Last week we started to look at the problem of azimuth alignment on olden compact cassette tape drives, and how this misalignment affects the sound. Let’s now have a look at how azimuth affects the frequencies.

OK, first, let’s model the trapezoidal filter. Let’s suppose, as a simplifying assumption, that the track is entirely under the head (that is, the head is longer than the track is wide, and is never so tilted that it’s entirely in the track). Or, in other words, we have this situation:

We suppose, initially, that we don’t really have control on the head’s azimuth. It may be mostly correct, but it may be wildly off. If it’s wildly off, the resulting projection of the head (in gray) gives the trapezoidal filter (orange):

Of course, we *want* the slant to be exactly zero, the head being perfectly vertical |. We see that if we vary the angle continuously, the window also changes quite gradually:

In the figure, the quarter circle indicates that the with of the head remains constant (something that might not be that clear in the diagram because 1) it’s hard in general comparing rectangle widths then rectangles have different orientations and 2) *Mathematica* makes its difficult to maintain correct aspect ratio in figures). But as you see, the center of the circle is the lower-right corner of the head against the track and the left side “rolls” on the circle in the right place, at exactly a head’s width from the other corner. So we’re sure the model computes the right projection.

Let’s suppose the azimuth is off by 5°, which is, methinks, rather large. Let’s see what happens. First, the filters looks like this:

In frequency domain, we have:

In green, the box filter, corresponding to a perfectly aligned head, in red, the head tilted 5°. Both are shifted relative to one another so that we can see something. In frequency space, both peaks align. What are we seeing? Superficially, we might think they look the same, but with a closer look, we see that even if the three main packs are of about the same amplitude, the red squiggle goes to zero much faster as we get further from the main peak. That means it has a lower high-frequency response than the green one. So, people claiming azimuth is important are right. It is. But how much is it, then?

Let’s have a look at a slant of 20°. That’s probably not as bad as it could get, but at this point I’d think your cassette deck is pretty much scrap.

The frequency analysis of the filter makes it even clearer: the red squiggle is *much* more compact than the green one, and therefore it responds even less to high frequencies.

So that settles the question about azimuth.

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But, how much should we worry about it? Well, it depends on how much that thing gets out of whack when you use the deck. If it’s a few degrees, say, 2 or less, that’s basically within the noise of the device. At 1° we get basically indistinguishable frequency responses. Lo!:

I can’t say right now that it doesn’t matter at all. First, we should compare the differences in dB knowing that a typical noise floor for those things is about -60 to -72dB (or in the range of 10 to 12 bits). If the difference in alignment causes a frequency shift of less than -60dB, then it’s harmless. Second, the width of the gap plays a lot in this. As its width goes to zero, *any* angle will cause a trapezoid window. The tolerable angle is therefore function of the gap width and the gap length. Will need more experimenting.