Wednesday, September 27, 2006

Harmonic Implications of the Overtone Series, Part VI

After four installments dealing with the harmonic implications of the overtone series, it only took one to outline the elementally simple contrapuntal implications the series prescribes. I could have spent a lot more time on the subject of counterpoint, but I am leaving town this weekend and I wish to bring this series to a conclusion before I depart. Besides, the elemental underlying laws of contrapuntal movement are enough to know, as learning to actually compose counterpont will almost always lead the student toward more traditional texts. Knowing the underlying fundamental laws governing contrapuntal motion will provide one with a good "BS Detector" so to speak.

There is one last musical element that the series has implications concerning, and that is rhythm (I have not worked through to any conclusions concerning melodic implications of the series as of yet, so that will have to await further inquisitions and inspirations).

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 which exists between the lowest perception of pitch and the fastest perception of rhythm. Further lowering and the former pitch will reemerge into the audio realm as a regular pulse, or a simple rhythmic continuity.

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. Each interval of the overtone series 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.

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 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 sapce. 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.

The perfect fifth's resultant is evidenced in Waltzes, and has been used in popular music as far back as the twelfth 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.


Well, that ends this series for the time being. I think that with a couple more revisions to the presentation order and the eye of a good editor, this will soon be in a formally publishable format. Atleast I have it web archived for copyright now.




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