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Pitch/Tempo Relationships - Bruce Sexauer September 2002

Lying in a bed in the ICU following my first heart attack, I had some time to think about my contributions to society, and what might pass for a legacy. I like to think I have made some great guitar over the years, maybe a few surpassingly so, but while I may well be a master of the form, I don’t see what I’ve added to it. In music, I have written some good songs, never released on a major label, and my playing, while competent, is unlikely to achieve the levels I strive for. I have, however, for several years been  expanding upon the idea  which is the subject of this essay, and which I have come to believe may be original to my thought, and which  I increasingly realize is irrefutably true, and may actually  prove to be of some utility.

The central idea is quite simple, and I will state it now. The pitch of a musical note, and the tempo of a musical composition, are mathematically related, as both are described by the same formula; pulses per time unit. In the case of pitch, it is describe in beats per second. An example is the standard of our musical universe; A = 440 beats per second. In the case of tempo, the description is in beats per minute, much like the beating heart. The pitch “A”, can also be described at twice as many, or half as many beats per second. These are called octaves, and can be generated at higher and lower intervals far beyond the limits of human hearing. So “A” can also be described as A = 880 beats per second (bps), or A = 1760 bps, which are higher octaves, or it can be described as A = 220 bps, or A = 110 bps or A = 55 bps, which are lower octaves, and for healthy ears still in the realm of human hearing.

Both higher and lower octaves can continue to be generated by  doubling or halving the bps value of the note, the pitch name, “A” in the example, will remain the same. As the note becomes higher in pitch it will eventually exceed the human ears ability to hear it, and eventually there will be a question as to whether or not it can be called sound at all, but this is not yet our concern. The lower octaves also keep  being the same note value as they are divided by two, but are no longer directly heard below 55bps by most people. Here are the next few pith descriptions in bps of extended lower octaves of “A”: A = 27.5 bps, A = 13.75, A = 6.875 bps, A = 3.4375 bps, A = 1.71875 bps. At around this point, it becomes unwieldy to describe “A” in parts of s second, as there are other, larger, time units at hand. Let’s try minutes, which being composed of 60 seconds, means we can multiply the bps figure by 60 and get a beats per minute (bpm) rating. 1.71875 (60) is 103.125 bpm, which could also be expressed by the integer/fraction 103 1/8 bpm. This pitch falls into to the common range of tempo for western music, and I suggest that it is one example of the tempo of “A”. An octave either way also falls into the standard range of tempo.  206 ¼ would be in the bebop or bluegrass range, and 51 9/16 has been known to polish many a belt buckle.

I therefore suggest that, whether it is useful or not, it is certainly possible to play music both in the key of “A”, and in the tempo of “A”. This is the original notion, which came to me something like twenty years ago, and which I have not yet seen expressed by any other source. Though it seems unlikely that no one else has ever thought of this, it does appear that no one has popularized the idea. More that one person with whom I have endeavored to share the concept has commented, “why bother”. As time has past I have found myself rising to this question, and have come up with several extensions of the idea which  may give it some practical application.


If the tune in the key of “A” is also played in the Tempo of “A”, then we might say that they are “in tune”.  And if the key is “A” and the tempo is slightly other than “A“, say 104 bpm, then we might say that the pitch/tempo is “out of tune”. Assuming then that we accept the notion of being in tune, or not, it seems it is possible to harmonize the key of the musical piece by carefully choosing the tempo. For instance the tempo could be based on the harmonic 5th of “A”, which is “E” (E = 77.26 or 154.52), or we could harmonize the key as the 5th  of the tempo; the key of “A” is the 5th of the tempo “D” (68.83 or 137.66). My reader may have thought that if it can’t be heard, it doesn’t matter. While it is certainly important to the musician to be able to hear whether or not pitches are in harmony, the casual listener is more likely to judge the music by the feeling that is generated, and this is the realm within which I suggest the merit of the concept be judged. Certainly musicians are not (yet) trained to be sensitive to pitch/tempo , but I do not believe this precludes the possibility of developing the ability.
When I am about to play a song, I like to take a moment to listen to the tonic pitch of the tune, and mentally divide the octaves down to the point where I can hear the beats, and then pick my tempo out with my body.  I certainly feel like this method is working, and improving with practice. I have not got a tool to test it, and  even when mentally hearing common intervals, I am wont to mistake fourths for fifths and fifths for octaves, but it seems I can pull tempos out of pitch, and better ears than mine ought to be able to do a better job yet, with practice.
Had I such a tool, which might be a .000 decimal readout metronome with a tap pad for recording the tempo input,  it would be interesting to make a study of popular tunes that “just feel right”, and see if they fall into pitch/tempo harmony. I wonder if different musical genre’s  might sound more natural with specific root tempos. Dominant for blues? Fourth for rock? Sixth for swing? Just a thought.

Rock Steady Time

Once a person had been trained to actually feel the relationship between pitch a tempo, the common problem of speeding up or slowing down of the tempo while playing music could become a thing of the past, at least for players of talent in that area.  Speeding up would be the same sort of musical defect as playing or singing sharp. Many players have the problem of speeding up, and the normal fix is to spend a lot of time with a metronome, which is monotonous, and not really a direct address to the problem. We humans are very subjectively located in the time continuum, and it often seems to be asking the impossible to keep immutably perfect time. Just a few players in my experience really excel at it. What is their secret? Maybe they have a ringing sound in their ears? Or maybe they are listening to the bass, which is already divided down to a pitch closely resembling a tempo. . .

Bruce Sexauer 8/20/2002