JP5716369B2 - Music signal synthesis method, program, and music signal synthesis apparatus - Google Patents

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, when a pseudo piano sound is desired to be generated, a musical tone signal is synthesized by executing a simulation calculation with a general-purpose computer in accordance with a physical model of strings and soundboards. A musical tone synthesizer using such a method is described in Patent Document 1, for example. Non-Patent Document 1 describes an example of attempting to incorporate a “string non-linear vibration mechanism” when synthesizing a pseudo piano sound.

Special table 2009-544995 gazette

A Modal-Based Real-Time Piano Synthesizer, PUBLISHED IN IEEE TRANS.ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 18, NO. 4, PP. 809-821, MAY 2010

  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 resonating” 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 includes a vibrating string, and a main body that supports the string by two string support ends, and the vibration of the string is transmitted through at least one end of the string support end. A musical sound signal synthesizing method for generating a musical tone signal of a sound emitted from an instrument having a three-dimensional structure according to input performance information , wherein a control unit exerts an influence on the string calculated according to the performance information First information representing a force is obtained, and the first information, the n-th derivative (n = 1, 2,...) Relating to the displacement on the mode coordinates of each natural vibration mode of the string or the time of displacement. ) And third information representing n-order derivatives (n = 1, 2,...) Related to displacement or displacement time on the mode coordinates of each natural vibration mode of the main body. By calculation using simultaneous ordinary differential equations expressing the relationship, A calculation step of calculating the third information from the first information, the control unit, based on the third information above is calculated, characterized in that it comprises a tone signal calculation process of calculating the tone signal A method for synthesizing a musical sound signal is provided.

  In another preferred embodiment, the string is plural.

Further, the present invention provides a three-dimensional structure including a string that is hit by a hammer, and a main body that supports the string by two string support ends and that transmits vibration of the string through at least one end of the string support end. A musical tone signal synthesis method for generating a musical tone signal of a sound emitted from a musical instrument having a structure in accordance with input performance information , wherein the control unit is initialized according to the performance information. First information representing the n-th derivative (n = 1, 2,...) Relating to the time of the nth-order derivative (n) relating to the displacement or displacement time on the mode coordinates of each natural vibration mode of the string. = 1, 2,..., And n-th derivative (n = 1, 2,...) Related to displacement or displacement time on the mode coordinates of each natural vibration mode of the main body. The simultaneous ordinary differential method representing the relationship of the third information representing The calculation using equation and calculation process of calculating said third information from the performance information, control unit, based on the third information above is calculated, and the tone signal calculation process of calculating the tone signal A musical sound signal synthesis method is provided.

  In another preferred embodiment, the hammer and the strings are plural.

  According to another aspect of the present invention, there is provided a three-dimensional computer having a vibrating string, and a main body that supports the string by two string support ends, and transmits the vibration of the string through at least one end of the string support end. A program for generating a musical tone signal of a sound emitted from a musical instrument having a structure in accordance with input performance information, wherein the computer represents a force exerted on the string calculated in accordance with the performance information. A second information representing the first information, the nth derivative (n = 1, 2,...) Related to the displacement or time of displacement on the mode coordinates of each natural vibration mode of the string; Simultaneously representing the relationship between the information and the third information representing the n-th derivative (n = 1, 2,...) Related to the displacement or time of displacement on the mode coordinates of each natural vibration mode of the main body. By using the differential equation, Calculating means for calculating said third information from the first information, based on the third information the calculated to provide a program for functioning as a tone signal calculating means for calculating said tone signal.

  According to another aspect of the present invention, there is provided a computer with a string hit by a hammer, and a main body that supports the string by two string support ends and transmits vibration of the string through at least one end of the string support end. A program for generating a musical tone signal of a sound emitted from a musical instrument having a three-dimensional structure according to input performance information, wherein the computer is initialized according to the performance information. First information representing the n-th derivative (n = 1, 2,...) Relating to the time of the nth-order derivative (n = 1, 2,..., And n-th derivative (n = 1, 2,...) Related to displacement or displacement time on the mode coordinates of each natural vibration mode of the main body. ) For the third information A calculation means for calculating the third information from the performance information by calculation using simultaneous ordinary differential equations representing the musical tone, and a musical signal calculation for calculating the musical tone signal based on the calculated third information A program for functioning as a means is provided.

  In addition, the present invention provides a three-dimensional musical instrument having a vibrating string, and a main body that supports the string by two string support ends and transmits vibration of the string through at least one end of the string support end. A musical tone signal generating device that generates a musical tone signal of a sound emitted from the sound according to input performance information, wherein first information representing a force acting on the string calculated according to the performance information is acquired. , The first information, the second information representing the nth derivative (n = 1, 2,...) Related to the displacement or time of displacement on the mode coordinates of each natural vibration mode of the string, and Use simultaneous ordinary differential equations representing the relationship of the third information representing the n-th derivative (n = 1, 2,...) Related to the displacement or displacement time on the mode coordinates of each natural vibration mode of the main body. The third information is obtained from the first information by the calculation performed. Calculating means for output, based on the third information the calculated to provide a musical tone signal generating apparatus characterized by comprising a tone signal calculating means for calculating said tone signal.

