Additive synthesis

Additive synthesis is a sound synthesis technique that creates timbre by adding sine waves together.

The timbre of musical instruments can be considered in the light of Fourier theory to consist of multiple harmonic or inharmonic partials or overtones. Each partial is a sine wave of different frequency and amplitude that swells and decays over time.

Additive synthesis generates its sound by adding the output of multiple sine wave generators. It may also be implemented using pre-computed wavetables or inverse Fast Fourier transforms.

Definitions

See also: Fourier series and Fourier analysis

Harmonic additive synthesis is closely related to the concept of a Fourier series which is a way of expressing a periodic function as a sum of sinusoidal functions with frequencies equal to integer multiples of a common fundamental frequency. These sinusoids are called harmonics, sometimes overtones, and most generally, partialss. In general, a Fourier series contains an infinite number of sinusoidal components, with no upper limit to the frequency of the sinusoidal functions and includes a DC component (one with frequency of 0 Hz). Frequencies outside of the human audible range can be omitted in additive synthesis. As a result only a finite number of sinusoidal terms with frequencies that lie within the audible range are modeled in additive synthesis.

A waveform or function is said to be periodic if

$y(t)=y(t+T)\$

for all $t\$ and for some period $T\$.

The Fourier series of a periodic function is mathematically expressed as:

${\begin{aligned}y(t)&={\frac {a_{0}}{2}}+\sum _{k=1}^{\infty }\left[a_{k}\cos(2\pi kf_{0}t)-b_{k}\sin(2\pi kf_{0}t)\right]\\&={\frac {a_{0}}{2}}+\sum _{k=1}^{\infty }r_{k}\cos \left(2\pi kf_{0}t+\phi _{k}\right)\\\end{aligned}}$

where

$f_{0}=1/T\,$ is the fundamental frequency of the waveform and equal the reciprocal of the period,

$k\,$ is the index over the summed harmonics, ranging from $1\,$ to $\infty \,$

$a_{k}=r_{k}\cos(\phi _{k})=2f_{0}\int _{0}^{T}y(t)\cos(2\pi f_{0}t)\,dt,\quad k\geq 0\,$

$b_{k}=r_{k}\sin(\phi _{k})=-2f_{0}\int _{0}^{T}y(t)\sin(2\pi f_{0}t)\,dt,\quad k\geq 1\,$

$r_{k}={\sqrt {a_{k}^{2}+b_{k}^{2}}}\,$ is the amplitude of the $k\,$th harmonic,

$\phi _{k}=\operatorname {atan2} (b_{k},a_{k})\,$ is the phase offset of the $k\,$th harmonic. atan2(&nbsp) is the four-quadrant arctangent function,

Being inaudible, the DC component, ${\frac {a_{0}}{2}}$, and all components with frequencies higher than some finite limit, $Kf_{0}$, shall be omitted in the expressions of additive synthesis.

Harmonic form

The simplest harmonic additive synthesis can be mathematically expressed as:

$y(t)=\sum _{k=1}^{K}r_{k}\cos \left(2\pi kf_{0}t+\phi _{k}\right)$, 1

where $y(t)\,$ is the synthesis output, $r_{k}\,$, $kf_{0}\,$, and $\phi _{k}\,$ are the amplitude, frequency, and the phase offset of the $k\,$th harmonic partial of a total of $K\,$ harmonic partials, and $f_{0}\,$ is the fundamental frequency of the waveform and the frequency of the musical note.

Time-dependent amplitudes

More generally, the amplitude of each harmonic can be prescribed as a function of time, $r_{k}(t)\,$, in which case the synthesis output is

$y(t)=\sum _{k=1}^{K}r_{k}(t)\cos \left(2\pi kf_{0}t+\phi _{k}\right)$. 2

Inharmonic form

Additive synthesis can also produce inharmonic sounds (which are non-periodic waveforms, within the time-frame given by the fundamental frequency) if the individual harmonics do not all have a frequency that is an integer multiple of the fundamental frequency. Inharmonic additive synthesis can be described as

$y(t)=\sum _{k=1}^{K}r_{k}(t)\cos \left(2\pi f_{k}t+\phi _{k}\right)$,

where $f_{k}\,$ is the constant frequency of $k\,$th partial.

Time-dependent frequencies

In the general case, the instantaneous frequency of a sinusoid is the derivative (with respect to time) of the argument of the sine or cosine function. If this frequency is represented in Hz, rather than in angular frequency form, then this derivative is divided by $2\pi \,$. This is the case whether the partial is harmonic or inharmonic and whether its frequency is constant or time-varying.

