Pitch/Tempo Relationships - Bruce SexauerSeptember 2002©All Rights Reserved
For several years, I have been expandingupon the idea which is the subject of this essay, and whichI 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.
Pitch & Tempo
The central idea is quite simple, and I willstate it now. The pitch of a musical note, and the tempo of amusical composition, are mathematically related, as both are describedby the same formula; pulses per time unit. In the case of pitch,it is describe in beats per second. An example is the standardof our musical universe; A = 440 beats per second. In the caseof tempo, the description is in beats per minute, much like thebeating heart. The pitch "A", can also be describedat twice as many, or half as many beats per second. These arecalled octaves, and can be generated at higher and lower intervalsfar beyond the limits of human hearing. So "A" can alsobe 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 forhealthy ears still in the realm of human hearing.
Both higher and lower octaves can continueto be generated by doubling or halving the bps value ofthe note, the pitch name, "A" in the example, will remainthe same. As the note becomes higher in pitch it will eventuallyexceed the human ears ability to hear it, and eventually therewill be a question as to whether or not it can be called soundat all, but this is not yet our concern. The lower octaves alsokeep being the same note value as they are divided by two,but are no longer directly heard below 55bps by most people. Hereare the next few pith descriptions in bps of extended lower octavesof "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 becomesunwieldy to describe "A" in parts of a second, as thereare other, larger, time units at hand. Let's try minutes, whichbeing composed of 60 seconds, means we can multiply the bps figureby 60 and get a beats per minute (bpm) rating. 1.71875 (60) is103.125 bpm, which could also be expressed by the integer/fraction103 1/8 bpm. This pitch falls into to the common range of tempofor western music, and I suggest that it is one example of thetempo of "A". An octave either way also falls into thestandard range of tempo. 206 14 would be in the bebop orbluegrass range, and 51 9/16 has been known to polish many a beltbuckle.
I therefore suggest that, whether it is usefulor not, it is certainly possible to play music both in the keyof "A", and in the tempo of "A". This is theoriginal notion, which came to me something like twenty yearsago, and which I have not yet seen expressed by any other source.Though it seems unlikely that no one else has ever thought ofthis, it does appear that no one has popularized the idea. Morethat one person with whom I have endeavored to share the concepthas commented, "why bother". As time has past I havefound myself rising to this question, and have come up with severalextensions of the idea which may give it some practicalapplication.
If the tune in the key of "A" isalso played in the Tempo of "A", then we might say thatthey 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 pieceby carefully choosing the tempo. For instance the tempo couldbe 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 thatif it can't be heard, it doesn't matter. While it is certainlyimportant to the musician to be able to hear whether or not pitchesare in harmony, the casual listener is more likely to judge themusic by the feeling that is generated, and this is the realmwithin which I suggest the merit of the concept be judged. Certainlymusicians are not (yet) trained to be sensitive to pitch/tempo, but I do not believe this precludes the possibility of developingthe ability.
When I am about to play a song, I like to takea moment to listen to the tonic pitch of the tune, and mentallydivide the octaves down to the point where I can hear the beats,and then pick my tempo out with my body. I certainly feellike this method is working, and improving with practice. I havenot got a tool to test it, and even when mentally hearingcommon intervals, I am wont to mistake fourths for fifths andfifths 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 jobyet, with practice.
Had I such a tool, which might be a .000 decimalreadout 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/tempoharmony. I wonder if different musical genre's might soundmore natural with specific root tempos. Dominant for blues? Fourthfor rock? Sixth for swing? Just a thought.
Rock Steady Time
Once a person had been trained to actuallyfeel the relationship between pitch and tempo, the common problemof speeding up or slowing down of the tempo while playing musiccould become a thing of the past, at least for players of talentin that area. Speeding up would be the same sort of musicaldefect as playing or singing sharp. Many players have the problemof speeding up, and the normal fix is to spend a lot of time witha metronome, which is monotonous, and not really a direct addressto the problem. We humans are very subjectively located in thetime continuum, and it often seems to be asking the impossibleto keep immutably perfect time. Just a few players in my experiencereally excel at it. What is their secret? Maybe they have a ringingsound in their ears? Or maybe they are listening to the bass,which is already divided down to a pitch closely resembling atempo.
Bruce Sexauer September 2002©AllRights Reserved
Bruce Sexauer is a luthier, currently engagedin building guitars under his name & for Schoenberg Guitars.He may be reached via e-mail at firstname.lastname@example.org.His website is http://www.sexauerluthier.com.