  Further, the present invention provides a three-dimensional structure including a string that is hit by a hammer, and a main body that supports the string by two string support ends and that transmits vibration of the string through at least one end of the string support end. A musical tone signal generating device for generating a musical tone signal of a sound emitted from a musical instrument having a structure in accordance with input performance information, wherein the hammer is initialized in accordance with the performance information, or n relating to a displacement time First information representing the first derivative (n = 1, 2,...), Nth derivative (n = 1, 2, regarding displacement or time of displacement on the mode coordinates of each natural vibration mode of the string. ,...) And second information representing n-order derivatives (n = 1, 2,...) Relating to displacement or displacement time on the mode coordinates of each natural vibration mode of the main body. Using simultaneous ordinary differential equations that represent the relationship of the three information Computation means for computing the third information from the performance information by computation, and a tone signal calculation means for computing the tone signal based on the calculated third information. A musical tone signal generating apparatus is provided.

  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 character which the sound emitted from the musical instrument emitted from the three-dimensional structure which has a string and a main body has are provided. can do.

It is a block diagram which shows the structure of the electronic musical instrument which concerns on embodiment of this invention. It is a figure explaining the relationship between the conversion part which concerns on embodiment of this invention, and a musical tone signal synthetic | combination part. It is a block diagram which shows the structure of the musical tone signal synthetic | combination part which concerns on 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 musical tone signal synthetic | combination part which has a calculating part which concerns on embodiment of this invention. It is a block diagram which shows the structure of the musical tone signal synthetic | combination part which concerns on the modification 1 of this invention. It is a block diagram which shows the structure of the electronic musical instrument which concerns on the modification 2 of this invention. It is a block diagram which shows the structure of the musical tone signal synthetic | combination part which concerns on the modification 2 of this invention. It is a block diagram which shows the structure of the electronic musical instrument which concerns on the modification 3 of this invention. It is a block diagram which shows the structure of the musical tone signal synthetic | combination part which concerns on the modification 3 of this invention. It is a block diagram which shows the structure of the electronic musical instrument which concerns on the modification 4 of this invention. It is a block diagram which shows the structure of the musical sound signal synthetic | combination part which concerns on the modification 4 of this invention.

<Embodiment>
[Configuration of electronic musical instrument 1]
FIG. 1 is a block diagram showing a configuration of an electronic musical instrument 1 according to an 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. A synthesizer 100 and a converter 110 that performs processing for converting performance information into a signal input to the musical sound signal synthesizer 100 are realized. The RAM 11d functions as a work area when the CPU 11a processes each data.

  The storage unit 12 is storage means such as a hard disk, for example, 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 a change in the key depression amount, the damper pedal depression amount, and the shift pedal depression amount (in addition, the hammer speed may be included) according to the time progress. Yes. These data may be loaded from an information storage medium DP (for example, a compact disk) or downloaded from a server via a network, and may not necessarily be stored in the storage unit 12.

  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 information output from the keyboard unit 15a may include information indicating the pushing acceleration.

  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. As described above, the keyboard unit 15a and the pedal unit 16 are operated to output the above information (performance information).

  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.

[Configuration of Conversion Unit 110]
Next, the tone signal synthesizing unit 100 and the converting unit 110 realized by the control unit 11 executing the control program will be described with reference to FIGS. Note that some or all of the components in the tone signal synthesizer 100 and the converter 110 described below may be realized by hardware.

FIG. 2 is a diagram for explaining the relationship between the conversion unit 110 and the musical sound signal synthesis unit 100. As shown in FIG. 2A, the conversion unit 110 acquires performance information output from the keyboard unit 15a and the pedal unit 16, and uses it in the musical sound signal synthesis unit 100 based on a conversion table stored in advance. Is converted into a signal to be output. The signal output from the conversion unit 110 in this way is input to the musical sound signal synthesis unit 100. As an input signal in the tone signal synthesizing unit 100, a signal generated according to information KS and KC indicating the key pressing amount output from the keyboard unit 15a (hereinafter referred to as an input signal 1 (e _K (nΔt))). , A signal indicating the hammer speed generated in accordance with the information KV, KC indicating the key pressing speed (may be the pressing acceleration) (hereinafter referred to as input signal 2 (V _H (nΔt))), pedal unit 16 According to the information PS, PC indicating the amount of depression of the damper pedal output from the signal PS, the signal generated according to the PC (hereinafter referred to as input signal 3 (e _P (nΔt))), the information PS, PC representing the amount of depression of the shift pedal 4 signals (hereinafter referred to as input signal 4 (e _S (nΔt))). Each of these four signals is input to the musical sound signal synthesizing unit 100 as a signal on a discrete time axis (t = nΔt; n = 0, 1, 2,...). These four signals may be those obtained by the control unit 11 reading out the musical tone control data stored in the storage unit 12 and converted by the conversion unit 110.

The conversion process in the conversion unit 110 will be described as a representative of the conversion process from the information KS to the input signal 1 (e _K (nΔt)). FIG. 2B is a diagram illustrating an example of a conversion table for the conversion unit 110 to convert the input signal 1 (“e _K ” in the figure) corresponding to the information KS acquired at a certain timing. . In this example, e _K starts to decrease from “1” when the key is pushed a certain amount from the rest position (rest), and reaches “0” at a stage a certain amount before the end position (end). It is decided to. Such a conversion table is provided corresponding to each input signal.

[Configuration of Music Signal Synthesis Unit 100]
FIG. 3 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. 4 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 (soundboards, columns, etc.) that radiate 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.

