only on one straight line.
(Assumption 16) The movement direction of the center of gravity of the hammer is orthogonal to both the direction of the central axis of the tip of the hammer (cylinder) and the direction of the central axis of the string (cylinder) in a static equilibrium state.
(Assumption 17) It is assumed that the direction in which the hammer deforms coincides with the movement direction of the center of gravity of the hammer.
(Assumption 18) The hammer compression force-compression amount relational expression is assumed to be given by a function whose exponent should be a positive real number.
(Assumption 19) It is assumed that there is no friction between the hammer tip and the string surface.
(Assumption 20) The action / reaction between the hammer and air is ignored.
(Assumption 21) For a string equipped with a damper, a resistance force by the damper that tries to stop the bending vibration of the string acts on a point on the central axis of the string (hereinafter referred to as a sound stop point). . (Assumption 22) The resistance force-speed relational expression of the damper is given by a linear expression.
(Assumption 23) The amplitude of the main body is assumed to be minute.
(Assumption 24) It is assumed that the main body can be treated approximately as a proportional viscous damping system.
(Assumption 25) The reaction that the main body receives from the air is ignored.
(Assumption 26) Air is assumed to be homogeneous.
(Assumption 27) It is assumed that the air pressure-volume strain relational expression is given by a linear expression.
(Assumption 28) It is assumed that the air has no vortex.

  Further, the right-handed system (x, y, z) is used to represent the object coordinates of the string in this example. Here, the x axis is aligned with the central axis of the string in a static equilibrium state, the bearing side support end is the origin (0, 0, 0), and the piece side support end is included in the region where x> 0. The direction of the x-axis is determined, and the direction of movement of the hammer center of gravity when struck is determined as the positive direction of the z-axis. Further, the right-handed system (X, Y, Z) is used to represent the body and air object coordinates. Time progress (time variable) is represented by t.

  Hereinafter, symbols representing each parameter described in this example will be described.

  The following “Equation 1” to “Equation 5” are information input in the calculation of each model. “Equation 1” is a parameter (time variation parameter) that varies with time. On the other hand, “Equation 2” to “Equation 5” are parameters that do not vary with the progress of time (time invariant parameters) and are preset.

  The following “Equation 1” indicates a parameter relating to performance, that is, a parameter corresponding to an input signal in the musical tone signal synthesis unit 100. Keys, strings, hammers, dampers, and main bodies indicate the components 21a, 21e, 21c, 21f, and 21j of the standard grand piano 21, respectively.

  The following “Equation 2” is a parameter relating to design.

  The following “Equation 3” is a parameter related to the design of the main body and the position of the observation point in the air.

  The following “Equation 4” is a parameter related to tuning.

  The following “Equation 5” is a parameter related to numerical calculation.

  The following “Equation 6” is information output by calculation of each model, that is, a musical tone signal.

  The following “Equation 7”, “Equation 8”, and “Equation 9” are other parameters necessary for calculation of each model.

Here, when there is one string corresponding to one hammer, β _{kk ′} is uniquely determined when Z _B , X _B , Y _B , and θ _H are given. When there are a plurality of strings corresponding to one hammer, β _{kk ′} is uniquely determined when Z _B , X _B , and Y _B are given.

  The following “Equation 10” is an explanation of an index described as a superscript in each parameter.

  Hereinafter, the processing contents of each unit of the tone signal synthesis unit 100 in this example will be described in order with reference to FIG. In the following description, if all the indexes are written, the formula becomes complicated and difficult to read.

In addition, “1” is set as an initial value (value at t = 0) in the variables e _K (t), e _P (t), and e _S (t). That is, the key (black key 15b, white key 15c) is not depressed, and the damper pedal 16a and shift pedal 16b are not depressed. Further, it is assumed that “0” is set as an initial value for all other variables relating to “t”.

The comparison unit 101 acquires the input key depression amount input signal 1 (e _K (nΔt)) and the damper pedal depression amount input signal 3 (e _P (nΔt)), and sets the smaller value as e _D (nΔt). Output. This is expressed by the following formula (1).

  The damper model calculation unit 102 calculates the damper model calculation unit 102-1 that calculates the damper corresponding to the first string (iw = 1), and the damper model calculation that calculates the damper corresponding to the first string (iw = 2). Part 102-2. In the following description, only the index of the string is different, so that the damper model calculation unit 102 will be described. When there are three or more strings, as described above, the damper model calculation unit 102-iw (iw = 3,4,...) Is replaced with the string (iw = 3,4,...). It may be provided in correspondence.

  The string model calculation unit 104 includes a string model calculation unit 104-1 for calculating the first string (iw = 1) and a string model calculation unit 104-2 for calculating the first string (iw = 2). In the following description, the string model calculation unit 104 is described because only the string index is different. When there are three or more strings, the string model calculation unit 104-iw (iw = 3, 4,...) May be provided in parallel to the main body model 105 as described above. . (The calculation of the string model calculation unit 104 will be described later.)

  The damper model calculation unit 102 acquires eD (nΔt) output from the comparison unit 101 and uk (xD, nΔt) (k = 1, 3) output from the string model calculation unit 104 as described later. Then, using these, fDk (nΔt) obtained as a result of the following calculation is output to the string model calculation unit 104.

  Hereinafter, the calculation in the damper model calculation unit 102 will be described.

