Medidas isotópicas y análisis elemental (Exp I y II)

There are a number of problems associated with the criteria for ‘good’ spacing used in much of the literature:

1. Methods such as that of Louden (1971), consider only the attempt to avoid modal degeneracy. An algorithm searches through a set of ratios, and selects the ones with the smallest standard deviation between modal frequencies. This method however, fails to consider the possibility that, even though modes may uniformly spaced, there may be no subjective improvement. Is the smallest standard deviation between eigenfrequencies the most perceptually relevant measure?

2. The metrics do not account for changes in perception as the frequency in- creases. It has been shown in Chapter 3 that a greater number of modes occur as the frequency increases. Over what range should we look at the uniformity of spacing? There is likely a difference in the perception of spaced resonances as a function of frequency. Indeed, it has been reported that we are more sensitive to frequency irregularity at higher frequencies (Avis et al., 2007; Kar- jalainen et al., 2004). Current aspect ratio metrics apply no specific weighting factor dependent on frequency.

3. A third problem with these criteria is the lack of account for positioning of the loudspeaker and listener in the room. It is assumed that each mode contributes equally, and with the same phase, to the overall response. In order for exci- tation and reception of every mode, the source must be placed in one corner, and the listener diagonally opposite. Furthermore, the response is significantly altered where two modes are out of phase with each other, as a result of the eigenfunctions, or ‘mode-shapes’ (Equation 3.12). The importance of these mode shapes will be fully explored in Chapter 5.

4. Figure 4.2 shows all modes having equal amplitude, which makes it tempting give them equal importance. However, in reality, modes of different types, axial, tangential and oblique will likely have different amplitudes, due to dif- ferent levels of damping. This is also unaccounted for by the spacing metrics. It is likely that the perception of two spaced resonances will differ under dif- fering damping conditions.

5. Finally, the volume of the room is often not considered. While Bolt (1946) does give a range of validity, Toole (2006) explains that it is rarely considered

in practice. A change in volume, whilst retaining the relative spacing, will alter the absolute spacing between modes.

These omissions form the rationale for the experiment in Section 4.4, investiga- ting the subjective response to the interaction between two modes. The experimental work addresses a number of the issues above, and the findings are extrapolated in order to discuss the results with regard to each of the concerns highlighted here. Before the test method is described, the interaction between two modes is further explored.

4.3 The Spacing of Two Resonances

In order to determine if a frequency and damping dependent optimal spacing can be defined, the simple case of two adjacent modes is studied, both analytically and subjectively. In the case of a single mode, the decay, analogous to the RT60

reverberation time concept in the diffuse field, can be related to its Q-factor by the approximate equation:

RTM ODAL =

2.2Q

f0 (4.2)

where Q is the quality factor of the mode and f0 the frequency.

We see that as the Q increases, the decay time, often described as ringing, increases. We observe the effect of adding a second resonance by simply summing them. Figure 4.4 shows a second resonance, first at 102Hz and then at 105Hz, where the Q factor is 30. Secondly, Figure 4.5 shows the same resonant frequencies, but where each has a Q factor of 50. The differences in these figures highlight the fact that we must not limit our understanding of a ‘good’ spacing simply to the frequencies at which two modes occur.

A simple visual investigation of the effect of altering the spacing between the two individual resonances reveals a clear impact on the temporal response. As the second frequency moves away from the first, the frequency response appears to flat- ten at the peak, and results in an apparent reduction of the overall decay time from the case where both resonances were at 100Hz. However, as the shift continues, the magnitude frequency response reveals a large dip and the resulting impulse res- ponse begins to show distinctive amplitude modulation. This is associated with the interaction between the two resonances and at these frequency differences, sound identical to first order beats as described in many psychoacoustic textbooks (eg. Howard and Angus (2001); Kuttruff (2007)). Where the Q is increased to 50 in

85 90 95 100 105 110 115 110 120 130 140 Magnitude Response Linear Frequency (Hz) Amplitude (dB) 0 200 400 600 800 1000 −1 −0.5 0 0.5 1 Impulse Response Time (ms) Normalised Amplitude (a) 100Hz 100Hz 85 90 95 100 105 110 115 110 120 130 140 Magnitude Response Linear Frequency (Hz) Amplitude (dB) 0 200 400 600 800 1000 −1 −0.5 0 0.5 1 Impulse Response Time (ms) Normalised Amplitude (b) 100Hz 102Hz 85 90 95 100 105 110 115 110 120 130 140 Magnitude Response Linear Frequency (Hz) Amplitude (dB) 0 200 400 600 800 1000 −1 −0.5 0 0.5 1 Impulse Response Time (ms) Normalised Amplitude (c) 100Hz 105Hz

85 90 95 100 105 110 115 110 120 130 140 Magnitude Response Linear Frequency (Hz) Amplitude (dB) 0 200 400 600 800 1000 −1 −0.5 0 0.5 1 Impulse Response Time (ms) Normalised Amplitude (a) 100Hz 100Hz 85 90 95 100 105 110 115 110 120 130 140 Magnitude Response Linear Frequency (Hz) Amplitude (dB) 0 200 400 600 800 1000 −1 −0.5 0 0.5 1 Impulse Response Time (ms) Normalised Amplitude (b) 100Hz 102Hz 85 90 95 100 105 110 115 110 120 130 140 Magnitude Response Linear Frequency (Hz) Amplitude (dB) 0 200 400 600 800 1000 −1 −0.5 0 0.5 1 Impulse Response Time (ms) Normalised Amplitude (c) 100Hz 105Hz

Figure 4.5, the onset of these beats appears at a closer spacing. Between two sinu- soids, the beating frequency is equal to the frequency between them. Therefore, the greater the spacing, the faster the beating effect.

Once again it is possible to fall into the trap of making assumptions, based upon a visual inspection of the figures, as to the perceived quality of an audio stimulus when passed through these resonant systems (assuming the audio material were to excite the corresponding frequency range). If a perceptual improvement is associa- ted with a shorter decay, then moving the second frequency away from the first is preferable. However, the introduction of beating is likely to be highly detectable to the listener and perhaps undesirable (Rasch and Plomp, 1999). Therefore, by taking into consideration both the modal frequencies and Q-factors, we see that the search for an ‘optimal’ spacing involves more than simply aligning the frequencies with uniform spacing.