Click here to purchase the entire book in PDF format. Horizontal plane Cardioids Unless you only record pop music and you never use your imagination, all of the graphs shown above don't really apply to what happens when you're recording. This is because, usually your microphone isn't pointing directly forward... you usually have more than one microphone and they're usually pointing slightly to the left or right of forward, depending on your configuration. Therefore, we have to think about what happens to the sensitivity pattern when you rotate your microphone. Figure 10.46 shows the sensitivity pattern of a cardioid microphone that is pointing 45$^\circ$ to the right. Notice that this plot essentially looks exactly the same as Figure 6.100, it's just been pushed to the side a little bit. Figure 10.46: Cartesian plot of the absolute value of the sensitivity pattern of a cardioid microphone on a decibel scale turned 45$^\circ$ to the right. Now let's consider the case of a pair of coincident cardioid microphones pointed in different directions. Figure 10.48 shows the plots of two polar patterns for cardioid microphones point at -45$^\circ$ and 45$^\circ$, giving us an included angle (the angle subtended by the microphones) of 90$^\circ$ as is shown in Figure 10.47. Figure 10.47: Diagram of two microphones with an included angle of 90$^\circ$. Figure 10.48: Cartesian plot of the absolute value of the sensitivity patterns of two cardioid microphones on a decibel scale turned $\pm$45$^\circ$. Figure 10.48 gives us two important pieces of information about how a pair of cardioid microphones with an included angle of 90$^\circ$ will behave. Firstly, let's look at the vertical difference between the two curves. Since this plot essentially shows us the output level of each microphone for a given angle, then the distance between the two plots for that angle will tell us the interchannel amplitude difference. For example, at an angle of incidence (to the pair) of 0$^\circ$, the two plots intersect and therefore the microphones have the same output level, meaning that there is an amplitude difference of 0 dB. This is also true at 180$^\circ$, despite the fact that the actual output levels are different than they are at 0$^\circ$ - remember, we're looking at the difference between the two channels and ignoring their individual output levels. In order to calculate this, we have to find the ratio (because we're thinking in decibels) between the sensitivities of the two microphones for all angles of incidence. This is done using Equation 10.2. $$\Delta Amp. = 20 * \log_{10} \left( \frac{S_1}{S_2} \right)$$ (11.2) where $$S_n = P_n + G_n * \cos \left( \alpha + \Omega_n \right)$$ (11.3) where $\Omega$ is the angle of rotation of the microphone in the horizontal plane. If we plot this difference for a pair of cardioids pointing at $\pm$45$^\circ$, the result will look like Figure 10.49. Notice that we do indeed have a $\Delta Amp.$ of 0 dB at 0$^\circ$ and 180$^\circ$. Also note that the graph has positive and negative values on the right and left respectively. This is because we're comparing the output of one microphone with the other, therefore, when the values are positive, the right microphone is louder than the left. Negative numbers indicate that the left is louder than the right. Figure 10.49: Interchannel amplitude differences for a pair of coincident cardioid microphones in the horizontal plane with an included angle of 90$^\circ$. Now, if you're using this configuration for a two-channel recording, if you like, you can feel free to try and make a leap from here back to Table 10.5 to make some conclusion about where things are going to wind up being located in the reproduced sound stage. For example, if you believe that a signal with a $\Delta Amp.$ of 2.5 dB results in a phantom image location of 10$^\circ$, then you can go to the graph in Figure 10.49, find out where the graph crosses a $\Delta Amp.$ of 2.5 dB then find the corresponding angle of incidence. This then tells you that an instrument located at that angle to the microphone pair will show up at 10$^\circ$ off-centre between the loudspeakers. This is, of course, if you're that one person in the world for whom Table 10.5 holds true (meaning you're probably also 172.3 cm tall, you have 2.6 children and two thirds of a dog, you live in Boise, Idaho and that your life is exceedingly... well... average...) We can do this for any included angle between the microphone pair, from 0$^\circ$ through to 180$^\circ$. There's no point in going higher than 180$^\circ$ because we'll just get a mirror image. For example, the response for an included angle of 190$^\circ$ is exactly the same as that for 170$^\circ$, just pointing towards the rear of the pair instead of the front. Figure 10.50: Interchannel amplitude differences for a pair of coincident cardioid microphones in the horizontal plane with an included angle of 0$^\circ$. Of course, if we actually do the calculation for an included angle of 0$^\circ$, we're only going to find out that the sensitivities of the two microphones are matched, and therefore the $\Delta Amp.