In the most general form, the frequency of each non-harmonic partial is a function of time, $f_{k}(t)\,$, yielding

$y(t)=\sum _{k=1}^{K}r_{k}(t)\cos \left(2\pi \int _{0}^{t}f_{k}(u)\ du+\phi _{k}\right)$. 3

Broader definitions

Additive synthesis has been used as an umbrella term for the class of sound synthesis techniques that sum simple elements to create more complex timbres, even when the elements are not sine waves. For example, F. Richard Moore listed additive synthesis as one of the "four basic categories" of sound synthesis along side subtractive synthesis, nonlinear synthesis, and physical modelling. In this general sense, the pipe organ and the Hammond organ may be considered as additive synthesizers. Summation of principal components and Walsh functions have also been classified as additive synthesis.

Implementation methods

Modern-day implementations of additive synthesis are mainly digital. (See section Discrete-time equations for the underlying discrete-time theory)

Oscillator bank synthesis

Additive synthesis can be implemented using a bank of sinusoidal oscillators, one for each partial.

Wavetable synthesis

Main article: Wavetable synthesis

In the case of harmonic, quasi-periodic musical tones, wavetable synthesis can be as general as time-varying additive synthesis, but requires less computation during synthesis. As a result, an efficient implementation of time-varying additive synthesis of harmonic tones can be accomplished by use of wavetable synthesis.

Group additive synthesis is a method to group partials into harmonic groups (of differing fundamental frequencies) and synthesize each group separately with wavetable synthesis before mixing the results.

Inverse FFT synthesis

An inverse Fast Fourier Transform can be used to efficiently synthesize frequencies that evenly divide the transform period. By careful consideration of the DFT frequency domain representation it is also possible to efficiently synthesize time varying sinusoids of arbitrary frequency using a series of overlapped inverse Fast Fourier Transforms.

Additive analysis/resynthesis

It is possible to analyze the frequency components of a recorded sound giving a "sum of sinusoids" representation. This representation can be re-synthesized using additive synthesis. One method of decomposing a sound into time varying sinusoidal partials is Fourier Transform-based McAulay-Quatieri Analysis.

By modifying the sum of sinusoids representation, timbral alterations can be made prior to resynthesis. For example, a harmonic sound could be restructured to sound inharmonic, and vice versa. Sound hybridisation or "morphing" has been implemented by additive resynthesis.

Additive analysis/resynthesis has been employed in a number of techniques including Sinusoidal Modelling, Spectral Modelling Synthesis (SMS), and the Reassigned Bandwidth-Enhanced Additive Sound Model. Software that implements additive analysis/resynthesis includes: SPEAR, LEMUR, LORIS, and SMSTools.

Relation to speech synthesis

Main article: Speech synthesis

During the 1970s additive synthesis was investigated in the context of speech synthesis research.

Sinewave synthesis, a technique for synthesizing speech by replacing the formants (main bands of energy) with pure tone whistles may be considered a 'non-harmonic additive re-synthesis for speech'.

Linear predictive coding (LPC) is an analysis and audio codec method for speech. In the decoder of an LPC audio codec, subtractive synthesis using filters, or sinewave synthesis using oscillators, are utilized to re-synthesize speech.

Timeline of additive synthesizers

In the following timeline, dates in parentheses are the release dates of commercial synthesizers, or in case of unique experimental machines, completion dates.

The drawbars on a Hammond organ, used to control the amplitude of each harmonic.

Historical precursors

Pipe organ (around 1450): Pipes of pipe organs can be combined in the manner of additive synthesis. The pipes that can be combined are mostly flute pipes, which create nearly sinusoidal or triangle waves. Pipes that generate other types of waveforms, for example square wave generating clarinet stops, are not suited to this purpose: they are used more as solo stops.

Telharmonium (1897): The Telharmonium, also known as the Dynamophone, was the first electronic musical instrument, predating the dynamic loudspeaker. The Telharmonium resembled and organ and allowed adjustment of the amplitudes of harmonics generated by tonewheels. Electronic amplification was not available at the time of its realization, which led to a bulky design using alternators. The first model weighed 7 tons.

Hammond organ (invented in 1934): The Hammond organ is an electronic organ that uses nine drawbars to mix several harmonics, which are generated by a set of tonewheels.

pre-1970s

ANS Synthesizer (1958): The development of the ANS synthesizer was led by its inventor, Evgeny Murzin. The machine ran a bank of 576 sine wave oscillators over a span of 8 octaves (at 1/6 semitone intervals). A second model (1964) extended the frequency range to 10 octaves and is still functional. The 720 oscillator frequencies are generated by 5 optical discs, photocells and band-pass amplifiers. Compositions are written in the form of spectrogram "scores" drawn on glass plates covered by black mastic, by scratching the mastic. Light that passes the scratches determines the oscillator amplitudes.