  The piano string in the initial state is in a state where vibration is suppressed by the damper. As the piano keys are pushed in, the corresponding dampers gradually move away from the corresponding strings and eventually the strings are completely released from the resistance of the dampers to prepare for hammering. In the piano, the degree of contact between the damper and the string can be changed not only by the key depression amount but also by the depression amount of the damper pedal, and the way of stopping the sound and the degree of string resonance can be finely controlled.

The mechanism of the damper in such a piano can be expressed concisely by the relational expression between the resistance force f _Dk (t) of the damper and the deformation amount u _k (x _D , t) of the damper shown in the following formula (2). it can.

In this example, by substituting e _D (nΔt) output from the comparison unit 101 into the equation (2), an amount “b _D e _D (nΔt)” corresponding to the viscosity coefficient of the damper is changed to a discrete time axis ( (t = nΔt; n = 0, 1, 2,...), the idea of sequentially changing the sound is to enable natural and continuous stop and string resonance control similar to a natural musical instrument piano.

  In the actual calculation, Expression (2) is incorporated into the equation of motion (Expression (16), Expression (18)) for each string mode in the string model calculation unit 104 described later. The above is the description of the damper model calculation unit 102.

The hammer model calculation unit 103 acquires the input signal 2 (V _H (nΔt)) and the input signal 4 (e _S (nΔt)), and outputs u ₁ output from the string model calculation unit 104 as described later. (X _H , nΔt) is acquired, and using these, f _H (nΔt) obtained as a result of the following calculation is output to the string model calculation unit 104.

  Hereinafter, the calculation in the hammer model calculation unit 103 will be described.

  When Newton's law of motion is applied to the assumptions regarding the physical model described above, Hammer's equation of motion is written as shown in Equation (3).

  The relational expression between the force exerted by the hammer tip on the string surface and the amount of compression of the hammer is as shown in Expression (4).

  However, when the hammer tip is in contact with the string surface, the equation (5) is applied, and when the hammer tip is separated from the string surface, the equations (6) and (7) are applied.

Now, if the right side of equation (3) is written as f (t) and dw _H (t) / dt is written as v _H (t), the second-order ordinary differential equation (equation (3)) for variable t is Then, by using the following forward Euler formula and trapezoid formula, a numerical solution is performed as shown in equation (8) on the discrete time axis (t = nΔt; n = 1, 2, 3,...). be able to.

That is, when a hammer speed V _H ((n−1) Δt) greater than “0” is given, v _H ((n−1) Δt), f ((n−1) Δt) in equation (8), By substituting V _H ((n−1) Δt), “0”, and W _H for w _H ((n−1) Δt), the hammer gravity center displacement w _H (nΔt) is immediately calculated. Eventually, when the condition of the hammer contact, that is, the expression (5) is satisfied, the output f _H ^[iw] (nΔt) to the string model is calculated from the expression (4).

Now, what is the mechanism of a shift pedal in a piano? When you depress it, the position of the hammer shifts to the treble side, changing the part that contacts the string of the hammer, or incomplete contact between the hammer and some strings In this example, by substituting the input signal 4 (e _S ^[iS] (nΔt)) into the equation (4), the elastic coefficient of the hammer is controlled. The equivalent amount K _H e _S ^[iS] (nΔt) is sequentially changed on the discrete time axis (t = nΔt; n = 0, 1, 2,. It enables natural and continuous tone control. The above is the description of the hammer model calculation unit 103.

The string model calculation unit 104 outputs f _Dk (nΔt) (k = 1, 3) output from the damper model calculation unit 102 and f _H (nΔt) output from the hammer model calculation unit 103, which are forces acting on the strings. In addition, as described later, u _Bk (nΔt) (k = 1, 2, 3) output from the main body model calculation unit 105 is acquired, and the results obtained by performing the following calculation using these are obtained. F _Bk (nΔt) (k = 1, 2, 3) is output to the main body model calculation unit 105, and u _k (x _D , nΔt) (k = 1, 3) is output to the damper model calculation unit 102. To do.

  Hereinafter, calculation in the string model calculation unit 104 will be described.

  When the Newton's law of motion is applied in the assumptions regarding the physical model described above, the equation of motion of the string is written as in equations (9), (10), and (11).

  However, in the equations (9) and (11), the nonlinear term due to the finite amplitude is omitted because the effect due to this is small. Similarly, in the formula (10), the force applied by the hammer in the chord axis direction is omitted because the effect of the force is small. Equation (9) is the bending vibration of the string corresponding to the movement direction of the hammer center of gravity, Equation (11) is the bending vibration of the string corresponding to the direction orthogonal to the movement direction of the hammer gravity center, and Equation (10) is Corresponds to the vertical vibration of the strings.

  Moreover, the string boundary condition is written as shown in equations (12) and (13).

  Conventionally, as a method for solving the unsteady vibration problem of strings that are simply supported at both ends (strings where the support ends do not move), `` string displacement '' is a Fourier sine series with an arbitrary time function as a coefficient. (For example, Reference 1: DEHall. Piano strings excitation. VI: Nonlinear modeling. J. Acoust. Soc. Am, Vol. 92, No. 1, pp. 95-105, 1992.).

  When this is written as an equation, equation (14) is obtained.

  The sine function of Equation (14) is the natural vibration mode of the string corresponding to the boundary condition of simple support at both ends. On the other hand, when the support end of a string moves like a piano, the natural vibration mode of a string cannot be easily obtained, so spatial discretization such as the finite element method or the difference method is required. It is common to rely on a solution that does this. However, these solutions (solutions in which the spatial function and the time function are not separated) are more error-prone to numerical computations on the time axis than the solutions using the natural vibration mode (solutions in which the spatial function and the time function are separated). Therefore, it is difficult to synthesize musical sounds for a long time with high accuracy using these solutions.

