Ejemplo 4.8. Variantes de las antífonas de Vísperas del Corpus
E- Bbc M1166/1967 , fols 70v-
2. Ejemplo citado por Bradshaw
As described in § 4.1, for a givenntracks, all the unique mixes exist on a hypersphere inRn, i.e. an (n−1)-sphere. To generate random track gains, random points in this space were determined. Forntracks, and mmixes,mpoints on a unit (n−1)-sphere (denoted asSn−1) were generated. Thentracks were first normalised according to perceived loudness, as defined in BS.1770-3 [32] and modified by Pestana et al. [158]. A number of methods can be used to generate a distribution of mixes. Two such methods are detailed here.
5.1.1 Method 1: uniform mixes
An easy way to pick random points on a hypersphere of arbitrary dimension is to generate n
Gaussian random variables x1,x2, ...,xn. Then the distribution of the vectors g, as defined by
Equation 5.1, is uniform over the surfaceSn−1[172, 173].
g=q 1 x21+x22+. . .+x2 n x1 x2 .. . xn (5.1)
For sufficiently large number of points m, this method will return virtually all possible mixes of then tracks. However, are uniformly generated mixes representative of real mixes? It was hypothesised that the generation of uniformly distributed mixes would likely produce many mixes that would not realistically be created by real mixers (see Chapter 6). As a consequence, the value ofmwould have to be very large in order to be comparable to the number of real mixes listed in Table 6.1 and constraints would need to be implemented in order to ensure that all instruments are presented with sufficient gain as to be audible.
5.1.2 Method 2: mixes close to arbitrary point
There are advantages to generating track gains according to some parametric distribution. For example, the value ofmcan be lower, greatly reducing the computation time required for feature- extraction and analysis. This method requires explicit parameters to be chosen. From § 2.2, as- suming that the better mixes aregenerallythe ones where the tracks areroughlyequal in perceived loudness, this method can be used to generate mixes distributed about the equal-loudness mix. The equal-loudness mix is determined as follows. When the gains of allntracks are equal, whatggives a point onSn−1, i.e. where theL2norm ofgis equal to 1?
r=1=|g| (5.2a) 1= s n
∑
i=1 g2i (5.2b) 12= n∑
i=1 g2i (5.2c) 1=ng2 (5.2d) n−2=g (5.2e)5.1. GENERATING RANDOMISED TRACK GAINS 115
For example, whenn=16,g=0.25. Applying this linear gaing to all n loudness normalised tracks would result in the equal-loudness mix, where theL2norm is equal to 1.
In selecting a suitable parametric distribution it is important to note that linear distributions, such as the normal distribution, are not appropriate as the domain in question is not linear but a spherical surface. Recall that a linear domain extends over the range[−∞+∞], while a circular domain is wrapped over a smaller range such as[0,2π). The statistics of such distributions are
described by a number of equivalent terms in the literature, such as circular, spherical or directional statistics. In order to generate random points close to the desired position on the(n−1)-sphere, points are generated from a von-Mises-Fisher distribution (vMF). The probability density function of the vMF distribution for a randomn-dimensional unit vectorxis given by
fn(x;µ,κ) =Cn(κ)eκ µ Tx
(5.3)
whereκ≥0,||µ||=1,n≥2 and the normalisation constantCn(κ)is given by
Cn(κ) =
κn/2−1
(2π)n/2In/2−1(κ)
(5.4)
HereIv is the modified Bessel function of the first kind at orderv. The parametersµ andκ are
called the mean direction and concentration parameter, respectively, andµT is the transpose of
µ. The greater the value ofκ the higher the concentration of the distribution around the mean
directionµ. The distribution is unimodal forκ>0 and is uniform on theSn−1forκ=0. Further
details can be found in Fisher [174] and Mardia and Jupp [175]. To generate points according to a vMF distribution the SphericalDistributionsRand1code was used based on the work of Chen et al. [176]. In the context of audio mixes,µ represents the mix about which others are distributed, akin
to the mean in a normal distribution. Theκ term represents the diversity of mixes generated.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 0.1 0.2 0.3 0.4 0.5 Tracks Gain
Figure 5.1:Boxplot of gain values for 1,000 mixes of 16 tracks, generated from vMF distribu- tion, designed to produce mixes around the equal-loudness mix.
Forn=8 tracks, as in § 4.5, the gains required for the equal-loudness mix are distributed around
5.1. GENERATING RANDOMISED TRACK GAINS 116
the following point,µ. This calculation is based on Equation 5.2.
µ = [0.3536 0.3536 0.3536 0.3536 0.3536 0.3536 0.3536 0.3536]
Previous studies have indicated that, while a good initial guess, presenting each track at equal loudness is not an ideal final mix. As discussed in the literature review (see § 2.2.1) and also was shown in Chapter 4, vocals are often the loudest element in a mix (in particular, see Fig. 4.13 and Table 4.4). To this equal loudness configuration, a vocal boost is added according to p.157 of [61], i.e. a boost of 6.54 dB. A sanity check was performed by audition of mixes generated with this boost and it was decided that, while it may be more than the authors’ own taste, such a boost is not unrealistic. This addition of 6.54 dB to the vocal track produces the following vector, where track 8 is vocals.
µ = [0.3536 0.3536 0.3536 0.3536 0.3536 0.3536 0.3536 0.7507]
If the previous vector was, then it is clear that this point is no longer on the unit 7-sphere. To project the point back onto the unit 7-sphere, the vector is divided by it’sL2norm, resulting in the
following.
µ = [0.2948 0.2948 0.2948 0.2948 0.2948 0.2948 0.2948 0.6259] (5.5)
This vector is the newµ on the unit 7-sphere about which mixes will be generated. The result is
shown in Figure 5.2. Each mix generated draws a gain value for each track such that theL2norm
is equal to 1. Note that the median values closely match the vectorµ, as expected. Of course,
there may not exist a mix which has these median values. This specific value ofκ was chosen to avoid generating negative gains, achieved through trial and error. Ignoring phase, a gain ofg
is perceptually equal to−g, meaning that the nature of the distribution would change if negative gains were included. Of course, for a distribution which produces negative gains the absolute value could be taken to avoid inverting the phase of the tracks.
OH1 OH2 Kick Snare Bass Gtr1 Gtr2 Vox
0 0.25 0.5 0.75 1 Gain
Figure 5.2:Boxplot of gain values for 1,000 mixes, generated from VMF distribution, with 6.54 dB boost to vocals (µ=Eqn. 5.5,κ=200)
5.2. GENERATING RANDOMISED EQUALISATION 117