2011.2.22 Tempo vs. pitch class
Musical pitches can be quantified by frequency: for instance, the canonical pitch is the A above middle C at 440 Hz. But tempo is also a frequency measure, just expressed in beats per minute rather than hertz. A pitch class is the collection of sound frequencies that are all related to each other by powers of 2 — i.e. they all have the same "pitch name" and differ by octaves. So "A" is a pitch class, and it includes 440 Hz (A4), 220 Hz (A3), 110 Hz (A2), 880 Hz (A5), and so on.
Meanwhile, tempos exhibit easy halftime or double-time relationships as well. For instance, dubstep usually runs at 140 bpm, but it frequently has an implied pulse of 70 bpm because of a halftime frequency of snare hits (a snare on or around the third beat of each measure rather than on the second and fourth beats). Such relationships are actually precisely the same as that between two pitches one octave apart from one another: two frequencies, one of which is half the frequency of the other. So 70 bpm can be said to be "one octave below" 140 bpm.
Furthermore, if you go low enough in pitch, you finally arrive at the frequency range more commonly associated with tempos, because the frequency is easily separated into discrete beats and is much too low to hear as a tone. For instance, A-4, that is, eight octaves below A4 (at 440 Hz), has a frequency of 440 ÷ 28 = 1.71875 Hz, i.e. more than one beat per second but less than two. If we multiply this by 60 to find beats per minute, we arrive at 103.125 bpm. So a tempo of 103.125 bpm is the very same frequency as A-4. Similarly, any tempo can be expressed as a pitch (adding or subtracting fractions of a semitone as necessary). If we place the twelve semitone pitch classes around a circle, we can place tempos on the same circle according to their pitch. Voilà:
I've shown tempos from 55 bpm to 195 bpm by intervals of 5 bpm here, and as you can see, such a wide range causes octave overlaps, for instance 140 and 70 bpm. I've also given tempo equivalents within that range for the semitone pitch classes. What's intriguing to me in looking around this chart is to place the ranges of various genres of electronic and dance music over it and to see how they correspond to the pitch classes. Take hip-hop, for instance: it can easily range from around 75 (the area around E) up to over 100 (the area around G#) and there are much slower and faster examples — in other words, it covers more than half the octave circle. Meanwhile, house mostly ranges from about 115 to about 135 bpm, i.e. the B to C# ranges — so its range is quite a bit less. Drum and bass has a range of about 160 to 190 bpm. Note that this is a larger bpm range than the one I gave for hip-hop (30 vs. around 25), but the d&b range in pitches is from the F area to the G area — a strict subset of the hip-hop range, one octave higher. Thus this circle is a useful illustration of the proportional relationships among tempos and tempo ranges.
One result I find quite interesting is that 120 bpm is close to C. I think of 120 bpm as being pretty much the most neutral tempo — the ideal moderato, so that any tempo below 120 seems "slow" to some degree and any above 120 "fast" — while C can be considered the most basic pitch, being the root of the diatonic major scale. That 120 is near C makes the tempo-pitch correspondences around the whole circle much more intuitive for me than they would otherwise be.
I also find it interesting to divide up the circle into even intervals to look at relationships among various tempos. For instance, if we take the quarter-circle B to D, i.e. around 116–137 bpm, we happen to cover the most common tempos of house and techno. The next quadrant, D to F range, about 138–163, covers the core of the "hardcore continuum" (breakbeat hardcore, oldschool jungle, garage, grime, dubstep). The next one up from that, F to G# or about 164–194, covers drum and bass pretty closely, while if we look at the corresponding octave below, about 82–97, we get the core hip-hop tempo range. Finally, going up from that lower-octave quadrant to the next one around, G# to B, or around 98–115, we get what could perhaps be considered the core of disco tempos, overlapped on the lower end by hip-hop and on the upper end by house.
After a number of such thought experiments, it occurred to me that there are a few positions on the circle that seem to be rather sparsely populated by electronic music, at least the genres I'm familiar with. For instance, the Eb–E area: even though I put it in the hardcore continuum range, the only time the 145–155 range has been much used by the hardcore continuum is when jungle was first speeding up from breakbeat hardcore tempos (140 and below) to modern drum and bass tempos (160 and above). Meanwhile, the A area (about 100–105 bpm) seems a little too fast for most hip-hop to interest itself in and decidedly slow for house music and even for disco (though there is plenty of disco in that area). It's interesting to me that these two ranges happen to be nearly halfway around the circle from one another. The relative sparsity of settlement in these areas piques a latent frontier spirit in me and makes me want to try my hand at producing music at such tempos, perhaps to bridge between seemingly disparate genres at tempos below and above them.