  Here, as a method of solving the unsteady vibration problem of a string when the support end moves with high accuracy and high speed, “displacement of string” is “Fourier sine series with an arbitrary time function as a coefficient” and “two support We present a new idea that is expressed by the sum of “displacement of straight line connecting ends”, that is, expressed by equation (15).

  At this time, Expression (15) satisfies the boundary condition expressions (12) and (13) for an arbitrary t. Note that the sine function in equation (15) is not a natural vibration mode in a strict sense, but here, for convenience, it will be referred to as a natural vibration mode.

  After substituting equation (15) into the partial differential equation (equations (9), (10), (11)), sin (ikπx / l) (ik = 1, 2,..., Mk; k = 1 , 2, 3) and integrating in the interval “0 ≦ x ≦ l”, the following second-order ordinary differential equations (Equations (16), (17), (18)) are derived.

Furthermore, the equation of motion for each string mode (Equations (16), (17), (18)) can be expressed as I _K × I _W ^[iK] × (2 × M ₁ ^{[iK ]} by using a bilinear sz transform. ^] + M ₂ ^[iK] ) (i _K = 1, 2,..., I _K ) parallelized second-order IIR (Infinite Impulse Response) filters can be described respectively. nΔt; n = 0, 1, 2,..., A _k ^{[iK] [iW] [mk]} (nΔt) (i _K = 1, 2,..., I _K ; i _W = 1 , 2,..., I _W ^[iK] ; m _k = 1, 2,..., M _k ^[iK] ; it can. However, in each time step, the calculation of Expression (16) and Expression (18) is preceded by the calculation of Expression (17), and the nonlinear term A _k ^[mk] (t) A _k included in the right side of Expression (17). ^[m'k] (t) (k = 1, 3) is treated like an external force term.

  The relational expression between the force exerted by the string on the string support end and the support end displacement is written as in Expressions (28) and (29).

  Further, by substituting equation (15) into equations (28) and (29), equations (30) and (31) are derived. However, the terms related to nonlinear terms and rotational inertia were omitted.

A _k ^[mk] (nΔt) calculated by the method described above is substituted into the equation obtained by substituting equations (30) and (31) into the following body-string physical coordinate conversion equation (equation (32)). If the value of (m _k = 1, 2,..., M _k ; k = 1, 2, 3) is substituted, the force F _Bk ^[iB] (nΔt) exerted on the string support end by the string, that is, the string An output from the model calculation unit 104 to the main body model calculation unit 105 can be calculated.

In addition, the displacement of the string hitting point and the sound stopping point is expressed as A _k ^[mk] (nΔt) (m _k = 1, 2,..., M in Equations (33) and (34) obtained from Equation (15). _k ; calculated by substituting the values of k = 1, 2, 3).

The u ₁ (x _H , nΔt) obtained here is output to the hammer model calculation unit 103 and is recursively substituted into Equation (5). Similarly, u _k (x _D ^[iD] , nΔt) is output to the damper model calculation unit 102, and again returns to equations (16) and (18) in the string model calculation unit 104 via equation (2). Assigned recursively. The above is the description of the string model calculation unit 104.

The main body model calculation unit 105 acquires F _Bk (nΔt) output from the string model calculation unit 104, and uses this to calculate A _C (nΔt) obtained as a result of the following calculation. The data is output to 106.

  Hereinafter, the calculation in the main body model calculation unit 105 will be described.

Due to the assumptions regarding the physical model described above, the equation of motion of the body is the output F _Bk ^{[iK] [iW] [iB]} (t) (i _K = 1, 2,..., I _K ; i _W ; = 1, 2,..., I _W ^[iK] ; i _B = 0, 1; k = 1, 2, 3) as input, the following second-order ordinary differential equation (formula (35 )).

  By the way, the main body of the piano is made of wood, metal, etc. Among them, wood has the feature that “the vibration damping ability of the high frequency component is larger than the low frequency component”, and this is the piano ( It is also the cause of the “sound that is comfortable in the ears and warm” that is peculiar to a musical instrument whose main body is made of wood. Such acoustic properties of wood can be explained by modeling wood as “a material having three-dimensional orthotropic anisotropy in both elastic modulus and structural damping coefficient” (for example, Reference 2: Japan). Society of Mechanical Engineers (ed.) Advanced Composite Materials, pp.68-70.

The main body model configured to include “a material having three-dimensional orthogonal anisotropy in both elastic modulus and structural damping coefficient” is also called a general structural damping system (non-proportional structural damping system or general hysteresis damping system). Therefore, the attenuation matrix cannot be diagonalized by real eigenvalue analysis (Reference 3), but here, by ignoring the off-diagonal terms of the attenuation matrix, the proportional structure damping system ( It is also referred to as a proportional hysteresis damping system).
(Reference 3: Akio Nagamatsu mode analysis. Bafukan, 1985)

  Further, the proportional damping system is approximated by a proportional viscous damping system, that is, the mode damping ratio is expressed by “mode structure damping coefficient / 2”. At this time, the natural angular frequency, the mode damping ratio, and the natural vibration mode included in the equation (35) are calculated by performing real eigenvalue analysis using a commercial finite element method software on an arbitrary three-dimensional body. can do. Here, the mode damping ratio should be an approximate mode damping ratio, but here, it is simply referred to as a mode damping ratio for convenience.