  3 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 output signal in the tone signal synthesis unit 100 is a tone signal (hereinafter referred to as a 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. .

  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 synthesis unit 100 shown in FIG. 3 has generality.

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 27 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) One end of the string is supported by a point on a bearing which is a part of the main body, and the other end is supported by a point on a piece which is a part of the main body. (The string shall not be constrained from rotating at the support end.)
(Assumption 9) The action / reaction between the string and air is ignored.
(Assumption 10) The shape of the portion in contact with 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 infinitely small, and the height of the cylinder is the other string To the extent that it does not interfere with.
(Assumption 11) When there are a plurality of strings corresponding to one hammer, the central axes of these strings in a static equilibrium state are on the same plane.
(Assumption 12) 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 13) 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 14) The center of gravity of the hammer moves only on one straight line.
(Assumption 15) 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 16) 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 17) 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 18) It is assumed that there is no friction between the hammer tip and the string surface.
(Assumption 19) The action / reaction between the hammer and air is ignored.
(Assumption 20) For a string equipped with a damper, the 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 21) The resistance force-speed relational expression of the damper is given by a linear expression.
(Assumption 22) It is assumed that the amplitude of the main body is very small.
(Assumption 23) It is assumed that the main body can be approximately treated as a proportional viscous damping system.
(Assumption 24) The reaction that the main body receives from the air is ignored.
(Assumption 25) Air is assumed to be homogeneous.
(Assumption 26) It is assumed that the air pressure-volume strain relational expression is given by a linear expression.
(Assumption 27) 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, in the case where there is one string corresponding to one hammer, if Z _B , X _B , Y _B , and θ _H are given, β _k′k is uniquely determined. When there are a plurality of strings corresponding to one hammer, β _k′k 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 signal 1 (e _K (nΔt)) and the input signal 3 (e _P (nΔt)), and outputs the smaller value as e _D (nΔt). This is expressed by the following formula (1).

[Damper model]
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 outputs e _D (nΔt) output from the comparison unit 101 and u _k (x _D , nΔt) (k = 1, 3) output from the string model calculation unit 104 as described later. And using these, f _Dk (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.
The above is the description of the damper model calculation unit 102.

[Hama model]
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.

When a hammer speed V _H (t) is given based on performance information, w _H (t) = W _H , dw _H (t) / dt = under the condition that the tip of the hammer is away from the string surface. The hammer state may be initialized as V _H (t).

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.

[String model]
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. Then, u ₁ (x _H , nΔt) is output to the hammer model calculation unit 103.

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

  When the Newton's law of motion is applied to the assumptions regarding the physical model described above, the equation of motion of the string is written as in the equations (8), (9), and (10).

  However, in the equations (8) and (10), the nonlinear term due to the finite amplitude is omitted because the effect due to this is small. Similarly, in the formula (9), the force applied by the hammer in the chord axis direction is omitted because the effect of the force is small. Equation (8) is the bending vibration of the string corresponding to the movement direction of the hammer center of gravity, Equation (10) is the bending vibration of the string corresponding to the direction orthogonal to the movement direction of the hammer gravity center, and Equation (9) is Corresponds to the vertical vibration of the strings.

  Further, the string boundary condition is written as in the equations (11) and (12).

  Now, “string displacement” is expressed by the sum of “relative displacement with respect to a straight line connecting two support ends” and “displacement of a straight line connecting two support ends”, and further, “two support ends. "Relative displacement with respect to a straight line connecting the two" is expressed by "a finite Fourier sine series with an arbitrary time function as a coefficient". That is, “string displacement” is expressed by equation (13). Here, the sine function included in Expression (13) is nothing but the natural vibration mode of the string when the displacement at the string support end of the string center axis is constrained. Further, “a linear displacement connecting two support ends” means “a static displacement of a string due to a displacement of a string support end”.

At this time, the expression (13) satisfies the boundary condition expressions (11) and (12) for an arbitrary time t.
After substituting equation (13) into the partial differential equation (equations (8), (9), (10)), 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 (14), (15), (16)) are derived.

  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 (25) and (26).

  Further, by substituting equation (13) into equations (25) and (26), equations (28) and (29) are derived. However, the terms related to nonlinear terms and rotational inertia were omitted.

  Further, the displacement of the string hitting point and the sound stopping point is written as in the equations (30) and (31) from the equation (13).

  The above is the description of the string model calculation unit 104.

[Body model]
The body model calculation unit 105 obtains 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. 106, and u _Bk (nΔt) (k = 1, 2, 3) is output to the string model calculation unit 104.

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

  Based on the assumptions regarding the physical model described above, the equation of motion of the main body can be described as a second-order ordinary differential equation (equation (32)) for each mode as follows.

  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 1: Japan). The Society of Mechanical Engineers (ed.). Advanced Composite Materials, pp.68-70. Gihodo Publishing, 1990.).

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 Document 2), 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 2: Akio Nagamatsu. Modal analysis. Baifukan, 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, mode damping ratio, and natural vibration mode included in Equation (32) are calculated by performing real eigenvalue analysis using a commercial finite element method software on the body of an arbitrary three-dimensional shape. 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.
The displacement of the string support end can be calculated by using the following formula (34).

  The above is the description of the main body model calculation unit 105.

