Editing Additive synthesis (section)

==Definitions==
{{See also|Fourier series|Fourier analysis}}
[[File:Additive synthesis.svg|250px|thumb|right|Schematic diagram of additive synthesis. The inputs to the oscillators are frequencies <math>f_k</math> and amplitudes <math>r_k</math>.]]
Harmonic additive synthesis is closely related to the concept of a [[Fourier series]] which is a way of expressing a [[periodic function]] as the sum of [[sine wave|sinusoidal]] functions with [[Frequency|frequencies]] equal to integer multiples of a common [[fundamental frequency]].  These sinusoids are called [[harmonic]]s, [[overtone]]s, or generally, [[Harmonic series (music)#Partial|partials]].  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 [[direct current|DC]] component (one with frequency of 0 [[Hertz|Hz]]).  [[Equal-loudness contour|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 function|periodic]] if

: <math> y(t) = y(t+P) </math>

for all <math> t </math> and for some period <math> P </math>.

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

: <math> \begin{align}
  y(t) &= \frac{a_0}{2} + \sum_{k=1}^{\infty} \left[ a_k \cos(2 \pi k f_0 t ) - b_k \sin(2 \pi k f_0 t ) \right] \\
       &= \frac{a_0}{2} + \sum_{k=1}^{\infty} r_k \cos\left(2 \pi k f_0 t + \phi_k \right) \\
   \end{align} </math>

where

* <math>f_0 = 1/P</math> is the [[fundamental frequency]] of the waveform and is equal to the reciprocal of the period,
* <math>a_k = r_k \cos(\phi_k) = 2 f_0 \int_{0}^P y(t) \cos(2 \pi k f_0 t)\, dt, \quad k \ge 0</math>
* <math>b_k = r_k \sin(\phi_k) = -2 f_0 \int_{0}^P y(t) \sin(2 \pi k f_0 t)\, dt, \quad k \ge 1</math>
* <math>r_k = \sqrt{a_k^2 + b_k^2}</math> is the [[amplitude]] of the <math>k</math>th harmonic,
* <math>\phi_k = \operatorname{atan2}(b_k, a_k)</math> is the [[phase (waves)|phase offset]] of the <math>k</math>th harmonic. [[atan2]] is the four-quadrant [[arctangent]] function,

Being inaudible, the [[direct current|DC]] component, <math>a_0/2</math>, and all components with frequencies higher than some finite limit, <math>K f_0</math>, are omitted in the following expressions of additive synthesis.

===Harmonic form===
The simplest harmonic additive synthesis can be mathematically expressed as:

{{NumBlk|:|<math>y(t) = \sum_{k=1}^{K} r_k \cos\left(2 \pi k f_0 t + \phi_k \right),</math>|{{EquationRef|1}}}}

where <math>y(t)</math> is the synthesis output, <math>r_k</math>, <math>k f_0</math>, and <math>\phi_k</math> are the amplitude, frequency, and the phase offset, respectively, of the <math>k</math>th harmonic partial of a total of <math>K</math> harmonic partials, and <math>f_0</math> is the [[fundamental frequency]] of the waveform and the [[Piano key frequencies|frequency of the musical note]].

===Time-dependent amplitudes===
{|class=wikitable align=right width=420px
|-
| [[File:Harmonic additive synthesis spectrum.png|280px]]
| <span style="font-size:85%;line-height:130%;">Example of harmonic additive synthesis in which each harmonic has a time-dependent amplitude. The fundamental frequency is 440&nbsp;Hz.</span>
[[File:Harmonic additive synthesis.ogg|noicon|150px]]
<span style="font-size:70%;line-height:130%;font-style:italic;">Problems listening to this file? See [[Media help]]</span>
|}
More generally, the amplitude of each harmonic can be prescribed as a function of time, <math>r_k(t)</math>, in which case the synthesis output is

{{NumBlk|:|<math>y(t) = \sum_{k=1}^{K} r_k(t) \cos\left(2 \pi k f_0 t + \phi_k \right)</math>.|{{EquationRef|2}}}}

Each [[Envelope (waves)|envelope]] <math>r_k(t)\,</math> should vary slowly relative to the frequency spacing between adjacent sinusoids.  The [[bandwidth (signal processing)|bandwidth]] of <math>r_k(t)</math> should be significantly less than <math>f_0</math>.