NED Synclavier PSMT + V/PK

1970s

Alles Machine: The Bell Labs Digital Synthesizer, commonly known as the Alles Machine, was a real-time digital sound synthesis system, developed at Bell Labs in 1970s following earlier non-real-time software experiments. It has been called the first true digital additive synthesizer. Based on this technology, several commercial synthesizers were developed, including Crumar/DKI GDS (1980), a reduced version DK Synergy (1982), and the Atari AMY sound chip.

EMS Digital Oscillator Bank (DOB) and Analysing Filter Bank: According to the Inside Story of Electronic Music Studios (London), Ltd. (EMS), in the early 1970s, David Cockerell and Peter Eastty developed a minicomputer controllable fully digital vocoder implementing real-time additive synthesis, following the previous analog version. It consisted of 64 digital oscillators with independent frequency and amplitude controls, 1024 point waveforms, and bank of 128 digital analysis filters to provide coefficients for the oscillators. (⇒See also #Additive analysis/resynthesis)

RMI Harmonic Synthesizer: Developed in 1974–76, the RMI Harmonic Synthesizer was an early digital additive / analog subtractive hybrid synthesizer marketed to consumers. The additive section consisted of 16 partials that could be mixed in an edit mode into a pre-calculated single-cycle stereo waveform. In play mode, the digital waveform was processed by analog modules such as a voltage-controlled filter controlled by an envelope generator, implementing subtractive synthesis. RMI Harmonic Synthesizer was used by synth pioneer Jean Michel Jarre, for his albums Oxygène and Equinoxe. A similar design (non-time-varying digital additive/analog subtractive hybrid synthesis) was also utilized in Oxford Synthesizer Company's OSCar (1983, 24 harmonics), Korg DSS-1 (1986, 128 harmonics), Kawai K3 (1986, 32 harmonics), Casio FZ-1 (1987, 48 harmonics).

New England Digital Synclavier: Synclavier was a programmable harmonic definable FM/additive synthesizer and sampler. Initially, it was not a real additive synth: one can construct a patch defining 16 partials per voice (as in the Casio FZ-1, Korg DSS-1, and Kawai K3) and apply dynamic enveloping, and FM operator with envelope, only with the partial timbre. Later, with the Synclavier software upgrade, one can specify several harmonic spectrums and crossfade between them in time.
Note that Synclavier's FM re-synthesis feature with fine grained time frame is sometimes considered as equivalent to Wavetable synthesis, and Wavetable synthesis under some conditions is equivalent to time-varying additive synthesis. (⇒ See section Wavetable synthesis)

Fairlight Quasar (1975-1977) and Fairlight CMI (1979-1985): Built by Fairlight, Quasar and CMI implemented harmonic additive synthesis and FFT-based additive resynthesis.

1980s

Crumar/DKI General Development System (GDS, 1980) and DK Synergy (1982): Descendants of Alles Machine, GDS (marketed in the US by Digital Keyboards, Inc.) and Synergy are user-definable phase modulation semi-algorithmic synthesizers with additive capabilities, operating 32 digital oscillators. This allows, for example, two-voice polyphony with 16 partials per voice.

Seiko Digital Sound System: Seiko introduced its Digital Sound System line of keyboard instruments in 1984, utilizing a 16-operator additive synthesis engine. Although the keyboards (the DS-101, DS-202, and DS-250) were not directly programmable, a separate programming device was available that allowed the user to create new presets.

Kurzweil K150 (1986) : The Kurzweil Music Systems K150 is an additive engine that trades off quantity of oscillators vs. polyphony and where one can program each partial individually with envelopes. Full programming (known as Fourier Synthesis option) is only possible using an old Apple II computer, and cannot be done from the front panel.

Kawai K5 (1987): While in Kurzweil K150 harmonics are controlled individually, in K5 manufactured by Kawai they are controlled in four groups. The more recent Kawai K5000 is also an additive synth, but combined with samples.

Kawai K5000S

1990s

Kawai K5000: More contemporary popular implementations of additive synthesis includes the Kawai K5000 series of synthesizers in the 1990s.

2000–

2000s (decade) saw the advent of software synthesizers such as discoDSP Vertigo, Camel Audio Alchemy and Cameleon 5000, Image-Line Morphine, Harmless and Harmour the VirSyn Cube, White Noise Audio Soft WNAdditive, and ConcreteFX Adder.

Discrete-time equations

This section reviews the theory underlying digital implementation of additive synthesis, appropriately described by discrete-time equations.