$ is 0 dB at all angles of incidence as is shown in Figure 10.50. This is true regardless of microphone polar pattern. Figure 10.51: Interchannel amplitude differences for a pair of coincident cardioid microphones in the horizontal plane with an included angle of 45$^\circ$. Figure 10.52: Interchannel amplitude differences for a pair of coincident cardioid microphones in the horizontal plane with an included angle of 135$^\circ$. Figure 10.53: Interchannel amplitude differences for a pair of coincident cardioid microphones in the horizontal plane with an included angle of 180$^\circ$. Note that, in the case of all included angles except for 0$^\circ$, the plot of $\Delta Amp.$ for cardioids goes to $\pm \infty$ dB because there will be one angle where one of the microphones has no output and because, on a decibel scale, something is infinitely louder than nothing. Also note that, in the case of cardioids, every value of $\Delta Amp.$ is duplicated at another angle of incidence. For example, in the case of an included angle of 90$^\circ$, $\Delta Amp.$ = +10 dB at angles of incidence of approximately 70$^\circ$ and 170$^\circ$. This means that sound sources at these two angles of incidence to the microphone pair will wind up in the same location in the reproduced sound stage. Remember, however, that we don't know the relative levels of these two sources because, all we know is their difference. It's quite probable that one of these two locations will result in a signal that is much louder than the other, but that isn't our concern just yet. Also note that there is one included angle (of 180$^\circ$) that results in a response characteristic that is symmetrical (at least in each polarity) around the $\infty$ dB point. Figure 10.54: Contour plot showing the difference in sensitivity in dB between two coincident cardioid microphones with included angles of 0$^\circ$ to 180$^\circ$, angles of rotation from -180$^\circ$ to 180$^\circ$ and a 0$^\circ$ angle of elevation. Note that Figure 10.49 is a horizontal ``slice'' of this contour plot where the included angle is 90$^\circ$. Subcardioids Looking back at Figures 6.102 and 6.103 we can see that the lowest sensitivity from a subcardioid microphone is 6 dB below its on-axis sensitivity. As a result, unlike a pair of cardioid microphones, the $\Delta Amp.$ of a pair of subcardioid microphones cannot exceed the $\pm$6 dB window, regardless of included angle or angle of incidence. Figure 10.55: Interchannel amplitude differences for a pair of coincident subcardioid microphones in the horizontal plane with an included angle of 45$^\circ$. Figure 10.56: Interchannel amplitude differences for a pair of coincident subcardioid microphones in the horizontal plane with an included angle of 90$^\circ$. As can be seen in Figures 10.55 through 10.58, a pair of subcardioid microphones does have one characteristic in common with a pair of cardioid microphones in that there are two angles of incidence for every value of $\Delta Amp.$ Figure 10.57: Interchannel amplitude differences for a pair of coincident subcardioid microphones with an included angle of 135$^\circ$ in the horizontal plane. Figure 10.58: Interchannel amplitude differences for a pair of coincident subcardioid microphones with an included angle of 180$^\circ$ in the horizontal plane. There is another aspect of subcardioid microphone pairs that should be remembered. As has already been discussed, in the case of subcardioids, it is impossible to have a value of $\Delta Amp.$ outside the $\pm6$ dB window. Armed with this knowledge, take a look back at Tables 10.5, 10.6 and 10.7. You'll note that in all cases of pair-wise panning, it takes more than 6 dB to swing a phantom image all the way into one of the two loudspeakers. Consequently, it is safe to say that, in the particular case of coincident subcardioid microphones, all of the the sound stage will be confined to a width smaller than the angle subtended by the two loudspeakers reproducing the two signals. As a result, if you want an image that is at least as wide as the loudspeaker aperture, you'll have to introduce some time differences between the microphone outputs by separating them a little. This will be discussed further below. Bidirectionals The discussion of pairs of bidirectional microphones has to start with a reminder of two characteristics of their polar pattern. Firstly, the negative polarity of the rear lobe can never be forgotten. Secondly, as we will see below, it is significant to remember that, in the horizontal plane, this microphone has two angles which have a sensitivity of 0 (or $-\infty$ dB). Figure 10.59 shows the $\Delta Amp.