Since the equation of motion (Equation (35)) for each mode of the main body can be described as M parallel second-order IIR filters by using the bilinear sz transform, like the strings, the discrete time axis (T = nΔt; n = 0, 1, 2,...) The value of A _C ^[m] (nΔt) (m = 1, 2,..., M) is sequentially calculated for each mode. And is output to the air model calculation unit 106.

For the displacement of the string support end, the value of A _C ^[m] (nΔt) (m = 1, 2,..., M) calculated by the above method is used as the physical coordinate-mode coordinate conversion formula (formula (36 )), The value can be calculated by substituting into the string-physical coordinate conversion formula (formula (37)).

U _Bk ^[iB] (nΔt) obtained here is output to the string model calculation unit 104, and the equations (16), (17), (18) and the equations (30), (31), (33), It is recursively assigned to (34). The above is the description of the main body model calculation unit 105.

The air model calculation unit 106 acquires A _C (nΔt) output from the main body model calculation unit 105, and outputs P (nΔt) obtained as a result of performing the following calculation using this.

  Hereinafter, calculation in the air model calculation unit 106 will be described.

  In principle, the unsteady sound pressure at an arbitrary observation point in the air radiated from a three-dimensional structure having an arbitrary shape is obtained by a method as shown in the following equation, that is, a sufficiently fine acoustic wave on the entire surface of the structure. After dividing into radiating elements (boundary elements), "impulse response function between the velocity of each acoustic radiating element of the structure and the sound pressure at the observation point in the air" and "velocity of each acoustic radiating element of the structure Is calculated for each element by the convolution integration with the element, and the total sum thereof is calculated (formula (38)).

However, to become a normal huge number acoustic radiating elements number I _G of the body necessary to synthesize a high-quality pseudo piano sounds, that the calculation of equation (38) it takes a lot of time Become. Therefore, the formula in equation (38) (39), (40) by substituting, changing the order of calculating the order and acoustic radiation element number I _G number of the sum to calculate the natural vibration mode of M sum, i.e., A calculation such as the equation (41) will be performed.

Natural vibration mode number M of the body necessary to synthesize a high-quality pseudo piano sound, as compared to the acoustic radiating element number I _G, normally, since much smaller, formula instead of formula (38) (41) That is, “impulse response between the velocity of each acoustic radiating element of the main body and the sound pressure at the observation point in the air” (“t” is “nΔt” (n = 0, 1 on the left side of Equation (40)). ,..., N ^[iP] -1)), but not "impulse response between the velocity on the mode coordinates of each natural vibration mode of the main body and the sound pressure at the observation point in the air" (formula (42 ) In the left side, “t” is “nΔt” (n = 0, 1,..., N ^[iP] −1)) is calculated in advance, thereby dramatically increasing the time required for synthesizing the pseudo piano sound. It can be shortened.

H ^{[iP] [iG]} (ω) included in the equation (43), that is, “the frequency response function between the velocity of each acoustic radiation element of the main body and the sound pressure at the observation point in the air” is arbitrary 3 It can be calculated by performing frequency response analysis using a commercial boundary element method software on a dimensional shape main body on a discrete frequency axis. Further, the equation (42) can be calculated by a general IFFT (Inverse Fast Fourier Transform) operation.

The differential coefficient included in the equation (41), that is, the “velocity on the mode coordinates of each natural vibration mode of the main body” is A _C ^[m] (nΔt) (m = 1, 2, .., M), that is, by integrating numerically differentiating “displacement of each natural vibration mode of the main body on the mode coordinates”, the integral included in the equation (41) is a general FIR (Finite Impulse Response). ) It can be calculated by the filter method.

Thus, from the equation (41), the output signal from the air model, that is, the sound pressure P ^[iP] (nΔt) (i on the discrete time axis (t = nΔt; n = 0, 1, 2,...) _P = 1, 2,..., I _P ) can be calculated sequentially and output as a musical sound signal.

  Here, by using a technique called “high-speed convolution” in which the convolution calculation in Expression (41) is performed in the frequency domain instead of the time domain, the speed can be further dramatically increased. The above is the description of the configuration of the tone signal synthesizer 100.

  In this way, the tone signal synthesis unit 100 has a rich and three-dimensional sound produced by three-dimensional vibration of the entire instrument, a ringing sound that can be heard when a string in the middle / low range is struck, a key press depth and a pedal. It is possible to generate a pseudo piano sound that realistically expresses the characteristics of a natural instrument piano sound, such as various musical nuances linked to the depth of depression. In addition, these characteristics can be controlled in the same way as a natural musical instrument piano.

Specifically, there are the following. For example, by changing parameters such as chord length (= distance between chord support points) or striking ratio (= “string length” / “distance between bearing side chord support end and strut point”), ringing sound It becomes possible to control the level. In the following, this ringing sound will be described in particular using equation (17). However, in order to make the explanation easy to understand, in the equation (17), the description will be made according to the equation (44) in which the displacement of the string support end, the displacement in the string y direction, and the internal viscous damping coefficient of the string are omitted. .

Equation (44) is an equation of motion of the longitudinal vibration of the string i _second- order natural vibration. By considering the right side as a periodic external force, it can be considered as an equation of motion of a one-degree-of-freedom viscous damping forced oscillation system. As is well known, the general solution of this equation is composed of the sum of a damped free vibration solution (a general solution of homogeneous equations) and a sustained forced vibration solution (a special solution of inhomogeneous equations). The property of the forced vibration solution is that the system vibrates at a frequency of a periodic external force, the amplitude increases as the frequency approaches the natural frequency of the system, and resonates when they coincide. Now, it is assumed that each natural vibration related to the bending vibration of the string is a harmonic vibration, that is, it is written as in Expression (45).

