The Levels of Rhythm in Music
As I prepare to make the final revisions to the first edition of The Musical Implications of the Harmonic Overtone Series (The editor did a great job cleaning up my linguistic crudities) I want to develop my thoughts about the rhythmic and formal implications of the series a bit more. Since these areas are the newest ones to come under my scrutiny, I am still evolving in this area. In fact, the musical ramifications of employing what the series does not imply as most perfect or natural is also something I'm beginning to make some progress on as well. Part of my low rate of posts over the past months has had to do with my working this stuff out in my head.
Pitch to rhythm form a natural continuum: It is only the human senses that divide them up, as there is an auditory deaf spot below the lowest sound percieved as pitch and the fastest beat percieved as rhythm. Within this auditory void, however, the frequency can still be percived through the sense of touch as vibration. The phenomenon we call rhythm is also manifested on at least three levels in music: The local level, which is the surface rhythm we usually think of as rhythm proper; the regional level, which is what is taught as harmonic rhythm but which can also be though of as functional rhythm; and finally, the global level, which are the various thematic periodicities that make up what we call form in music.
The first two phenomena are generally well understood by educated musicians, but the formal implications of the series are something new (So far as I have been able to determine). Nature defines what musical proportions are by the ratios in the series, so using these ratios as formal proportions is something so natural that musicians have been doing this intuitively for over a milennium. Taking the octave's 2:1 ratio, for example, we discover the basis for the A, A, B and A, B, A song forms that have been ubiquitous, probably since antediluvian times (I am speaking particularly of instances in which the sections are of equal length, of course). Higher order ratios from the series can be employed as well, and the perfect fifth's 3:2 ratio is found in many A, A, B, A forms in which the B is twice the length of the A. Recognizing that these simple forms are intuited out implications of the harmonic series allows us to inform our intuitions and thereby gain more control of the compositional processes by applying these implications intentionally. We can continue to progress up the series' ratios, of course, but note that as you do you are following a parabolic curve that inches ever closer to a 1:1 proportion. The ratios within the first seven partials can achieve most of the formal effects that a composer could desire, and remember: There is absolutely nothing wrong with using that unison proportion either, since it is perfectly musical as defined by the series.
Of course, the distribution of climaxes - pitch climaxes, dynamic climaxes, &c. - is also a rhythmic and a proportional phenomenon as well. Traditionally - due to extra-musical aesthetic considerations - many of history's great composers have used the golden mean, which can be traced back to the early sixteenth century. While this irrational proportion has a compelling logical arguement - a + b is to a as a is to b - and is found in many natural phenomena - and obviously manifests itself in several living creatures- it is not a musical proportion because it is not found in the harmonic series (I'm sure it's circa 1.618 proportion could be approximated with non-adjacent positions using harmonics beyond the seventh partial, but that is a stretch, which is the point here). What the series most directly implies is the logical point for the main climax of a piece is at the 2/3 point: The proportion of the perfect fifth. Coincidentally (or not), this yields 1.5 versus the golden mean's 1.6: Not enough of a difference to be perceptable (!). Again, higher order ratios can also be employed and to good effect: As the climax gets closer to the end, the abruptness of the end can be used as a disturbing or even a shocking effect. The reverse is also true: You could place the main climax before the half-way point according to any series ratio's inverse, and have everything after be one long denouement. The possibilities are limited only by the imagination.
As an aside at this point, I'd like to mention an online music discussion I dropped in on a while back. The topic was the question of why pitch was notated so precisely while duration wasn't. Obviously, this is because the human perception of pitch is very precise, while the perception of time is profoundly vague (in comparison). Aside from the local level, where rhythmic patterns can be percieved with great precision, the regional harmonic or functional rhythms are less well percieved, and the overall global proportions - in large works especially - may only be sensed intuitively, if at all (For most listeners). This explains why a main climax at the golden ratio or the perfect fifth's ratio is probably not a perceptable difference for all but a few highly acute listeners... which begs the question, then what difference does it make if we apply musical or extra-musical ratios to broader rhythmic and formal elements? I suppose I'd have to admit that this gets into the realm of musical philosophy at this point, but it's important to me, and here's why:
If we imagine a race of beings who can percieve rhythm on all levels and formal proportions with the same accuracy as we humans can percieve pitch, how do you think a main climax at the golden mean point would sound to them versus one at the 2/3's point? It would sound dissonant, or at least "out of tune" is how it would sound: They would "prefer" the perfect fifth's ratio. To prove the un-musical nature of the golden ratio, all you have to do is turn it into an interval: It falls nerly exactly between a major and minor sixth, and sounds horribly out of tune. Our imaginary race of super-time-percievers would notice this.
As I mentioned previously, I am also beginning to intellectually pursue the implications of using in music elements that the series implies are non-musical or at least less than perfectly musical. Though I'm just beginning to apply this thinking to the five purely musical elements - harmony, counterpoint, melody, rhythm, and form - it is obvious that series-implied voice leading paradigms can be dispensed with for expressive purposes - and indeed must be dispensed with to achieve certain effects (See Chopin or any jazz composer) - and harmonically vilifiable structures can also have their place.
To be continued...
I'm getting really tired of this cold weather.
Pitch to rhythm form a natural continuum: It is only the human senses that divide them up, as there is an auditory deaf spot below the lowest sound percieved as pitch and the fastest beat percieved as rhythm. Within this auditory void, however, the frequency can still be percived through the sense of touch as vibration. The phenomenon we call rhythm is also manifested on at least three levels in music: The local level, which is the surface rhythm we usually think of as rhythm proper; the regional level, which is what is taught as harmonic rhythm but which can also be though of as functional rhythm; and finally, the global level, which are the various thematic periodicities that make up what we call form in music.
