Resultados de experimento IIIb: Evaluación de la germinación de Panicum virgatum

In order to answer the question posed, the two spaced resonances were artificially modelled. Resonances were generated using the Green’s Function described in Sec- tion 3.3.3. It is common to supply this equation with an array of modal frequencies for a full room model. However, here, a fixed array (eg. Ên[100,102]) of the two

modes was fed into the equation, with the source and receiver eigenfunctions as unity, to obtain the system’s response.

Equation 3.27 produces a complex output in the frequency domain. This res- ponse was transformed to the time domain, giving the impulse response of the system in question. It is common practice in psychoacoustic testing of this nature to convolve the system response with an input stimulus such as a test tone or mu- sical refrain to determine the perception of that stimulus to that system response.

However, in this case, the impulse itself is played to the subject as the test stimulus, in order to obtain absolute thresholds between decay reduction and the onset of bea- ting. A number of stimuli were considered during pilot testing, such as noise, and also sine bursts. However, these were rejected on the grounds that there would be unnecessary masking effects. Single frequency decaying sine tones were considered, but the decay length of the tone would in some cases be responsible for masking the decay of the resonance itself. Furthermore, a tone of increasing bandwidth would be needed to excite the full frequency range under test as the spacing increased. Using a musical stimulus, whilst realistic in a listening scenario, does not allow absolute thresholds to be detected. As such, an optimal value measurement from the impulse itself, corresponding to the ‘worst case scenario’, was found to be appropriate.

4.4.1 Method

As mentioned, both the frequency and the decay must be considered to better un- derstand spacing. Therefore, two independent variables were chosen - the frequency of the first mode (63, 125 and 250Hz) and Q-factor. Four Q-factors (10, 20, 30, and 50) were chosen to represent a broad range, typical in listening conditions. The spacing of the second resonance was adjusted by way of a slider on a graphical user interface (Figure 4.6). Samples were generated immediately each time the sli- der was moved, reducing the resolution error which would be inherent if playing back pre-processed samples with a set interval. All programming was carried out in MATLAB_•R. Subjects were asked to adjust the slider to that point where the

shortest decay time before the audible degradation of beats occurred, thereby cor- responding to the aforementioned definition of ‘optimal spacing’. Prior to the test, explanation of the differences in presentation sounds (long decay, shorter decay, and beating effect) were explained and demonstrated, along with images in the time domain. No time domain images were displayed during the actual tests to avoid bias.

The dependent variable was therefore the frequency spacing required by the subject. Eleven subjects were tested, in quiet studio conditions. Each subject was a student or lecturer of Music Technology at the University of Huddersfield and was given time to practice before the test commenced. The block of twelve tests was randomised, and repeated three times by each subject.

Figure 4.6: Screenshot showing the Graphical User Interface for the optimal spacing test

4.4.2 Calibration

A laptop computer was separated from the monitor and mouse by way of an acoustic screen. Stimuli were auditioned over a pair of Sennheiser HD-650 headphones. The presentation levels of the three frequencies were weighted to ensure that the percei- ved level of each sample was the same. Each was presented according to the 90dB equal loudness contour (Robinson and Dadson, 1956). The calibration was achieved by placing the headphones on a Neumann K900 dummy head and measuring the output level of full amplitude sine tones at each of the test frequencies. Scaling values were then applied to ensure correct playback level. These were: 63Hz - 0.551, 125Hz - 0.299, 250Hz - 0191.

Each impulse was filtered with a third order Butterworth low pass filter at 1kHz, and windowed with a Tukey window providing a 12 millisecond opening and closing time in order to remove any spectral artefacts from the beginning and end of the signal which may give audible clues to the listener. The sound card used was a professional M-Audio Firewire 410 and was considered to be a negligible source of replay error.

4.4.3 Results

Figure 4.7 shows the mean spacing across 11 subjects and reveals a clear trend. As the Q-factor increases, the optimal spacing needed to provide the shortest decay reduces. This result makes good sense, as the higher Q-factors are associated with a narrower bandwidth. Therefore they must be spaced closer together for the interac- tion of the two to produce a smooth response. Furthermore, being ‘more discrete’, the further apart they move, the more obvious the beating effect as each beat has a greater amplitude.

When comparing the test frequencies, it is seen that higher frequencies require a greater spacing between the two resonances. Interestingly, this puts the perceived

Figure 4.7: Subjective optimal mean spacing across Q-Factor and frequency optimum in direct contradiction to the natural effect within a room, as the actual spacing of modes decreases with increased frequency. Somewhat counteracting these results, we see that the level of uncertainty shown by the standard deviation error bars also increases with frequency, suggesting that the concept of an optimal spacing becomes less meaningful at higher frequencies. An increase in standard deviation is also observed with decreasing Q, suggesting increasing difficulty in hearing modal effects as Q decreases. Again, this result makes sense - the reduced decay times of lower Q tests make it more difficult to accurately perceive the length of that decay compared to another or to notice the beating effect. It is noted that no direct comparisons of decay were made between cases. The ‘optimal’ spacing only refers to that frequency and that Q. It is not the case that with the spacings obtained, all impulses are equal in their perceived quality. For example, at 63Hz and Q=50, even with the optimal spacing of 0.5Hz, the perceived ringing is longer than 250Hz at Q=10.

Analysis of variance was carried out to ascertain the level of significance across the variable parameters. ANOVA shows that both Q Factor and modal frequency are highly significant at the 1% level (p<0.01), which indicates the success of systematic testing. Figure 4.7 reveals that there is less difference across the three frequencies at

0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 4 Modal Bandwidth (Hz)

Optimal Subjective Spacing (Hz)

Mean Subjective Optimal Spacing across Bandwidth

Figure 4.8: Mean subjective optimal spacing presented in ascending modal band- width

low Q (shorter decay times). The large overlap between frequencies suggests there is no significant difference between frequencies at these levels of Q.

Although both factors are themselves highly significant, it is useful at this point to note that they can be related in terms of a single value - modal bandwidth. The relationship is shown in Equation 4.3

Bw =

f

Q (4.3)

Table 4.1 and Figure 4.8 consider each of the 12 test scenarios in ascending bandwidth (the bandwidth of the first mode is reported, with the second mode’s bandwidth assumed to be the same due to the relatively small spacing). The results again show a clear trend.

Bandwidth (Hz) 1.26 2.10 2.50 3.15 4.17 5.00 Modal Spacing (Hz) 0.50 0.66 0.65 1.11 1.41 1.43 Bandwidth (Hz) 6.25 6.30 8.33 12.50 12.50 25.00 Modal Spacing (Hz) 1.99 2.92 2.44 3.17 3.92 4.00

0 50 100 150 200 250 300 0 20 40 60 80 100 Frequency (Hz)

Optimal spacing as % of bandwidth

Optimal frequency spacing for two adjacent modes

Q=50 Q=20 Q=20 Q=10

Figure 4.9: Optimal Spacing across ascending bandwidth for the four different Q Factors tested