ground up. By adding together pure frequencies (sine waves) of varying frequencies and amplitudes, we can precisely define the timbre of the sound that we want to create. Definitions 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 sinusoidal functions with frequencies equal to integer multiples of a common fundamental frequency. These sinusoids are called harmonics, overtones, or generally, 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 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 for all and for some period . The Fourier series of a periodic function is mathematically expressed as: where is the fundamental frequency of the waveform and is equal to the reciprocal of the period, is the amplitude of the th harmonic, is the phase offset of the th harmonic. atan2 is the four-quadrant arctangent function, Being inaudible, the DC component, , and all components with frequencies higher than some finite limit, , are omitted in the following expressions of additive synthesis. Harmonic form The simplest harmonic additive synthesis can be mathematically expressed as: where is the synthesis output, , , and are the amplitude, frequency, and the phase offset, respectively, of the th harmonic partial of a total of harmonic partials, and 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, , in which case the synthesis output is Each envelope should vary slowly relative to the frequency spacing between adjacent sinusoids. The bandwidth of should be significantly less than . Inharmonic form Additive synthesis can also produce inharmonic sounds (which are aperiodic waveforms) in which the individual overtones need not have frequencies that are integer multiples of some common fundamental frequency. While many conventional musical instruments have harmonic partials (e.g. an oboe), some have inharmonic partials (e.g. bells). Inharmonic additive synthesis can be described as where is the constant