Musical Implications of the Harmonic Overtone Series: Chapter VIII
Rhythmic Implications of the Series
It might seem counter-intuitive to expect that sound would have rhythmic implications, but it isn't: If you take any given pitch and lower it to under about 15-20Hz (Cycles per second, remember), it will dissappear from human hearing and enter the auditory void - our "deaf spot" which in some ways corresponds to the visual blind spot we all have but seldom notice - which exists between the lowest frequency perceptable as pitch and the fastest frequency perceptable as rhythm. Further lowering and the former pitch will reemerge into auditory perception as a regular pulse, or a simple rhythmic continuity. While in the auditory void between pitch an rhythm, the cyclical repetition is perceptable to touch as vibration.
Now, if you go through the same process with two pitches involved in a harmonic relationship, when they reemerge into hearing, it will be in the form of a rhythmic palindrome instead of an even repeating pulse. This rythmic effect is the result of the interference between the two wave periodicities, exactly as harmony itself is the result of periodic interference between waveforms which have a certain ratio relationship. Joseph Schillinger brought these to my attention, and he also provided the notational methodology for displaying them in visual terms that musicians can understand and perform.
It is interesting to me to be able to see harmonic relationships as rhythmic resultants of interference, but I must admit that I have not employed them in composition all that much. Instead, I am developing a rhythmic methodology which allows harmonic and contrapuntal implications present in the music itself to create rhythmic vartiety. Schillinger insisted that these resultants could be applied to virtually every aspect of music, and if you are seriously interested in his thoughts on the matter, I'd suggest you read his Theory of Rhythm from Volume I of The Schillinger System of Musical Composition for yourself. Frankly, I think there is something rather forced and less than organic about his theorietical methodologies, and I believe that is the result of his continually distancing himself from the implications of the harmonic overtone series, while simultaneously drawing a lot of ideas from it. Schillinger was quite a contradictory individual.
Each interval of the overtone series - its ratio - will result in a unique rhythmic palindrome, and I have demonstrated that for the seven consonant intervals below.
The process for figuring this out in musical notation is mechanically simple.
The first step is to multiply the two terms together to get the number of indivisible units: For the octave that is 2 * 1= 2. Then you must decide what base unit to use: I chose the quarter note. So, in the top space of the upper stave in each system you will see the result of the multiplication process in quarter notes.
The second step is to put the minor generator down below the indivisible unit total: For the octave the minor generator is two quarter notes (I define the minor generator in music notation as the smaller rhythmic unit - in this case the quarter note - and not the smaller number of the ratio, which would be one in this instance: If memory serves, Schillinger does it the other way around).
The third step is to put the major generator underneath the minor generator, and for the octave the major generator is a half note.
Finally, you get the rhythmic resultant of interference between the two terms by dropping plumblines at each attack: For the octave that simply results in an accented quarter note followed by an unaccented quarter note. The first attack in each resultant is accented because that is the place - and, the only place - in the resultant where the two periodicities coincide or have simultaneous attacks. The resultants are on the bottom staves of each system.
The ratio of the perfect fifth is 3:2, and 3 * 2= 6, so there are six quarter notes in the top space of the second example. The minor generator is three, so that comes out to three half notes in the second space. Then, the major generator is two, which comes out to a pair of dotted half notes, as you see in the bottom space. Finally, the rhythmic resultant is an accented half note (Where the two terms attack simultaneously), followed by a pair of quarter notes, and finally an unaccented half note, or 2 + 1 + 1 + 2.
As you can see, as you proceed up the series the rhythmic resultants become more complex. If the ratios are superparticular (The terms differing by one), or if the ratio's terms differ by an odd number, the rhythmic pallindrome will be divisible through it's axis of symmetry, as is the case with every example among the consonances except for the major sixth: It has as its axis of symmetry a dotted half note, as its terms differ by an even number (Two). The minor sixth is divisible through its axis of symmetry because its terms differ by an odd number (Three).
The resultant for the octave is the most common in all of music, and is represented all too well by the Rock and Roll "backbeat" that is so mind-numbingly ubiquitous in popular music.
The perfect fifth's resultant is evidenced in Waltzes, and has been used in popular forms as far back as the twelfth or thirteenth century. Using the first half repeated as an antecedant and the second half in the penultimate measure of a phrase as the consequent, you get this for an eight measure phrase:
||:2 + 1|2 + 1|2 + 1|2 + 1|2 + 1|2 + 1|1 + 2|3:||
As I said, this rhythmic resultant of interference as implied by the overtone series was first intuited hundreds of years ago. The rest remain little explored, and almost nothing has been done with the more complex concords. Personally, I do not start out pieces with a rhythmic conception first (At least, that has been the case so far), so I'm probably not the guy who is best equipped to explore these implications of the series. But, they are there, just awaiting a fertile mind to figure out how to effectively apply them.
I should also note that resultants can be calculated for ratios with more than two terms. A major triad would be the resultant for 6:5:4, for example, and the overtone chord itself would be the four terms of 7:6:5:4. I will add a second page of resultants to demonstrate this for all of the triads and tetrads in the final book version of this series.
Oh sure; happens every day.