[Following equations of motion]
Next, an example of solving the equation of motion in each of the above models will be described. In the following description, the equation of motion of the hammer (the above equation (3)), the equation of motion for each mode of the string (the above equations (14), (15), (16)), and the equation of motion for each mode of the main body (the above equation ( 32)) will be collectively referred to as “hammer-string-body motion equation”. By substituting the above equations (2), (4), (5), (6), (28), (29), (30), (31), (34) into these equations of motion, variables _{f Dk} that represents the interaction between the structure _{^{[iD] (t), f}} H [iW] (t), w e (t), f Bk [iB] (t), u 1 (x H, t), If u _k (x _D ^[iD] , t) and u _Bk ^[iB] (t) are deleted, the “hammer-string-body equation of motion” gives the displacement w _H (t) of the center of gravity of the hammer, each string Displacement A _k ^[mk] (t) (m _k = 1, 2,..., M _k ; k = 1, 2, 3) on the mode coordinates of the natural vibration mode, each natural vibration mode mode of the main body This is a simultaneous nonlinear ordinary differential equation related to the displacement A _C ^[m] (t) (m = 1, 2,..., M) on the coordinates. Now, by setting a state before performance, that is, a stationary state as an initial condition, the problem dealt with here can be considered as a so-called “initial value problem of simultaneous nonlinear ordinary differential equations”. The “initial value problem of simultaneous nonlinear ordinary differential equations” can be converted into a problem in which simultaneous nonlinear algebraic equations are sequentially solved on a discrete time axis by using some numerical integration method (reference document 3).
(Reference 3: The Japan Society of Mechanical Engineers. Fundamentals and applications of numerical integration. Corona, 2003.)
Below are some examples of solutions.

[How to solve the whole]
First, we show how to solve the equations of motion for the hammer model, string model, and body model all together. Applying the Newmark β method to the above “hammer-string-body motion equation” (simultaneous nonlinear ordinary differential equation), “hammer center of gravity acceleration or acceleration increment”, “mode coordinates of each natural vibration mode of the string Simultaneous accelerations and acceleration increments "and" acceleration or acceleration increments on the mode coordinates of each natural vibration mode of the main body "can be derived and simultaneous nonlinear algebraic equations can be derived. Here, the reason for writing “acceleration or acceleration increment” is that the numerical integration method known as the Newmark β method has two algorithms: an algorithm that makes acceleration an unknown quantity, and an algorithm that makes acceleration increment an unknown quantity. This is due to the existence of streets.

  The Newton method is applied to the simultaneous nonlinear algebraic equations, or the linear linear algebraic equations are derived by the interval linearization method (Reference 3), and then the direct method (for example, LU decomposition method) or the iterative method (for example, conjugate method). By applying the gradient method, the calculation unit 120 described below can sequentially determine the unknown amount on the discrete time axis. Thus, the structure in the case of calculating by the method of solving the whole collectively is demonstrated using FIG.

FIG. 5 is a block diagram showing a configuration of the tone signal synthesis unit 100 having the calculation unit 120 according to the embodiment of the present invention. The musical sound signal synthesis unit 100 that performs calculations by a method that solves the whole is provided with a comparison unit 101, a calculation unit 120, and an air model calculation unit 106Z.
The calculation unit 120 performs a calculation using the “hammer-string-main body equation of motion” that summarizes the calculations in the damper model calculation unit 102, the hammer model calculation unit 103, the string model calculation unit 104, and the main body model calculation unit 105. Do. The arithmetic unit 120 obtains e _D (nΔt) from the comparison unit 101, obtains the input signal 2 (V _H (nΔt)) and the input signal 4 (e _S (nΔt)). The above-mentioned unknown amount is sequentially calculated and determined by an operation using the “hammer-string-body motion equation”. Here, information (d / dt (A _C (nΔt))) indicating “the speed of each natural vibration mode of the main body on the mode coordinates” among the unknown quantities is output to the air model calculation unit 106Z.

  Here, the “velocity on the mode coordinates of each natural vibration mode of the main body” is “the nth order derivative (n = 1, 2,... Regarding the time of displacement on the mode coordinates of each natural vibration mode of the main body”.・) 」. The velocity can be easily calculated as a numerical derivative of the displacement when the displacement is known and as a numerical integral of the acceleration when the acceleration is known.

[Method to solve for each partial structure]
Subsequently, a method of solving the equation of motion of the hammer model, the string model, and the body model for each partial structure (hereinafter, the hammer model calculation unit 103, the string model calculation unit 104, and the body model calculation unit 105 are collectively referred to as a partial structure). Indicates. In this method, the variables f _H ^[iW] (t), f _Bk ^[iB] (t), u ₁ representing the interaction between the substructures eliminated in the explanation of the “hammer-string-body motion equation” above. (X _H , t), u _k (x _D ^[iD] , t), u _Bk ^[iB] (t) values are calculated explicitly, and these values are transferred between the partial structures. The calculation is advanced every time.

  When this solution is used, when solving the equation of motion of the hammer (the above equation (3)), an unknown quantity relating to the string and the body is included, and the kinematic mode equation of motion (the above equations (14), (15), ( When solving 16)), unknown quantities related to the main body are included, but the unknown quantities related to the strings and the main body are extrapolated from past values, etc., and are calculated repeatedly. The calculation can be carried out stably. The following are three examples based on the difference in the numerical integration method.