The first two phenomena are generally well understood by educated musicians, but the formal implications of the series are something new (So far as I have been able to determine). Nature defines what musical proportions are by the ratios in the series, so using these ratios as formal proportions is something so natural that musicians have been doing this intuitively for over a milennium. Taking the octave's 2:1 ratio, for example, we discover the basis for the A, A, B and A, B, A song forms that have been ubiquitous, probably since antediluvian times (I am speaking particularly of instances in which the sections are of equal length, of course). Higher order ratios from the series can be employed as well, and the perfect fifth's 3:2 ratio is found in many A, A, B, A forms in which the B is twice the length of the A. Recognizing that these simple forms are intuited out implications of the harmonic series allows us to inform our intuitions and thereby gain more control of the compositional processes by applying these implications intentionally. We can continue to progress up the series' ratios, of course, but note that as you do you are following a parabolic curve that inches ever closer to a 1:1 proportion. The ratios within the first seven partials can achieve most of the formal effects that a composer could desire, and remember: There is absolutely nothing wrong with using that unison proportion either, since it is perfectly musical as defined by the series.
Of course, the distribution of climaxes - pitch climaxes, dynamic climaxes, &c. - is also a rhythmic and a proportional phenomenon as well. Traditionally - due to extra-musical aesthetic considerations - many of history's great composers have used the golden mean, which can be traced back to the early sixteenth century. While this irrational proportion has a compelling logical arguement - a + b is to a as a is to b - and is found in many natural phenomena - and obviously manifests itself in several living creatures- it is not a musical proportion because it is not found in the harmonic series (I'm sure it's circa 1.618 proportion could be approximated with non-adjacent positions using harmonics beyond the seventh partial, but that is a stretch, which is the point here). What the series most directly implies is the logical point for the main climax of a piece is at the 2/3 point: The proportion of the perfect fifth. Coincidentally (or not), this yields 1.5 versus the golden mean's 1.6: Not enough of a difference to be perceptable (!). Again, higher order ratios can also be employed and to good effect: As the climax gets closer to the end, the abruptness of the end can be used as a disturbing or even a shocking effect. The reverse is also true: You could place the main climax before the half-way point according to any series ratio's inverse, and have everything after be one long denouement. The possibilities are limited only by the imagination.
As an aside at this point, I'd like to mention an online music discussion I dropped in on a while back. The topic was the question of why pitch was notated so precisely while duration wasn't. Obviously, this is because the human perception of pitch is very precise, while the perception of time is profoundly vague (in comparison). Aside from the local level, where rhythmic patterns can be percieved with great precision, the regional harmonic or functional rhythms are less well percieved, and the overall global proportions - in large works especially - may only be sensed intuitively, if at all (For most listeners). This explains why a main climax at the golden ratio or the perfect fifth's ratio is probably not a perceptable difference for all but a few highly acute listeners... which begs the question, then what difference does it make if we apply musical or extra-musical ratios to broader rhythmic and formal elements? I suppose I'd have to admit that this gets into the realm of musical philosophy at this point, but it's important to me, and here's why:
If we imagine a race of beings who can percieve rhythm on all levels and formal proportions with the same accuracy as we humans can percieve pitch, how do you think a main climax at the golden mean point would sound to them versus one at the 2/3's point? It would sound dissonant, or at least "out of tune" is how it would sound: They would "prefer" the perfect fifth's ratio. To prove the un-musical nature of the golden ratio, all you have to do is turn it into an interval: It falls nerly exactly between a major and minor sixth, and sounds horribly out of tune. Our imaginary race of super-time-percievers would notice this.
As I mentioned previously, I am also beginning to intellectually pursue the implications of using in music elements that the series implies are non-musical or at least less than perfectly musical. Though I'm just beginning to apply this thinking to the five purely musical elements - harmony, counterpoint, melody, rhythm, and form - it is obvious that series-implied voice leading paradigms can be dispensed with for expressive purposes - and indeed must be dispensed with to achieve certain effects (See Chopin or any jazz composer) - and harmonically vilifiable structures can also have their place.
To be continued...
I'm getting really tired of this cold weather.
posted by Hucbald at 11:27 AM
2 Comments:
“Obviously, this is because the human perception of pitch is very precise, while the perception of time is profoundly vague (in comparison).”
Hmm… I wonder if this is really the case.
Say the smallest rhythmic inflection that can be perceived is in the range of tens of milliseconds, and the smallest perceivable difference in pitch is somewhere in the region of a few cents. I’m probably missing something here, but how can these two things be directly compared especially since these are so heavily culturally dependent (e.g. listeners/performers of go-go funk which has rock-solid rhythm vs. contemporary performances of Romantic works with heavy rubato; or choruses in which the pitch content can, when measured, can vary as much as 100 cents).
S, tig
Actually, you have it exactly right about the local level, where rhythmic patterns are percieved quite accurately by trained musicians. Back in my Jazz and R&R band days, for example, I liked to work with drummers and bassists who had a good connection so the groove was happening: That takes great rhythmic precision and a good sense of time. Where the perception of rhythm and duration becomes fuzzy is at the regional level, where harmonic rhythms are not nearly as precisely percieved, and on the global level, where in large sophisticated works the form may even remain a mystery unless you have a score to look at.
Cheers
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