  At this time, the inside of the right side {} of Expression (44) is derived as Expression (46).

Now, with i ₂ fixed, paying attention to the sequence formed by the term cos2π (f ₁ ^[m1] + f ₁ ^{[m1 + i2]} ) t included in the equation (46), “the second (2m ₁ + i of this sequence) ₂ ) When the deviation of the secondary frequency f ₁ ^[m1] + f ₁ ^{[m1 + i2] from} the harmonic series frequency is calculated, when i ₂ is small, the value is “bending vibration number (2m ₁ + i ₂ )”. It can be confirmed that the frequency is about 1/4 of the deviation from the harmonic sequence frequency of the second natural frequency f ₁ ^{[2m1 + i2]} . By analyzing the piano sound of a natural instrument, it is known that "the partial sequence of piano has a sub-sequence whose frequency deviation from the harmonic sequence is about 1/4 of the main sequence" Therefore, the sequence created by the term applies to the subsequence. As i ₂ increases, this “deviation amount” gradually increases.

In addition, the sequence formed by the term cos2π (f ₁ ^[m1] + f ₁ ^[i2 −m1 ^] ) t included in Equation (46) also contributes to the formation of the subsequence, provided that the degree of contribution is first. It can be understood that it is smaller than the term.

The formula obtained by substituting the formula (46) into the formula (44) is that the (2m ₁ + i ₂ ) th order frequency f ₁ ^[m1] + f ₁ ^{[m1 + i2] of the} subsequence is the i th of the longitudinal vibration of the string. resonance phenomenon when matching the _secondary natural frequency represents that occur. This is a characteristic phenomenon of the piano sound of a natural instrument, “There are sub-sequences in the partial sequence of pianos whose frequency deviation from the harmonic sequence is about 1/4 of the main sequence” In addition, “when the frequency of the odd-order partial sound of the sub-sequence matches the odd-order natural frequency of the longitudinal vibration of the string, or the frequency of the even-order partial sound of the sub-sequence is When it matches the even-order natural frequency, the level of the sub-sequence partial sound increases and this becomes a ringing sound. When the sum of the natural frequencies matches the odd-order natural frequency of the longitudinal vibration of the string, or the sum of a set of odd-order natural frequencies of the bending vibration of the string, or a set of even-order natural frequencies Ringing sound when the sum of is equal to the even natural frequency of the longitudinal vibration of the string. For but generated "it (reference 4) and gives a mathematical description.
(Reference 4: J. Ellis. Longitudinal model in piano strings: Results of new research. Piano Technicians journal, pp.16-23, May 1998.)

Furthermore, with respect to the roaring phenomenon such as “Rin Ling” and “Hin Hin”, for example, the frequency of the sub-sequence 15th (= 7 + 8 = 2 × 7 + 1) order and the sub-sequence 15th (= 6 + 9 = 2 × 6 + 3) order is Since it is slightly different, it can be explained that the difference in frequency produces a sag. Incidentally, the formula terms contained in ^{_{(44) cos2π (f 1 [}} m1] -f 1 [m1 + i2]) t and ^{_{cos2π (f 1 [m1] -f}} 1 [i2 - m1]) t is a bending vibration This indicates the presence of a partial sound having a frequency slightly higher than the natural frequency.

  When the material constant of the string is fixed, the natural frequency of the longitudinal vibration of the string depends only on the string length from the equation (20). Note that this is not the case for wound strings (strings in which a copper wire is wound around a steel core wire) that is normally used in the bass range of a piano.

  In the range of about 30th to 40th keys of a standard 88-key piano, the frequency and string of the 15th (= 7 + 8 = 2 × 7 + 1) th sub-sequence are caused by the setting of the string length. In some cases, the fundamental natural frequency of the vertical vibration is close. Even in such a case, setting the stringing ratio to “7” or “8” can avoid an excessive increase in ringing sound level.

  This is because the 15th (= 7 + 8 = 2 × 7 + 1) th subsequence is a product of the 7th and 8th natural vibrations of the bending vibration, but the stringing ratio is “7” or “8”. This is because the “7” th order or “8th” order natural vibration of the bending vibration is lost, and the 15th (= 7 + 8 = 2 × 7 + 1) th subsequence is not generated. Even in this case, the fifteenth (= 6 + 9 = 2 × 6 + 3) order of the subsequence still exists, but these do not resonate with the fundamental natural vibration of the longitudinal vibration. When the time frequency analysis of the piano sound of a natural instrument is performed, the peak of the natural vibration of the longitudinal vibration when it does not coincide with the subsequence (this is the free vibration solution when the right side is set to “0” in equation (17)). ) Rapidly decays and is not observed as persistent. As a cause of this attenuation, it is presumed that a portion due to friction at the string support portion is large. In the string model of the present invention, “external friction distributed over the entire string”, that is, a term including the external viscous damping coefficient b in Expression (10), and “local external friction in the string support portion” is substituted. Yes.

  As described above, the generation mechanism of ringing sound and the design factors (string length, string striking ratio) that control the level have been explained, but the vertical vibration of the string itself has little ability to radiate sound in the air. In order for the ringing sound to be heard as a sound, in addition to the “string non-linear (finite amplitude) vibration mechanism” just described, the “three-dimensional coupled vibration mechanism between the string and the body” (“attachment angle of the string to the body”) Needless to say, design factors such as “frame shape” are included) and “three-dimensional acoustic radiation mechanism of the main body” (including “frame shape” is also included).