A “method for deriving a difference equation” will be described as a first example.
The central difference method is applied to the equation of motion of the hammer (the above equation (3)), the equation of motion for each mode of the string (the above equations (14), (15), (16)) and the equation of motion of each mode of the main body. A series of difference equations is derived by applying the bilinear sz transformation method to (Equation (32) above). Each difference equation can be solved by a general second-order IIR filter operation. In this method, “displacement of the center of gravity of the hammer”, “displacement of each natural vibration mode of the string on the mode coordinates”, and “displacement of the natural vibration mode of the main body on the mode coordinates” are defined as unknown quantities. Is sequentially determined on the discrete time axis.

As a second example, the “Galarkin method” will be described.
For the equation of motion of the hammer (the above equation (3)), the equation of motion for each mode of the string (the above equations (14), (15), (16)) and the equation of motion for the mode of the body (the above equation (32)) By applying the Galerkin method (Reference 4) using a cubic function related to time as a test function, the acceleration and jerk on the mode coordinates of each natural vibration mode of the string An algorithm is obtained in which “acceleration” and “acceleration and jerk on mode coordinates of each natural vibration mode of the main body” are unknown quantities and their values are sequentially determined on a discrete time axis. Here, when the Galerkin method using a quaternary function as a test function instead of a cubic function with respect to time is used, an algorithm having acceleration, jerk, and jerk as unknown quantities is obtained.
(Reference 4: Yukio Kagawa. Vibration / acoustic engineering by finite element method / Basics and applications. Baifukan, 1981.)

As a third example, the “Newmark β method” will be described.
For the equation of motion of the hammer (the above equation (3)), the equation of motion for each mode of the string (the above equations (14), (15), (16)) and the equation of motion for the mode of the main body (the above equation (32)) By applying the Newmark β method, “acceleration or acceleration increment of hammer center of gravity”, “acceleration or acceleration increment on mode coordinates of each natural vibration mode of string”, and “natural vibration mode of main body” An algorithm that sequentially determines the values on the discrete time axis with the acceleration or acceleration increment on the mode coordinates as an unknown quantity can be obtained.

[Intermediate method between solving the whole and solving each partial structure]
An intermediate method between the above-described “method of solving the whole as a whole” and “method of solving each partial structure” can also be used. For example, the hammer model and the string model may be collected together and the body model may be solved separately, or the hammer model may be solved first and the string model and the body model may be solved together.

  In addition, as described above, “displacement on the mode coordinate of each natural vibration mode of the main body”, “displacement on the mode coordinate of each natural vibration mode of the main body”, and “displacement on the mode coordinate of each natural vibration mode of the main body” which are unknown quantities For each of these displacements, there may be acceleration, jerk, etc. depending on the solution. Furthermore, considering that the velocity can be easily calculated as a numerical derivative of displacement or as a numerical integral of acceleration, “displacement of hammer center of gravity”, “displacement of each natural vibration mode of the string on the mode coordinates”, and Each displacement of “displacement on the mode coordinates of each natural vibration mode of the main body” may be an n-order derivative (n = 1, 2,...) Related to the time of displacement. The same can be said for other displacements. For example, the displacement of the string support end may be an nth derivative (n = 1, 2,...) Related to the time of the displacement.

[Air model (acoustic radiation calculation)]
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.

  The unsteady sound pressure at an arbitrary observation point in the air radiated from the main body having an arbitrary three-dimensional shape can be obtained by the method shown in the following equation: Convolution integration of "impulse response function between sound pressures at the observation point in the middle" and "velocity on the mode coordinates of each natural vibration mode of the main body" is performed for each natural vibration mode of the main body, and the sum of them is calculated. It can be calculated by the method of

H ^{[iP] [iG]} (ω) included in the equation (37), 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. Equation (36) can be calculated by a general IFFT (Inverse Fast Fourier Transform) operation, and the integral included in Equation (35) can be calculated by a general FIR (Finite Impulse Response) filter method.

Thus, Ac ^[m] (nΔt) (m = 1, 2,..., M) or Ac ^[m] (nΔt) (m = 1, 2,...) Output from the body model calculation unit 105. , M) using the derivative with respect to time, the output signal from the air model, that is, the sound pressure P ^[iP] (t = nΔt; n = 0, 1, 2,...) nΔt) (i _P = 1, 2,..., I _P ) can be calculated sequentially and output as a musical tone signal.

Here, by using a technique called fast convolution in which the convolution operation in the equation (35) is performed in the frequency domain instead of the time domain, the speed can be further dramatically increased. At this time, the IFFT calculation included in the high-speed convolution may be performed after taking the sum of the frequency domain convolution results for each main body natural vibration mode, and does not need to be performed for each natural vibration mode of the main body.
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 (15). However, in order to make the explanation easy to understand, in the equation (15), the description will be made according to the equation (38) in which the displacement of the string support end, the displacement in the string y direction, and the internal viscosity damping coefficient of the string are omitted. .

Equation (38) is a motion equation of the longitudinal vibration second i _2-order natural vibration of the strings, by regarding the right and periodic external force, it can be thought of as the equation of motion of the single-degree-of-freedom viscous damping forced vibration 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 (39).

  At this time, the inside of the right side {} of Expression (38) is derived as Expression (40).