$ of the absolute value of a pair of bidirectional microphones with an included angle of 90$^\circ$. In this case, the absolute value of the microphone's sensitivity is used in order to avoid errors when calculating the logarithm of a negative number. This negative is the result of angles of incidence which produce sensitivities of opposite polarity in the two microphones. For example, in this particular case, at an angle of incidence of +90$^\circ$ (to the right), the right microphone sensitivity is positive while the left one is negative. However, it must be remembered that the absolute value of these two sensitivities are identical at this location. A number of significant characteristics can be seen in Figure 10.59. Firstly, note that there are now four angles of incidence where the $\Delta Amp.$ reaches $-\infty$ dB. This is due to the fact that, in the horizontal plane, bidirectional microphones have two null points. Secondly, note that the pattern in this plot is symmetrical around these infinite peaks, just as was the case with a pair of cardioid microphones at 180$^\circ$, apparently resulting in four angles of incidence which result in sound sources located at the same phantom image location. This, however, is not the case due to polarity differences. For example, although sound sources located at 30$^\circ$ and 60$^\circ$ (symmetrical around the 45$^\circ$ location) appear to result in identical sensitivities, the 30$^\circ$ location produces similar polarity signals whereas the 60$^\circ$ location produces opposite polarity signals. Finally, it is significant to note that the response of the microphone pair is symmetrical front-to-back, with a left/right and a polarity inversion. For example, a sound source at +10$^\circ$ results in the same $\Delta Amp.$ as a sound source at -170$^\circ$, however, the rear source will be have a negative polarity in both channels. Similarly, a sound source at +60$^\circ$ will have the same $\Delta Amp.$ as a sound source at -120$^\circ$, however the former will be positive in the ``right'' channel and negative in the ``left'' whereas the opposite is the case for the latter. Figure 10.59: Interchannel amplitude differences for a pair of coincident bidirectional microphones in the horizontal plane with an included angle of 90$^\circ$. Various other included angles for a pair of bidirectional microphones results in a similar pattern as was seen in Figure 10.59, with a ``skewing'' of the response curves. This can be seen in Figures 10.60 and 10.61. It is also important to note that Figures 10.60 and 10.61 are mirror images of each other. This, however does not simply mean that the pair can be considered to be changed from ``pointing'' from the front to the side in this case. This is again due to the polarity differences between the two channels for specific various angles of incidence. Figure 10.60: Interchannel amplitude differences for a pair of coincident bidirectional microphones in the horizontal plane with an included angle of 45$^\circ$. Figure 10.61: Interchannel amplitude differences for a pair of coincident bidirectional microphones in the horizontal plane with an included angle of 135$^\circ$. There is one final configuration worth noting in the specific case of bidirectional microphones; when the included angle is 180$^\circ$. As can be seen in Figure 10.62, this results in the absolute values of the sensitivities being matched at all angles of incidence. Remember however, that in this particular case, this means that the two channels are exactly matched and opposite in polarity - theoretically, you wind up with exactly the same signal as you would with one microphone split to two channels on the mixing console and the polarity switch (frequently incorrectly referred to as the ``phase'' switch) engaged on one of the two channels. Figure 10.62: Interchannel amplitude differences for a pair of coincident bidirectional microphones in the horizontal plane with an included angle of 180$^\circ$. Figure 10.63: Contour plot showing the difference in sensitivity in dB between two coincident bidirectional microphones with included angles of 0$^\circ$ to 180$^\circ$, angles of rotation from -180$^\circ$ to 180$^\circ$ and a 0$^\circ$ angle of elevation. Hypercardioids Not surprisingly, the response of a pair of hypercardioid microphones looks like a hybrid of the bidirectional and cardioid pairs. As can be seen in Figure 10.64, there are four infinite peaks in the value of $\Delta Amp.