  By the way, in the development field of natural instrument pianos, the improvement of piano sound is nothing but finding the overall optimal solution of a complex system called the piano. This is particularly inefficient for a huge acoustic structure having many design factors and error factors, such as pianos, which have variations in physical properties of natural materials and variations in work by humans such as sounding. Since the present invention quantitatively clarifies the causal relationship between the specification (cause) and the sound (result) of the piano, it contributes to the improvement of piano development efficiency as a design simulator. In addition to the above, an advantage of the musical sound synthesis method based on the physical model is that a supernatural effect (for example, a remarkably large piano that is difficult to produce in reality) can be virtually generated.

Second Embodiment
FIG. 4 is a block diagram showing a configuration of an electronic musical instrument 1A according to the second embodiment of the present invention. The electronic musical instrument 1A is, for example, an electronic piano, and includes a control unit 11A, a storage unit 12A, a user operation unit 13A, a performance operation unit 15A, and a sound emission unit 17A. These units are connected to each other via a bus 18A. Since the user operation unit 13A, the sound emission unit 17A, and the bus 18A have the same functions as the user operation unit 13, the sound emission unit 17, and the bus 18 in the electronic musical instrument 1 according to the first embodiment, description thereof is omitted.

  Unlike the performance operation unit 15 in the first embodiment, the performance operation unit 15A has the shift pedal 16b removed. Therefore, the pedal position sensor 16Ac detects the depression amount of the damper pedal 16a. Other configurations in the performance operation unit 15A have the same functions as those of the performance operation unit 15 in the first embodiment, and thus the description thereof is omitted.

Unlike the storage unit 12 in the first embodiment, the storage unit 12A stores the force f _H (t) exerted by the hammer tip on the string surface. This value indicates a value in a state where the shift pedal 16b in the first embodiment is not depressed (rest position).

  Unlike the control unit 11 in the first embodiment, the control unit 11A realizes a tone signal synthesis unit 100A that does not use the hammer model calculation unit 103 among the tone signal synthesis units 100 realized by executing a control program. .

FIG. 5 is a block diagram showing the configuration of the tone signal synthesizer 100A. As shown in FIG. 5, the musical tone signal synthesis unit 100A does not have the hammer model calculation unit 103. Chord model calculation unit 104A-1,104A-2 does not acquire the _f H (t) which is output from the hammer model calculation unit 103 obtains the _f H (t) stored in the storage unit 12A. Other configurations in the musical tone signal synthesis unit 100A have the same functions as those of the musical tone signal synthesis unit 100 in the first embodiment, and thus description thereof is omitted.

  As described above, the electronic musical instrument 1A according to the second embodiment is assumed to be an electronic piano including only the damper pedal 16a as a pedal.

<Third Embodiment>
FIG. 6 is a block diagram showing a configuration of an electronic musical instrument 1B according to the third embodiment of the present invention. The electronic musical instrument 1B is, for example, an electronic piano, and includes a control unit 11B, a storage unit 12B, a user operation unit 13B, a performance operation unit 15B, and a sound emission unit 17B. These units are connected to each other via a bus 18B. Since the user operation unit 13B, the sound emission unit 17B, and the bus 18B have the same functions as the user operation unit 13, the sound emission unit 17, and the bus 18 in the electronic musical instrument 1 according to the first embodiment, the description thereof is omitted.

  Unlike the performance operation unit 15 in the first embodiment, the performance operation unit 15B has the damper pedal 16a removed. Therefore, the pedal position sensor 16Bc detects the depression amount of the shift pedal 16b. Other configurations in the performance operation unit 15B have the same functions as those of the performance operation unit 15 in the first embodiment, and thus description thereof is omitted.

Unlike the storage unit 12 in the first embodiment, the storage unit 12B stores the resistance force f _Dk (t) of the damper. This value indicates a value when the damper pedal 16a in the first embodiment is not depressed (rest position).

  Unlike the control unit 11 in the first embodiment, the control unit 11B includes the comparison unit 101 and the damper model calculation units 102-1 and 102-2 in the musical sound signal synthesis unit 100 realized by executing the control program. An unused tone signal synthesis unit 100B is realized.

FIG. 7 is a block diagram showing the configuration of the tone signal synthesizer 100B. As shown in FIG. 7, the tone signal synthesis unit 100B does not include the comparison unit 101 and the damper model calculation units 102-1 and 102-2. Chord model calculation unit 104B-1,104B-2 does not acquire the _f Dk output from the damper model calculation section 102 (t), we obtain the stored _f Dk (t) in the storage unit 12B. Other configurations in the musical tone signal synthesis unit 100B have the same functions as those in the musical tone signal synthesis unit 100 in the first embodiment, and thus description thereof is omitted.

  As described above, the electronic musical instrument 1B according to the third embodiment is assumed to be an electronic piano provided with only the shift pedal 16b as a pedal.

<Fourth embodiment>
FIG. 8 is a block diagram showing a configuration of an electronic musical instrument 1C according to the fourth embodiment of the present invention. The electronic musical instrument 1C is, for example, an electronic piano, and includes a control unit 11C, a storage unit 12C, a user operation unit 13C, a performance operation unit 15C, and a sound emission unit 17C. These units are connected to each other via a bus 18C. Since the user operation unit 13C, the sound emission unit 17C, and the bus 18C have the same functions as the user operation unit 13, the sound emission unit 17, and the bus 18 in the electronic musical instrument 1 according to the first embodiment, description thereof is omitted.