===Inharmonic form===
Additive synthesis can also produce [[Inharmonicity|inharmonic]] sounds (which are [[aperiodic]] waveforms) in which the individual overtones need not have frequencies that are integer multiples of some common fundamental frequency.<ref name=smith05>
{{Cite book     
 | last1        = Smith III | first1 = Julius O.
 | last2        = Serra     | first2 = Xavier
 | year         = 2005
 | chapter      = Additive Synthesis
 | chapter-url   = https://ccrma.stanford.edu/~jos/parshl/Additive_Synthesis.html
 | title        = PARSHL: An Analysis/Synthesis Program for Non-Harmonic Sounds Based on a Sinusoidal Representation
 | url          = https://ccrma.stanford.edu/~jos/parshl/
 | series       = Proceedings of the International Computer Music Conference (ICMC-87, Tokyo), Computer Music Association, 1987
 | publisher    = [[Center for Computer Research in Music and Acoustics|CCRMA]], Department of Music, Stanford University
 | access-date   = 11 January 2015
}} ([https://ccrma.stanford.edu/STANM/stanms/stanm43/stanm43.pdf online reprint])</ref><ref name=smith11>
{{Cite book     
 | last         = Smith III | first = Julius O.
 | year         = 2011
 | chapter      = Additive Synthesis (Early Sinusoidal Modeling)
 | chapter-url   = https://ccrma.stanford.edu/~jos/sasp/Additive_Synthesis_Early_Sinusoidal.html
 | title        = Spectral Audio Signal Processing
 | url          = https://ccrma.stanford.edu/~jos/sasp/
 | publisher    = [[Center for Computer Research in Music and Acoustics|CCRMA]], Department of Music, Stanford University
 | isbn         = 978-0-9745607-3-1
 | access-date   = 9 January 2012
}}</ref> While many conventional musical instruments have harmonic partials (e.g. an [[oboe]]), some have inharmonic partials (e.g. [[bell (instrument)|bells]]).  Inharmonic additive synthesis can be described as

: <math>y(t) = \sum_{k=1}^{K} r_k(t) \cos\left(2 \pi f_k t + \phi_k \right),</math>

where <math>f_k</math> is the constant frequency of <math>k</math>th partial.
{|class=wikitable align=right width=420px
|-
| [[File:Inharmonic additive synthesis spectrum.png|280px]]
| <span style="font-size:85%;line-height:130%;">Example of inharmonic additive synthesis in which both the amplitude and frequency of each partial are time-dependent.</span>
[[File:Inharmonic additive synthesis.ogg|noicon|150px]]
<span style="font-size:70%;line-height:130%;font-style:italic;">Problems listening to this file? See [[Media help]]</span>
|}

===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 [[hertz]], rather than in [[angular frequency]] form, then this derivative is divided by <math>2 \pi</math>.  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 non-negative function of time, <math>f_k(t)</math>, yielding

{{NumBlk|:|<math>y(t) = \sum_{k=1}^{K} r_k(t) \cos\left(2 \pi \int_0^t f_k(u)\ du + \phi_k \right).</math>|{{EquationRef|3}}}}

===Broader definitions===
''Additive synthesis'' more broadly may mean sound synthesis techniques that sum simple elements to create more complex timbres, even when the elements are not sine waves.<ref>
{{cite book     
 | last         = Roads | first = Curtis
 | author-link   = Curtis Roads
 | year         = 1995
 | title        = The Computer Music Tutorial
 | url=https://archive.org/details/computermusictut00road
 | url-access=limited
 | publisher    = [[MIT Press]]
 | isbn         = 978-0-262-68082-0
 | page         = [https://archive.org/details/computermusictut00road/page/n152 134]
}}</ref><ref name="MooreFoundationsCM">
{{cite book     
 | last         = Moore | first = F. Richard
 | year         = 1995
 | title        = Foundations of Computer Music
 | publisher    = [[Prentice Hall]]
 | isbn         = 978-0-262-68082-0
 | page         = 16
}}
</ref> For example, F. Richard Moore listed additive synthesis as one of the "four basic categories" of sound synthesis alongside [[subtractive synthesis]], nonlinear synthesis, and [[physical modelling synthesis|physical modeling]].<ref name="MooreFoundationsCM"/> In this broad sense, [[pipe organ]]s, which also have pipes producing non-sinusoidal waveforms, can be considered as a variant form of additive synthesizers. Summation of [[Principal component analysis|principal components]] and [[Walsh functions]] have also been classified as additive synthesis.<ref>
{{cite book     
 | last         = Roads | first = Curtis
 | author-link   = Curtis Roads
 | year         = 1995
 | title        = The Computer Music Tutorial
 | url=https://archive.org/details/computermusictut00road
 | url-access=limited
 | publisher    = [[MIT Press]]
 | isbn         = 978-0-262-68082-0
 | pages        = [https://archive.org/details/computermusictut00road/page/n168 150]&ndash;153
}}</ref>