$, similar to bidirectional pairs, however the slope of the peaks are skewed further left and right as in the case of cardioids. Figure 10.64: Interchannel amplitude differences for a pair of coincident hypercardioid microphones in the horizontal plane with an included angle of 90$^\circ$. Again, similar to the case of bidirectional microphones, changing the included angle of the hypercardioids results in a further skewing of the response curve to one side or the other as can be seen in Figures 10.65 and 10.66. Figure 10.65: Interchannel amplitude differences for a pair of coincident hypercardioid microphones in the horizontal plane with an included angle of 45$^\circ$. Figure 10.66: Interchannel amplitude differences for a pair of coincident hypercardioid microphones in the horizontal plane with an included angle of 135$^\circ$. Figure 10.67 shows the interesting case of hypercardioid microphones with an included angle of 180$^\circ$. In this case the maximum sensitivity point in the rear lobe of each microphone is perfectly aligned with the maximum sensitivity point in the other microphone's front lobe. However, since the rear lobe has a sensitivity with an absolute value that is 6 dB lower than the front lobe, the value of $\Delta Amp.$ remains outside the $\pm6$ dB window for the larger part of the 360$^\circ$ rotation. Figure 10.67: Interchannel amplitude differences for a pair of coincident hypercardioid microphones in the horizontal plane with an included angle of 180$^\circ$. Figure 10.68: Contour plot showing the difference in sensitivity in dB between two coincident hypercardioid microphones with included angles of 0$^\circ$ to 180$^\circ$, angles of rotation from -180$^\circ$ to 180$^\circ$ and a 0$^\circ$ angle of elevation. Spaced omnidirectionals In the case of spaced omnidirectional microphones, it is commonly assumed that the distance to the sound source is adequate to ensure that the impinging sound can be considered to be a plane wave. In addition, it is also assumed that there is no difference in signal levels due to differences in propagation distance to the transducers. In reality, for widely spaced microphones and/or for sound sources closely located to any microphone, neither of these assumptions is correct, however they will be used for this discussion. The difference in time of arrival of a sound at two spaced microphones is dependent both on the separation of the transducers $d$ and the angle of rotation around the pair $\vartheta$. Figure 10.69: Spaced omnidirectional microphones showing the microphone separation $d$, the angle of rotation $\vartheta$ and the resulting extra distance $D$ to the further microphone. The additional distance, $D$, travelled by the sound wave to the further of the two microphones, shown in Figure 10.69, can be calculated using Equation 10.4. $$D = d \sin \vartheta$$ (11.4) where $d$ is the distance between the microphone capsules in cm. The additional time $\Delta Time$ required for the sound to travel this distance is calculated using Equation 10.5. $$\Delta Time = \frac{10 D}{c}$$ (11.5) where $\Delta Time$ is the interchannel time difference in ms, $\vartheta$ is the angle of incidence of the sound source to the pair, and $c$ is the speed of sound in m/s. This time of arrival difference is plotted for various microphone separations in Figures 10.70 through 10.73. Note that the general curve formed by this calculation is a simple sine wave, scaled by the separation between the microphones. Also note that the value of $\Delta Time$ is 0 ms for sound sources located at 0$^\circ$ and 180$^\circ$ and a maximum for sound sources at 90$^\circ$ and -90$^\circ$. As was mentioned early in the section on interchannel amplitude differences between coincident directional microphones, one might be tempted to draw conclusions and predictions regarding image locations based on the values of $\Delta Time$ and the values listed in the tables and figures in Section 10.3.1. Again, one shouldn't be hasty in this conclusion unless you consider your listeners to be average. Figure 10.70: Interchannel time differences for a pair of spaced microphones in the horizontal plane with a separation of 0 cm. Figure 10.71: Interchannel time differences for a pair of spaced microphones in the horizontal plane with a separation of 15 cm. Figure 10.72: Interchannel time differences for a pair of spaced microphones in the horizontal plane with a separation of 30 cm. Figure 10.73: Interchannel time differences for a pair of spaced microphones in the horizontal plane with a separation of 45 cm. Figure 10.74: Interchannel time differences vs. microphone separation for a pair of spaced microphones in the horizontal plane.