  Unlike the performance operation unit 15 in the first embodiment, the performance operation unit 15C has the pedal unit 16 removed. Therefore, there is no pedal position sensor. Other configurations in the performance operation unit 15C have the same functions as those of the performance operation unit 15 in the first embodiment, and thus description thereof is omitted.

Unlike the storage unit 12 in the first embodiment, the storage unit 12C stores the resistance force f _Dk (t) of the damper and the force f _H (t) exerted by the hammer tip on the string surface. These values indicate values when the damper pedal 16a and the shift pedal 16b are not depressed (rest position) in the first embodiment.

  Unlike the control unit 11 in the first embodiment, the control unit 11C includes a comparison unit 101, damper model calculation units 102-1 and 102-2, among the musical sound signal synthesis units 100 realized by executing a control program. And the musical tone signal synthesis unit 100C that does not use the hammer model calculation unit 103 is realized.

FIG. 9 is a block diagram showing the configuration of the tone signal synthesizer 100C. As shown in FIG. 9, the tone signal synthesis unit 100C does not include the comparison unit 101, the damper model calculation units 102-1 and 102-2, and the hammer model calculation unit 103. The string model calculation units 104C-1 and 104C-2 do not acquire f _Dk (t) output from the damper model calculation unit 102 and f _H (t) output from the hammer model calculation unit 103, but store them. F _Dk (t) and f _H (t) stored in the unit 12C are acquired. The other configuration of the tone signal synthesis unit 100C has the same function as that of the tone signal synthesis unit 100 in the first embodiment, and a description thereof will be omitted.

  As described above, the electronic musical instrument 1C according to the third embodiment is assumed to be an electronic piano that does not include the damper pedal 16a and the shift pedal 16b.

  As mentioned above, although embodiment of this invention was described, this invention can be implemented in various aspects as follows.

<Modification 1>
In the above-described embodiment, for example, in order to function as the electronic musical instrument 1 that generates sound in response to the operation of the keyboard unit 15a and the pedal unit 16, the musical tone signal synthesis processing is performed in real time, but according to the musical tone control data. In the case of generating sound, non-real time processing may be used.

  In this case, for example, using the musical tone control data for one song, the “variation data on the time axis of the speed on the mode coordinates of each natural vibration mode of the instrument body” is calculated first, It can also be said that the convolution calculation of the data and “data of impulse response or frequency response between the velocity on the mode coordinates of each natural vibration mode of the main body and the sound pressure at the observation point in the air” is performed later. This means that tone synthesis can be easily performed when only the position of the observation point is changed.

<Modification 2>
In the embodiment described above, the tone signal synthesis process is a synthesis process of a tone signal simulating a piano sound. However, the tone signal synthesis process is not limited to a piano, and supports strings that vibrate and strings. Any musical instrument (eg, harpsichord, koto, guitar, etc.) may be used as long as it has a three-dimensional structure having a main body that emits sound into the air by being transmitted. When a column (corresponding to a piano piece) is provided between both ends of a string that is stretched like a koto, one end of the string support end is a column.

<Modification 3>
The control program in the above-described embodiment is provided in a state stored in a computer-readable recording medium such as a magnetic recording medium (magnetic tape, magnetic disk, etc.), an optical recording medium (optical disk, etc.), a magneto-optical recording medium, or a semiconductor memory. Can do. It is also possible to provide a communication unit that can be connected to a network and download it via a network such as the Internet.

DESCRIPTION OF SYMBOLS 1 ... Electronic musical instrument, 11, 11A, 11B, 11C ... Control part, 11a ... CPU, 11b ... DSP, 11c ... ROM, 11d ... RAM, 11e ... Signal interface, 11f ... Internal bus, 12, 12A, 12B, 12C ... Storage unit 13, 13A, 13B, 13C ... User operation unit, 13a ... Operation panel, 13b ... Mouse, 13c ... Operation switch, 13d ... Keyboard, 14 ... Display unit, 15, 15A, 15B, 15C ... Performance operation unit, 15a ... Keyboard part, 15b ... Black key, 15c ... White key, 15d ... Key position sensor, 15e ... Key speed sensor, 16 ... Pedal part, 16a ... Damper pedal, 16b ... Shift pedal, 16c, 16Ac, 16Bc ... Pedal position sensor , 17, 17A, 17B, 17C ... sound emitting part, 17a ... digital-analog converter, 17b ... speaker, 18 18A, 18B, 18C ... bus, 21 ... grand piano, 21a ... key, 21b ... keyboard, 21c ... hammer, 21d ... action mechanism, 21e ... string, 21ea ... piece, 21eb ... bearing, 21f ... damper, 21h ... cabinet, 21j ... Main unit, 100, 100A, 100B, 100C ... Music signal synthesis unit, 101 ... Comparison unit, 102-1 and 102-2 ... Damper model calculation unit, 103 ... Hammer model calculation unit, 104-1, 104A-1, 104B-1, 104C-1, 104-2, 104A-2, 104B-2, 104C-2 ... string model calculation unit, 105 ... body model calculation unit, 106 ... air model calculation unit

Claims (10)

