7.1 Additive Synthesis

In the domain of additive synthesis (see section 2.2), application of spectral envelopes consists of a simple change of the amplitude of the input partials, while leaving their frequencies untouched. The simplest way is to replace the amplitude of a partial at frequency fwith the value of the envelope v(f) at that frequency. If the input partials and the modulating spectral envelope are from different sounds, this is called cross synthesis , since it crosses the characteristics of two distinct sounds: the partial structure (presence and frequency location, or absence of partials and their development in time) of the input sound, and the modulating spectral envelope estimated from the other sound.

Instead of imposing the spectral envelope, a gradual mixture between the spectral envelope of the input sound (given by the partial amplitudes), and another spectral envelope is possible. This is governed by a mix factor m, between 0 for the original and 1 for the imposed spectral envelope. If a_i is the amplitude of partial i at frequency f_i, and v(f_i) the value of the envelope at that frequency, then the new amplitude a'_i is

a'_i = (1 - m) a_i + m v(f_i)

One step further to flexibility is taken by considering the mix factor being a function m(f) dependent of frequency. This way, e.g. only the high frequency part of the input sound can be changed to adopt the spectral envelope characteristics of the modulator.

Note that this can be generalized to n spectral envelopes v_j(f) and n mix functions m_j(f) with their sum $m_s(f) = \sum_{j=1}^n m_j(f) \le 1$, such that the proportion of the original is 1 - m_s(f_i):

$$a'_i = (1 - m_s(f_i)) a_i + \sum_{j=1}^n m_j(f_i) v_j(f_i)$$

If the envelope isn't represented directly but as filter coefficients, these have to be converted to the spectral envelope first, as shown in chapter 3 in equations (3.3) and (3.10) for LPC and cepstrum coefficients, respectively.