Now, with i ₂ fixed, paying attention to the sequence created by the term cos2π (f ₁ ^[m1] + f ₁ ^{[m1 + i2]} ) t included in the equation (40), “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 generated by the term cos2π (f ₁ ^[m1] + f ₁ ^[i2-m1] ) t included in the equation (40) also contributes to the formation of the subsequence, provided that the degree of contribution is It can be understood that it is smaller than the term.

The formula obtained by substituting the formula (40) into the formula (38) 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 (ref 5), is intended to provide a mathematical description.
(Reference 5: 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, formula (4 0) term included in ^{_{cos2π (f 1 [m1] -f}} 1 [m1 + i2]) t and ^{_{cos2π (f 1 [m1] -f}} 1 [i2-m1]) t is bending vibration This indicates the presence of a partial sound having a frequency slightly higher than the natural frequency.

  Now, when the material constant of the string is fixed, the longitudinal vibration natural frequency of the string depends only on the string length from the equation (18). 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.

  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.

<Modification>
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, the string model calculation unit 104 outputs f _Dk (nΔt) (k = 1, 3) output from the damper model calculation unit 102 and the hammer model calculation unit 103 as forces acting on the strings. Although f _H (nΔt) and u _Bk (nΔt) (k = 1, 2, 3) output from the main body model calculation unit 105 were acquired, f _Dk (nΔt) (k = 1, 3). As for one or both of f and f _H (nΔt), those calculated by other calculation methods may be acquired. In the embodiment, the air model calculation unit 106 calculates the musical sound signal P (nΔt) by calculation using the air model based on A _C (nΔt) output from the main body model calculation unit 105. The tone signal P (nΔt) may be calculated by other calculation methods.
When both f _Dk (nΔt) (k = 1, 3) and f _H (nΔt) are calculated by other calculation methods, and the musical sound signal P (nΔt) is calculated by a calculation method using other than the air model The configuration of will be described with reference to FIG.

  FIG. 6 is a block diagram showing a configuration of a musical tone signal synthesis unit 100A according to the first modification of the present invention. The tone signal synthesis unit 100A includes a force calculation unit 107 instead of the comparison unit 101, the damper model calculation unit 102, and the hammer model calculation unit 103 in the embodiment, and calculates a tone signal instead of the air model calculation unit 106 in the embodiment. Part 108.

The force calculation unit 107 corresponds to f _Dk (nΔt) (k = 1, 3) and f _H (nΔt) based on each input signal output from the conversion unit 110 and input to the musical sound signal synthesis unit 100A. Information is calculated and output to the string model calculation unit 104A.
In calculating the information corresponding to f _H (nΔt), the force calculation unit 107 does not acquire u ₁ (x _H , nΔt) from the string model calculation unit 104A, and determines u ₁ (x _H , nΔt) is used. The force calculation unit 107 may calculate u ₁ (x _H , nΔt) by a predetermined calculation formula based on each input signal.
The force calculation unit 107 calculates the information corresponding to f _Dk (nΔt) (k = 1, 3) from the string model calculation unit 104A by u _k (x _D , nΔt) (k = 1, 3). ) Is not used, and a predetermined u _k (x _D , nΔt) (k = 1, 3) is used. The force calculation unit 107 may calculate u _k (x _D , nΔt) (k = 1, 3) based on a predetermined calculation formula based on each input signal.

Although the force calculation unit 107 has been described as being replaced with the comparison unit 101, the damper model calculation unit 102, and the hammer model calculation unit 103 in the embodiment, the hammer model calculation unit 103 has the same configuration as the embodiment, and the comparison unit 101 and the damper model calculation unit 102 may be replaced. Conversely, the force calculation unit 107 may be configured such that the comparison unit 101 and the damper model calculation unit 102 have the same configuration as that of the embodiment, and the hammer model calculation unit 103 is replaced. That is, of the results in the string model calculation, u ₁ (x _H , nΔt) and u _k (x _D , nΔt) (k = 1, 3) used for calculating the force acting on the string in the embodiment. The force acting on the string may be calculated without using one or both.

The tone signal calculation unit 108 calculates a tone signal P (nΔt) based on A _C (nΔt) output from the main body model calculation unit 105. The musical tone signal calculation unit 108 only needs to calculate the musical tone signal P (nΔt) using a predetermined calculation formula using A _C (nΔt). At this time, the musical sound signal P (nΔt) may not indicate the unsteady sound pressure at an arbitrary observation point in the air, and may indicate vibration at an arbitrary position of the main body. The musical tone signal calculation unit 108 calculates the musical tone signal P (nΔt) based on u _Bk (nΔt) (k = 1, 2, 3) output from the main body model calculation unit 106 to the string model calculation unit 104A. May be.

[Modification 2]
In the embodiment described above, an electronic musical instrument having a configuration in which the shift pedal 16b is removed may be used. The configuration in this case will be described with reference to FIGS.

  FIG. 7 is a block diagram showing a configuration of an electronic musical instrument 1B according to the second modification 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 embodiment, the description thereof is omitted.

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

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

  Unlike the control unit 11 in the embodiment, the control unit 11B implements a tone signal synthesis unit 100B that does not use the hammer model calculation unit 103 among the tone signal synthesis units 100 that are implemented by executing a control program.

