Physical modelling synthesis

Modelling attempts to replicate laws of physics that govern sound production, and will typically have several parameters, some of which are constants that describe the physical materials and dimensions of the instrument, while others are time-dependent functions describing the player's interaction with the instrument, such as plucking a string, or covering toneholes.

For example, to model the sound of a drum, there would be a mathematical model of how striking the drumhead injects energy into a two-dimensional membrane. Incorporating this, a larger model would simulate the properties of the membrane (mass density, stiffness, etc.), its coupling with the resonance of the cylindrical body of the drum, and the conditions at its boundaries (a rigid termination to the drum's body), describing its movement over time and thus its generation of sound.

Similar stages to be modelled can be found in instruments such as a violin, though the energy excitation in this case is provided by the slip-stick behavior of the bow against the string, the width of the bow, the resonance and damping behavior of the strings, the transfer of string vibrations through the bridge, and finally, the resonance of the soundboard in response to those vibrations.

In addition, the same concept has been applied to simulate voice and speech sounds.[1] In this case, the synthesizer includes mathematical models of the vocal fold oscillation and associated laryngeal airflow, and the consequent acoustic wave propagation along the vocal tract. Further, it may also contain an articulatory model to control the vocal tract shape in terms of the position of the lips, tongue and other organs.

Although physical modelling was not a new concept in acoustics and synthesis, having been implemented using finite difference approximations of the wave equation by Hiller and Ruiz in 1971, it was not until the development of the Karplus-Strong algorithm, the subsequent refinement and generalization of the algorithm into the extremely efficient digital waveguide synthesis by Julius O. Smith III and others, and the increase in DSP power in the late 1980s[2] that commercial implementations became feasible.

Yamaha contracted with Stanford University in 1989[3] to jointly develop digital waveguide synthesis; subsequently, most patents related to the technology are owned by Stanford or Yamaha.

The first commercially available physical modelling synthesizer made using waveguide synthesis was the Yamaha VL1 in 1994.[4][5]

While the efficiency of digital waveguide synthesis made physical modelling feasible on common DSP hardware and native processors, the convincing emulation of physical instruments often requires the introduction of non-linear elements, scattering junctions, etc. In these cases, digital waveguides are often combined with FDTD,[6] finite element or wave digital filter methods, increasing the computational demands of the model.[7]

Examples of physical modelling synthesis:

• Karplus-Strong string synthesis
• Digital waveguide synthesis
• Mass-Interaction physical networks
• Formant synthesis
• Articulatory synthesis

• Korg OASYS and Korg Kronos – STR-1 Plucked string
• Korg OASYS PCI
• Korg Prophecy
• Korg SOLO-TRI (an expansion board for the Trinity with the synth engine of the Prophecy)
• Korg Z1
• Korg MOSS-TRI (a expansion board for the Trinity with the synth engine of the Z1) and EXB-MOSS (a multi timbral expansion board for the Triton and the KARMA workstation with the synth engine of the Z1)
• Yamaha VL1, VP1 and VL7
• Yamaha VL70m, PLG-100VL and 150VL (VL70m in the form of a plug-in card that can be installed into any of several Yamaha keyboards, tone modules, and the SW1000XG high-end PC midi sound card)
• Yamaha EX5, EX5R
• Technics WSA1/WSA1R
• Clavia Nord Modular G2
• Alesis Fusion
• Roland V-Piano
• Pianoid
• Physis Unico
• Physis Piano (made in Italy, with a full touch controlled user interface)
• Hartmann Neuron and Neuron VS

While not purely a hardware synth, the Yamaha DS-XG sound cards included hardware-assisted software VL physical modelling along with the Yamaha XG, wave audio, and 3D gaming sound capabilities of the chipset. But as they were not fully compatible with the AC-97 and later AC-98 standards, these chipsets have not been manufactured in nearly a decade.

The WSA1 (and its rackmounted counterpart WSA1R) was Technics' first and only try at high-end synthesizers. It featured 64 voices of polyphony with a combination of sample playback (for initial transients) and DSP acoustic modelling. Launched in 1995 with an MSRP of $5,000 (USD), the WSA1 was not a commercial success; only about 600 keyboards and 300 rack models were ever made, and most were sold at highly discounted prices.

Various Roland synthesizer models (V-Synth, V-Combo, XV-5080, Fantom, etc.) use COSM ("Composite Object Sound Modeling") physical modeling techniques to replicate guitars, brass and other instruments. COSM has been superseded by "SuperNatural", also based on physical modeling techniques. Introduced first in 2008 as part of the ARX expansion boards for Fantom hardware synthesizers, "SuperNatural" modeling is used in Roland's V-Drums (TD-30, TD-15, TD-11), V-Accordions (FR-7, FR-8) and various synth models (Jupiter 80, Integra 7, FA-08, JD-Xi, etc.) Later this has been expanded to ACB ("Analogue Circuit Behaviour"), using similar physical modeling techniques as before, which were incorporated into Roland's latest line of AIRA hardware synthesizer products (TB-3, System-1, System-1m, System-8), as well as their 'Boutique' line of hardware modules (JP08, JX03, JU06). While the Roland ESC2 chip inside the TD-30 and Integra-7 sound modules were marketed as "SuperNatural" modelling, the same ESC2 chip inside latest Roland "AIRA" and Boutique Products (System-1, System-1m, System-8, SH-01A, D-05, etc) was marketed as "ACB" or DCB (“Digital Circuit Behaviour”, in case of the D-05) modelling technology.

• SWAM-S Bowed Strings by Audio Modeling (based on the Digital Waveguide Synthesis and on the SWAM technology)
• Pianoteq by Modartt (Various Pianos based on physical modelling synthesis)
• MODO by IK Multimedia (Electric basses based on physical modelling synthesis)
• Arché by Expressive E (Bowed string instruments based on physical modelling synthesis)
• Iron Axe by Xhun Audio (Electric guitar based on physical modelling synthesis)
• Sculpture by Apple, built into their Logic Pro X and Mainstage programs (wood, nylon, steel, and glass fundamental models, capable of a huge variety of sounds)
• Madrona Labs Kaivo (Physical Modeling and granular synthesis)

• Hiller, L.; Ruiz, P. (1971). "Synthesizing Musical Sounds by Solving the Wave Equation for Vibrating Objects". Journal of the Audio Engineering Society.

• Karplus, K.; Strong, A. (1983). "Digital synthesis of plucked string and drum timbres". Computer Music Journal. Computer Music Journal, Vol. 7, No. 2. 7 (2): 43–55. doi:10.2307/3680062. JSTOR 3680062.

• Julius O. Smith III (December 2010). Physical Audio Signal Processing.

• Cadoz, C.; Luciani A; Florens JL (1993). "CORDIS-ANIMA : a Modeling and Simulation System for Sound and Image Synthesis: The General Formalism". Computer Music Journal. Computer Music Journal, MIT Press 1993, Vol. 17, No. 1. 17/1 (1).

• "The next generation, part 1". Future Music. No. 32. Future Publishing. June 1995. p. 80. ISSN 0967-0378. OCLC 1032779031.

• Julius. O Smith III's A Basic Introduction to Digital Waveguide Synthesis
• Music synthesis approaches sound quality of real instruments — Stanford University's 1994 news release