A musical tone signal composition for generating a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted a force from at least a first direction and a string supporting end that supports the string in accordance with performance information that is input. A method,
First information representing a force exerted on the string calculated according to the performance information is acquired, and three equations of motion having an interaction through the string support end, wherein the first information is used. A first equation of motion representing a first bending vibration of the string in the first direction, a second equation of motion representing a second bending vibration of the string in a direction different from the first direction, and a longitudinal direction of the string A string model calculation process for calculating second information indicating the first bending vibration, the second bending vibration, and the longitudinal vibration of the string based on a third equation of motion representing vibration;
A musical sound signal synthesis method comprising: calculating a musical sound signal based on the second information. The music signal synthesis method according to claim 1, wherein the third equation of motion is expressed using the first bending vibration and the second bending vibration. A musical tone signal composition for generating a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted a force from at least a first direction and a string supporting end that supports the string in accordance with performance information that is input. A method,
First information representing a force exerted on the string calculated according to the performance information is acquired, and two equations of motion having an interaction through the string support end, wherein the first information is used. Based on the first equation of motion representing the first bending vibration of the string in the first direction and the third equation of motion representing the longitudinal vibration of the string, the first bending vibration and the longitudinal vibration of the string are shown. A string model calculation process for calculating second information;
A musical sound signal synthesis method comprising: calculating a musical sound signal based on the second information. A musical tone signal composition for generating a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted a force from at least a first direction and a string supporting end that supports the string in accordance with performance information that is input. A method,
First information representing a force exerted on the string calculated according to the performance information is acquired, and two equations of motion having an interaction through the string support end, wherein the first information is used. Based on a first equation of motion representing a first bending vibration of the string in the first direction and a second equation of motion representing a second bending vibration of the string in a direction different from the first direction, A string model calculation process for calculating second information indicating the first bending vibration of the string and the second bending vibration;
A musical sound signal synthesis method comprising: calculating a musical sound signal based on the second information. Causing a computer to generate a musical tone signal of a sound emitted from a musical instrument having a string that vibrates under a force applied in at least a first direction and a string support end that supports the string in accordance with input performance information A program,
The computer,
First information representing a force exerted on the string calculated according to the performance information is acquired, and three equations of motion having an interaction through the string support end, wherein the first information is used. A first equation of motion representing a first bending vibration of the string in the first direction, a second equation of motion representing a second bending vibration of the string in a direction different from the first direction, and a longitudinal direction of the string A string model calculating means for calculating second information indicating the first bending vibration, the second bending vibration and the longitudinal vibration of the string based on a third equation of motion representing vibration;
The program for functioning as a musical tone signal calculation means for calculating the musical tone signal based on the second information. Causing a computer to generate a musical tone signal of a sound emitted from a musical instrument having a string that vibrates under a force applied in at least a first direction and a string support end that supports the string in accordance with input performance information A program,
The computer,
First information representing a force exerted on the string calculated according to the performance information is acquired, and two equations of motion having an interaction through the string support end, wherein the first information is used. Based on the first equation of motion representing the first bending vibration of the string in the first direction and the third equation of motion representing the longitudinal vibration of the string, the first bending vibration and the longitudinal vibration of the string are shown. A string model calculating means for calculating second information;
The program for functioning as a musical tone signal calculation means for calculating the musical tone signal based on the second information. Causing a computer to generate a musical tone signal of a sound emitted from a musical instrument having a string that vibrates under a force applied in at least a first direction and a string support end that supports the string in accordance with input performance information A program,
The computer,
First information representing a force exerted on the string calculated according to the performance information is acquired, and two equations of motion having an interaction through the string support end, wherein the first information is used. Based on a first equation of motion representing a first bending vibration of the string in the first direction and a second equation of motion representing a second bending vibration of the string in a direction different from the first direction, A string model calculating means for calculating second information indicating the first bending vibration of the string and the second bending vibration;
The program for functioning as a musical tone signal calculation means for calculating the musical tone signal based on the second information. Generation of a musical tone signal for generating a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted a force from at least a first direction and a string supporting end that supports the string in accordance with input performance information A device,
First information representing a force exerted on the string calculated according to the performance information is acquired, and three equations of motion having an interaction through the string support end, wherein the first information is used. A first equation of motion representing a first bending vibration of the string in the first direction, a second equation of motion representing a second bending vibration of the string in a direction different from the first direction, and a longitudinal direction of the string A string model calculating means for calculating second information indicating the first bending vibration, the second bending vibration and the longitudinal vibration of the string based on a third equation of motion representing vibration;
A musical tone signal generating device comprising: musical tone signal calculating means for calculating the musical tone signal based on the second information. Generation of a musical tone signal for generating a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted a force from at least a first direction and a string supporting end that supports the string in accordance with input performance information A device,
First information representing a force exerted on the string calculated according to the performance information is acquired, and two equations of motion having an interaction through the string support end, wherein the first information is used. Based on the first equation of motion representing the first bending vibration of the string in the first direction and the third equation of motion representing the longitudinal vibration of the string, the first bending vibration and the longitudinal vibration of the string are shown. A string model calculating means for calculating second information;
A musical tone signal generating device comprising: musical tone signal calculating means for calculating the musical tone signal based on the second information. Generation of a musical tone signal for generating a musical tone signal of a sound emitted from a musical instrument having a string that vibrates by being exerted a force from at least a first direction and a string supporting end that supports the string in accordance with input performance information A device,
First information representing a force exerted on the string calculated according to the performance information is acquired, and two equations of motion having an interaction through the string support end, wherein the first information is used. Based on a first equation of motion representing a first bending vibration of the string in the first direction and a second equation of motion representing a second bending vibration of the string in a direction different from the first direction, A string model calculating means for calculating second information indicating the first bending vibration of the string and the second bending vibration;
A musical tone signal generating device comprising: musical tone signal calculating means for calculating the musical tone signal based on the second information.

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