FIG. 8 is a block diagram showing the configuration of the tone signal synthesizer 100B. As shown in FIG. 8, the musical tone signal synthesis unit 100B does not have the hammer model calculation unit 103. Chord model calculation unit 104B-1,104B-2 does not acquire the _f H _(nΔt) outputted from the hammer model calculation unit 103 acquires the stored _f H _(nΔt) in the storage unit 12B. Other configurations in the musical tone signal synthesis unit 100B have the same functions as those of the musical tone signal synthesis unit 100 in the embodiment, and thus description thereof is omitted.

Note that even without the structure of the hammer model calculation unit 103 is not present, the tone signal synthesis section 100 in the embodiment, the fixed e S _(nΔt) = 1 _(shift pedal fixed to the rest position) to a state of being Thus, a case where there is no shift pedal may be realized.

[Modification 3]
In the above-described embodiment, an electronic musical instrument having a configuration in which the damper pedal 16a is removed may be used. The configuration in this case will be described with reference to FIGS.

  FIG. 9 is a block diagram showing a configuration of an electronic musical instrument 1C according to Modification 3 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 those of the user operation unit 13, the sound emission unit 17, and the bus 18 in the electronic musical instrument 1 according to the embodiment, description thereof is omitted.

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

Unlike the storage unit 12 in the embodiment, the storage unit 12C stores the resistance force f _Dk (nΔt) of the damper. This value indicates a value in a state where the damper pedal 16a in the embodiment is not depressed (rest position).

  Unlike the control unit 11 in the embodiment, the control unit 11C does not use the comparison unit 101 and the damper model calculation units 102-1 and 102-2 in the tone signal synthesis unit 100 realized by executing the control program. The musical tone signal synthesis unit 100C is realized.

FIG. 10 is a block diagram showing the configuration of the tone signal synthesizer 100C. As shown in FIG. 10, the musical tone signal synthesis unit 100C does not include the comparison unit 101 and the damper model calculation units 102-1 and 102-2. Chord model calculation unit 104C-1,104C-2 does not acquire the _f Dk output from the damper model calculation section 102 (n.DELTA.t), acquires the stored _f Dk _(nΔt) in the storage unit 12C. Other configurations in the tone signal synthesis unit 100C have the same functions as those in the tone signal synthesis unit 100 in the embodiment, and thus the description thereof is omitted.

  Note that even if the comparison unit 101 and the damper model calculation units 102-1 and 102-2 do not exist, the musical sound signal synthesis unit 100 in the embodiment is fixed to ep (nΔt) = 1 (the damper pedal is at the rest position). A case where there is no damper pedal may be realized by setting the state to be fixed.

[Modification 4]
In the embodiment described above, an electronic musical instrument having a configuration in which the damper pedal 16a and the shift pedal 16b are removed may be used. The configuration in this case will be described with reference to FIGS.

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

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

Unlike the storage unit 12 in the embodiment, the storage unit 12D stores the resistance force f _Dk (nΔt) of the damper and the force f _H (nΔt) 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 embodiment.

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

FIG. 12 is a block diagram showing the configuration of the tone signal synthesis unit 100D. As shown in FIG. 12, the tone signal synthesis unit 100D 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 104D-1 and 104D-2 do not acquire f _Dk (nΔt) output from the damper model calculation unit 102 and f _H (nΔt) output from the hammer model calculation unit 103, but store them. F _Dk (nΔt) and f _H (nΔt) stored in the unit 12D are acquired. Other configurations in the musical tone signal synthesis unit 100D have the same functions as those in the musical tone signal synthesis unit 100 in the embodiment, and thus description thereof is omitted.

Even if the comparison unit 101, the damper model calculation units 102-1 and 102-2, and the hammer model calculation unit 103 do not exist, e _S (nΔt) = 1 in the musical sound signal synthesis unit 100 in the embodiment. A case where the shift pedal and the damper pedal are not present may be realized by fixing (the shift pedal is fixed at the rest position) and fixed at ep (nΔt) = 1 (the damper pedal is fixed at the rest position). .

[Modification 5]
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 “speed data on the time axis for each natural vibration mode of the instrument main body” is calculated first, and the data and the “natural vibration of the main body” are calculated. It can also be said that the convolution operation with “impulse response or frequency response data between the mode and 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 6]
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 7]
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. Further, the electronic musical instrument 1 may download the control program via a network.

DESCRIPTION OF SYMBOLS 1,1B, 1C, 1D ... Electronic musical instrument, 11, 11B, 11C, 11D ... Control part, 11a ... CPU, 11b ... DSP, 11c ... ROM, 11d ... RAM, 11e ... Signal interface, 11f ... Internal bus, 12, 12B, 12C, 12D ... storage unit, 13, 13B, 13C, 13D ... user operation unit, 13a ... operation panel, 13b ... mouse, 13c ... operation switch, 13d ... keyboard, 14 ... display unit, 15, 15B, 15C, 15D ... 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, 16Bc , 16Cc ... pedal position sensor, 17, 17B, 17C, 17D ... sound emission part, 17a ... digital-analog converter, 17 ... Speaker, 18, 18B, 18C, 18D ... 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 body, 100, 100A, 100B, 100C, 100D ... tone signal synthesis unit, 101 ... comparison unit, 102-1, 102-2 ... damper model calculation unit, 103 ... hammer model calculation unit, 104 -1,104A-1,104B-1,104C-1,104D-1,104-2,104A-2,104B-2,104C-2,104D-2 ... string model calculation unit, 105 ... main body model calculation unit 106 ... Air model calculation unit 107 ... Force calculation unit 108 ... Music signal calculation unit 110 ... Conversion unit 120 ... Calculation unit

