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The index functions (Fl) is intended primarly to describe the
variation of timbre with time. However the loudness is also a function of
the index and care must be taken in the appreciation of the amplitude
value (P5) that will often have to be variable and not fixed as in figure 5.
Some automatic correction could be calculated by the means of a normalization function as discussed by Moorer (3). However it requires storage for
a new function and computation time, it complicates the instrument and it
is only approximate. Moreover this variation of loudness often sounds
natural and the simplest way to use this instrument is to use if necessary
the PS input as a post compensation of loudness.
The transfer function (F3) generates the distortion and consequently it determines the harmonic content of the output as a function of
the input. In general the timbre becomes richer as the index increases,
but this is highly dependent on the choise of this transfer function.
However some general remarks can be made: in the standard use of the non
linear distorsion the do term (the constant of the polynomial) should be
set to zero to avoid a click at the beginning and end of a note especially
with softer (small index value) notes; the odd and even harmonics are
independent and the maximum number of harmonics is equal to the degree of
the polynomial (but the higher harmonics may be insignificant). When the
order is small one can use the direct equalities (3) to predict this
evolution. With larger values one needs the aid of a computer program displaying the evolution of the spectrum as a function of the index value.
THE CHOICE OF A TRANSFERT FUNCTION.
The characteristics of the non linear distortion are entirely
determined by the coefficients of the polynomial. But there are many ways
to calculate these coefficients.
1.- Evaluation from a steady-state spectrum.
Relation 4 allows us to calculate the 0 coefficients (the polynomial)
starting from the H ones (the spectrum) given an index value XO. We have
seen that the distortion of a cosine wave gives only cosine components,
the amplitude of which are real but can be positive or negative. So different distortions, hence different timbre variations, can be used which
verify the initial conditions, only by changing the signs of some components.
An abruptly limited spectrum or a rich spectrum (more than 20 harmonics)
often gives rise to irregularities in the evolution of the sound with
the index, which is akin to the Gibbs phenomenon.
2.- Evaluation from a continuous transfer functior
In this case, the general form of the transfer function is known,
and the problem is to get a polynomial approximation of it, in order to
obtain a band limited spectrum and avoid foldover. A limited development
of the function approximates it very well around the origin (small values
of the index) but usually not in other areas. One can also take the complete transfer function, calculate the spectrum for a specific index
value and then use the first procedure with the first N harmonics. This
approximates well the spectrum for that value of the index, but abrupt
bandwidth limitation causes the ripple effect previously described and
it may prove useful to attenuate little by little the last partials values
to get smoother evolutions with almost identical spectra (this is in effect
a kind of windowing). Other classical algorithms can be used, such as the
least mean square approximation.
3.- Direct evaluation.
An experimental and/or intuitive choice of the coefficients of
the polynomial is also a good strategy: there is a strong but subtle
connection between the regularity of variation of the coefficients (and
their sign) and the homogenity of the resulting timbre. Choosing to affect
the same sign to all odd order coefficients, and doing so for even order
coefficients (though this sign may be different of the other one) produces
a very steep transfer function, which may produce too brassy sounds. A
good practise can be to alternate the sign of successive odd order coefficients (say Al, AS, A9 positive, and A3, A7, All negative) and do the same
for even order coefficients.
SOME EXAMPLES.
Examples 1 to 4 use the previously described Music V instrument,
except that "amplitude" input (for correction of loudness) comes from an
oscillator scanning once the compensation function F4.
Example 1. Brilliant sounds.
Many non linear distortion create impulse like waveforms (figure 6).
In this case, the timbre becomes richer as the index increases.