Harmonic form

In the discrete-time form the equation for harmonic additive synthesis (2) can be written as

$y[n]=\sum _{k=1}^{K}r_{k}[n]\cos \left({\frac {2\pi kf_{0}}{f_{\mathrm {s} }}}n+\phi _{k}\right)$

where

$y[n]\,$ is the output sample at discrete time $n\,$,

$f_{0}\,$ is the fundamental frequency of the waveform or the note frequency,

$f_{\mathrm {s} }\,$ is the sampling frequency,

$k\,$ is an index over the summed harmonics, ranging from $1\,$ to $K\,$

$r_{k}[n]$ is the amplitude envelope of the $k\,$th harmonic at discrete time $n\,$,

$\phi _{k}\,$ is a constant phase offset.

In discrete-time additive synthesis, the frequency of the highest harmonic must be less than the Nyquist frequency $f_{\mathrm {s} }/2\,$ to prevent aliasing. This implies that $K<\mathrm {floor} (f_{\mathrm {s} }/2f_{0})\,$.

If $r_{k}[n]\,$ is constant, as in (1), the discrete-time equations can be written in a form suitable for generation with an inverse Fast Fourier transform,

$y[n]=\sum _{k=1}^{K}\left[a_{k}\cos \left({\frac {2\pi kf_{0}}{f_{\mathrm {s} }}}n\right)-b_{k}\sin \left({\frac {2\pi kf_{0}}{f_{\mathrm {s} }}}n\right)\right]$,

where

$a_{k}=r_{k}\cos(\phi _{k})\,$ and

$b_{k}=r_{k}\sin(\phi _{k})\,$.

Inharmonic form

The time-dependent frequency $f[n]\,$ of a sinusoid $\cos(\theta [n])\,$ at the time of sample $n\,$ can be defined in an implementation-friendly way as an angle increment

$\theta [n]=\theta [n-1]+{\frac {2\pi f[n]}{f_{\mathrm {s} }}}\,$

where $(2\pi f[n])/f_{\mathrm {s} }\,$ is the angle increment at the time of sample $n\,$. It is also the case that the angle is

$\theta [n]={\frac {2\pi }{f_{\mathrm {s} }}}\sum _{i=-\infty }^{n}f[i]\,$.

By replacing the $k\,$th harmonic frequency, $kf_{0}\,$, with a time-varying and general (not necessarily harmonic) frequency, $f_{k}[n]\,$ (the time-dependent frequency of the $k\,$th partial at the time of sample $n\,$ ), and generalising to a time-dependent phase offset $\phi _{k}[n]\,$, the equation for the sample output is

$y[n]=\sum _{k=1}^{K}r_{k}[n]\cos \left({\frac {2\pi }{f_{\mathrm {s} }}}\sum _{i=1}^{n}f_{k}[i]+\phi _{k}[n]\right)\,$.

If $f_{k}[n]=kf_{0}\,$ with constant $f_{0}\,$ and $\phi _{k}[n]\,$ is constant, all partials are harmonic and these more general equations reduce to the harmonic case above.

The time-dependent phase offset term $\phi _{k}[n]\,$ of each partial can be absorbed into the time-dependent frequency term, $f_{k}[n]\,$ by the substitution

$f_{k}[n]\leftarrow f_{k}[n]+{\frac {f_{\mathrm {s} }}{2\pi }}\left(\phi _{k}[n]-\phi _{k}[n-1]\right)\,$.

If that substitution is made, all of the $\phi _{k}[n]\,$ phase terms can be set to zero with no loss of generality (retaining the initial phase value at time $n=0\,$ ) and the expressions of inharmonic additive synthesis can be simplified to the discrete-time form of (3),

$y[n]=\sum _{k=1}^{K}r_{k}[n]\cos \left({\frac {2\pi }{f_{\mathrm {s} }}}\sum _{i=1}^{n}f_{k}[i]+\phi _{k}[0]\right)\,$ .

If this constant phase term (at time $n=0\,$) is expressed as $\phi _{k}[0]={\frac {2\pi }{f_{\mathrm {s} }}}\sum _{i=-\infty }^{0}f_{k}[i]\,$, the general expression of additive synthesis can be further simplified:

${\begin{aligned}y[n]&=\sum _{k=1}^{K}r_{k}[n]\cos \left({\frac {2\pi }{f_{\mathrm {s} }}}\sum _{i=-\infty }^{n}f_{k}[i]\right)\\&=\sum _{k=1}^{K}r_{k}[n]\cos \left(\theta _{k}[n]\right)\\\end{aligned}}$,

where $\theta _{k}[n]=\theta _{k}[n-1]+2\pi f_{k}[n]/f_{\mathrm {s} }\,$ for all $n\,$, and $\theta _{k}[0]=\phi _{k}[0]\,$.

See also

• Subtractive synthesis
• Speech synthesis
• Harmonic series (music)
• Fourier series

External links

• Digital Keyboards Synergy