Claims (8)

A musical tone generated from a three-dimensional musical instrument having a vibrating string, and a main body that supports the string by two string support ends, and the vibration of the string is transmitted through at least one end of the string support end. A musical sound signal synthesis method for generating a signal in accordance with input performance information,
The control unit obtains first information representing the force acting on the string calculated according to the performance information, and the first information, displacement or displacement of each natural vibration mode of the string on the mode coordinates Second information representing the n-th order derivative (n = 1, 2,...) Relating to the time and the n-th derivative relating to the displacement or time of displacement on the mode coordinates of each natural vibration mode of the main body ( a calculation process for calculating the third information from the first information by an operation using simultaneous ordinary differential equations representing the relationship of the third information representing n = 1, 2,.
A musical tone signal synthesis method , wherein the control unit comprises a musical tone signal calculation step of calculating the musical tone signal based on the calculated third information. The music signal synthesizing method according to claim 1, wherein the string is plural. It is emitted from a three-dimensional musical instrument having a string to be struck by a hammer and a main body that supports the string by two string support ends and that transmits vibration of the string through at least one end of the string support end. A musical tone signal synthesis method for generating musical tone signals according to input performance information,
Control unit, the first information indicating the initialized by displacement or displacement time for n order derivative of the hammer (n = 1, 2, · · ·) in response to said performance information, each unique said chord Second information representing the n-th derivative (n = 1, 2,...) Relating to the displacement or time of displacement on the mode coordinates of the vibration mode, and the mode coordinates of each natural vibration mode of the main body The third performance is calculated from the performance information by an operation using simultaneous ordinary differential equations representing the relationship of the third information representing the n-th derivative (n = 1, 2,...) Relating to the displacement or the displacement time. A calculation process for calculating information;
A musical tone signal synthesis method , wherein the control unit comprises a musical tone signal calculation step of calculating the musical tone signal based on the calculated third information. The music signal synthesis method according to claim 3, wherein the hammer and the strings are plural. The computer emits a three-dimensional musical instrument having a vibrating string, and a main body that supports the string by two string support ends, and the vibration of the string is transmitted through at least one end of the string support end. A program for generating a musical tone signal according to input performance information,
The computer,
First information representing a force exerted on the string calculated according to the performance information is acquired, and the first information, n on the mode coordinates of each natural vibration mode of the string or n on the displacement time Second information representing the second derivative (n = 1, 2,...) And the nth derivative (n = 1, n) related to the displacement or displacement time on the mode coordinates of each natural vibration mode of the main body. 2,...)) Calculating means for calculating the third information from the first information by calculation using a simultaneous ordinary differential equation representing the relationship of the third information.
A program for functioning as a tone signal calculation means for calculating the tone signal based on the calculated third information. A three-dimensional musical instrument having a string hit by a hammer and a main body that supports the string with two string support ends and transmits vibration of the string through at least one end of the string support end. A program for generating a musical tone signal of a sound emitted from the sound according to the performance information input,
The computer,
First information representing the n-th derivative (n = 1, 2,...) Relating to the displacement of the hammer or the time of displacement that is initialized according to the performance information, the mode of each natural vibration mode of the string Second information representing n-order derivatives (n = 1, 2,...) Relating to displacement or displacement time on the coordinates, and displacement or displacement of each natural vibration mode of the main body on the mode coordinates The third information is calculated from the performance information by an operation using a simultaneous ordinary differential equation representing the relationship of the third information representing the n-th derivative (n = 1, 2,...) Relating to time. Computing means;
A program for functioning as a tone signal calculation means for calculating the tone signal based on the calculated third information. A musical tone generated from a three-dimensional musical instrument having a vibrating string, and a main body that supports the string by two string support ends, and the vibration of the string is transmitted through at least one end of the string support end. A musical sound signal generating device for generating a signal according to input performance information,
First information representing a force exerted on the string calculated according to the performance information is acquired, and the first information, n on the mode coordinates of each natural vibration mode of the string or n on the displacement time Second information representing the second derivative (n = 1, 2,...) And the nth derivative (n = 1, n) related to the displacement or displacement time on the mode coordinates of each natural vibration mode of the main body. 2,...)) Calculating means for calculating the third information from the first information by calculation using a simultaneous ordinary differential equation representing the relationship of the third information.
A musical sound signal generating device comprising: a musical sound signal calculating means for calculating the musical sound signal based on the calculated third information. It is emitted from a three-dimensional musical instrument having a string to be struck by a hammer and a main body that supports the string by two string support ends and that transmits vibration of the string through at least one end of the string support end. A musical tone signal generating device for generating a musical tone signal according to input performance information,
First information representing the n-th derivative (n = 1, 2,...) Relating to the displacement of the hammer or the time of displacement that is initialized according to the performance information, the mode of each natural vibration mode of the string Second information representing n-order derivatives (n = 1, 2,...) Relating to displacement or displacement time on the coordinates, and displacement or displacement of each natural vibration mode of the main body on the mode coordinates The third information is calculated from the performance information by an operation using a simultaneous ordinary differential equation representing the relationship of the third information representing the n-th derivative (n = 1, 2,...) Relating to time. Computing means;
A musical sound signal generating device comprising: a musical sound signal calculating means for calculating the musical sound signal based on the calculated third information.

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