Experimento IIIb: Evaluación de la germinación de Panicum virgatum L en

The modal density is essentially a measure of the number of modes occurring within a given bandwidth. This can be obtained by simply calculating all eigenfrequencies and summing over the given range. Furthermore, it has been shown in Chapter 3 that Equation 3.30 represents a statistical average for the expected number of modes at a specific frequency. It is this equation which is used in this chapter to calculate the room parameter known as modal density. The concept of metrics, used in scoring

rooms based upon one or more objective measures was introduced in the previous chapter. Unsurprisingly, there have also been a number of metrics suggested in terms of the modal density. Two such examples are widely quoted. Firstly, the ‘Bonello Criterion’ (Bonello, 1981), which aims to guide room design through a set of criteria including that where modal degeneracy occurs, there should be at least five modal frequencies within that third octave band. The implication here is that a greater density may offer a subjective improvement to counteract the effects of degeneracy. Bonello explains that it is through personal experience that he comes to his conclusions. In a study 28 years later, Welti highlights that the criterion assumes certain conditions which are almost never met, though he does call the method ‘intuitively satisfying’ (Welti, 2009). It is for this reason – that such a typical metric may not offer a subjective improvement and yet remains enticing for use due to its simplicity – that a thorough subjective study of the modal density is necessary at this time.

The second widely quoted objective quantity derived from the density is that of the ‘Schroeder Frequency’ (Schroeder, 1996). This frequency indicates the point where sufficient modes exist, within the bandwidth of a single mode, that the fre- quency response can be assumed to be statistical in nature. Specifically, above this frequency, the average spacing between adjacent maxima is equal to 6.7/RT60, with

the same result regardless of the room. The frequency is therefore traditionally said to define a transition between the ‘modal’ and ‘statistical’ regions in a given room (Toole, 2008). This transition frequency is most often determined by Equation 3.31. Note that the constant 2000 was changed in Schroeder’s 1996 paper from his ori- ginal stated value of 4000. This has the effect of lowering the crossover frequency to the point where just three modes are present within the bandwidth of one mode as opposed to ten. This change is interesting in itself and no real justification is given. Again, we are left with only an implication that this may have a subjective relevance.

It can be argued that it is widely believed that when above this transition fre- quency (fc), we are listening within ‘diffuse sound-field’ conditions, and therefore

individual effects associated with discrete resonances are no longer perceived. It must be stressed here that this objective measure reveals the transition to a room where the frequency response can be considered statistical in nature. Schroeder’s papers make no reference to any subjective effects of reaching this frequency. Howe- ver, tracing the history of room acoustics, one can see that the use of the Schroeder Frequency has led to an ever increasing polarisation of modal and diffuse sound fields despite some warnings (e.g. Toole (2006)). As this gap widens, the assumption that

perception of audio falling within these two regions is also polarised seems to have proliferated. Evidence for this can be seen from attendance at major audio and acoustic conferences and lectures, but also within the literature. For example, many research papers use this crossover frequency as a limiting point for their investiga- tions into the effects of low frequency resonances. The work of Avis et al. (2007) which investigates the perception of room modes uses the Schroeder Frequency as the point of transition when forming binaural room models. In their ‘Room Sizing and Optimization’ paper, Cox et al. (2004) also state that the frequency range un- der investigation can be “guided by the Schroeder Frequency”. A further example of its use in this way can be seen in Blaszak (2007). Finally, Toole (2006) states the importance of the crossover region as a “real phenomenon” which needs to be better understood.

As the size of an enclosure increases, the Schroeder Frequency (fc) decreases.

Therefore, in large rooms such as concert halls,fc is typically very low, often below

the 20Hz threshold of our hearing. However, spaces such as control rooms, with typically small volumes (i.e. 100m3_{), are classified as having modal regions at fre-}

quencies not only above 20Hz, but well into the range of most musical situations (i.e. RT60= 1.28s,V = 75m3, fc = 261Hz - middle C).

Regardless of room size, the modal density naturally increases with frequency. Eventually many hundreds of modes exist within just a few Hertz. It is this increase in modal density that underpins the definition of the Schroeder Frequency. It is argued here that, just as we have seen that we should not rely on solely objective measures of modal spacing, we should not rely on them for density. Rather, we must ask the question: is there a threshold above which enough modes exist such that we no longer perceive any degradation? Is it in fact possible to determine a ‘subjective counterpart to the Schroeder Frequency?

To further emphasise a misunderstanding of the density parameter, it is noted that it is often either assumed or implied (e.g. in diagrams such as Figure 5.1) that as a large number of modes are concentrated in a given frequency range, as occurs with an increase in volume and/or frequency, the overall magnitude frequency response becomes ‘flatter’ and thus is commonly associated with better quality reproduction. We have seen from the spacing work that an addition of multiple modes may in some cases appear to smooth the response. With many modes sharing a similar bandwidth, any energy present through say, a bass note, will excite each of these modes, and dissipate through them. However, the previous chapter also revealed that the interactions between modes must not be considered simplistically. Phase interactions between closely spaced modes may in fact result in a highly irregular

Figure 5.1: A typical representation of the transition frequency, taken from Howard and Angus (2001)

frequency response and actually degrade the audio. Therefore, a higher density may in fact compound such problems.