Harmonic Implications of the Overtone Series, Part V
With harmonic canon explained, the natural next step is to explain the contrapuntal implications of the overtone series.
My presentation is the other way around from the historical development of Western Art Music - modal counterpoint having preceeded the harmonically driven variety - but there are reasons for this: 1) The harmonic implications of the series are more important to tonal/modal musicians today (Most of whom are involved with the jazz and popular generas), and 2) The art of counterpoint was really not perfected until the harmonic implications of the series were incorporated into it.
Whereas a harmonic continuity can be thought of as a stratum in which the chord tones transform in a manner prescribed by the root motion type - there is only one "most natural transformation" for any given root motion - counterpoint is just exactly the other way around: It is the interplay between the expected and unexpected transformations of the chord tones that lends vigor and freedom to the process so that two or more melodic trajectories can be combined to good effect.
In the first example above I have presented an extended version of the harmonic overtone series with the intervallic ratios and interval names added: I went all the way up to where the minor second appears (I could not fit the entire annotated series onto one system, so it continues on the second set of staves).
Along with contrapuntal developments in early Western modal music, the intervals were classified by theorists, and various rule-sets were developed to teach students how to write in the polyphonic styles. Like any early groping theoretical attempts in virtualy any science, these rule-sets were clumsy and sometimes even contradictory. For a time, thirds were considered consonances, but sixths - their octave displacement inversions - were not, for instance. Also, perfect fifths were not allowed to move in parallel, but perfect fourths - their octave inversions - were.
As with any true solution, the answer to both interval classification and parallel prohibitions is simple, and is explained by the implications of the series.
1) All adjacent intervals contained within the first seven partials of the series - plus their compounds and inversions - are consonances:
A) The Perfect Octave at 2:1
B) The Perfect Fifth at 3:2
C) The Perfect Fourth at 4:3
D) The Major Third at 5:4
E) The Minor Third at 6:5 (And 7:6)
F) The Major Sixth at 5:3
G) The Minor Sixth at 8:5
2) All adjacent intervals after the seventh partial - plus their compounds and inversions - are dissonances:
A) Major and Minor Seconds
B) Major and Minor Sevenths
C) Major and Minor Ninths
Note: I didn't bother with the ratios for the dissonances, as the laws that govern them are so simple that they are not required.
3) Consonances which are superparticular ratios* (Or even particulars) in both octave inversions are Perfect Consonances:
A) Perfect Octave/Perfect Unison
B) Perfect Fifth/Perfect Fourth
* In superparticular ratios, the difference between the two terms is one, as in 2:1 (2- 1= 1).
4) Consonances which are superparticular ratios in only one position are Imperfect Consonances:
A) Major Third/Minor Sixth= 8:5
B) Minor Third/Major Sixth= 5:3
The First Law of Contrapuntal Motion: Parallel Perfect Consonances are Prohibited.
The reason for this is twofold: 1) The two sets of overtones for pitches in perfect relationships interweave so well as to cancel out any sense of melodic independence - the two pitches blend so well as to become one in the ear of the listener - and, 2) in the diatonic system all octaves and unsons are perfect, and only the tri-tones are different among the fifths and fourths: There is almost no oportunity for variety with parallel perfect intervals.
In so-called simple counterpoint, it has been taught that parallel perfect fourths are OK in three or more voices in situations where the voices will not be inverted at the octave (Which would produce parallel perfect fifths), but this is really wrong. Nonetheless, it's a convention, and I'm sure I'll never be able to stamp the notion out. I would only point out though, that if you take octave inversion as being a natural feature of all correctly written counterpoint then parallel perfect fourths must be proscribed. I hold to the view that all correctly written counterpoint inverts at the octave, naturally, but I too write in the relaxed "simple" style as well.
Interestingly - to me anyway - the allowance for parallel perfect fourths in traditional simple counterpoint and traditional harmonic voice leading (Which is just a simplified form of counterpoint) comes from the harmonic implications of the series.
Remember those parallel perfect fifths in the transformations between diatonic tetrads involved in super-progressions and super-regressions? There you have it: In another inversion, those would be parallel perfect fourths. Composers just intuitively "borrowed" a feature of harmonic transformation to relax the laws of contrapuntal motion.
The Second Law of Contrapuntal Motion: Parallel Imperfect Consonances are Allowed.
The only thing limiting the number of parallel thirds and sixths you can string together is taste. There are never more than two major thirds/minor sixths in succession, or three minor thirds/major sixths succession in the diatonic system, so there is plenty of variety in the genders one hears, and the overtones of two pitches in imperfect relationships do not blend so well as to allow the tones to completely unify in the ear.
The Third Law of Contrapuntal Motion: Parallel Dissonances are Proscribed.
Intervals in imperfect but consonant relationships are sonorous in quality, and so mediate between the perfect consonances and the dissonances. To modern ears, minor sevenths and major ninths may sound "pretty" in a harmonic context, but in a contrapuntal context, they are not allowed by the implications of the series. Personally, I hate the details of musical acoustics - I find the topic mind-numbingly boring and I can never retain any of it - but I have studied the subject enough to realize how the implications of the series make these three laws sensical from an acoustic standpoint.
*****
From these three simple laws of musical motion everything else about counterpoint can be extrapolated: Any intervallic sequence may be justified by oblique or contrary stepwise motion; Parallel unequal fourths and fifths are allowed; Parallel name-only dissonances are allowed if the sonority is the acoustical equivalent of an imperfect consonance, &c.
The idea that dissonances must be resolved is really a stylistic affectation, as are things like prohibitions against "direct fifths" (But, "horn fifths" are OK (?!)) &c. I have written counterpoint in the jazz swing style, and I can attest to this fact.
What is happening in counterpoint, as I mentioned earlier, is that the composer is playing with the active tone/passive tone dichotomy inherent in the series. In harmony the root progressions mandate which transformation is most logical (But, composers have traditionally used contrapuntal "surprises" in that contex too, obviously), but in counterpoint it is the less certain transformations which are exploited to achieve independence between the melodic trajectories.
By dispensing with the ponderous rule-sets that are usually taught, counterpoint can be presented in a simplified form that allows for the student's own style to develop. After all, the rule-set of so-called modal counterpoint merely explains aspects of the personal style of Palestrina, and the rule-set of so-called tonal-counterpoint only explains the personal style of Bach. If the student wishes to learn these styles (As I did), he is certainly free to do so, but the underlying laws governing musical motion in counterpoint as implied by the overtone series are elementally simple.
*****
It would not be a good idea to allow me to get ahold of that quill pen.
My presentation is the other way around from the historical development of Western Art Music - modal counterpoint having preceeded the harmonically driven variety - but there are reasons for this: 1) The harmonic implications of the series are more important to tonal/modal musicians today (Most of whom are involved with the jazz and popular generas), and 2) The art of counterpoint was really not perfected until the harmonic implications of the series were incorporated into it.
Whereas a harmonic continuity can be thought of as a stratum in which the chord tones transform in a manner prescribed by the root motion type - there is only one "most natural transformation" for any given root motion - counterpoint is just exactly the other way around: It is the interplay between the expected and unexpected transformations of the chord tones that lends vigor and freedom to the process so that two or more melodic trajectories can be combined to good effect.
In the first example above I have presented an extended version of the harmonic overtone series with the intervallic ratios and interval names added: I went all the way up to where the minor second appears (I could not fit the entire annotated series onto one system, so it continues on the second set of staves).
Along with contrapuntal developments in early Western modal music, the intervals were classified by theorists, and various rule-sets were developed to teach students how to write in the polyphonic styles. Like any early groping theoretical attempts in virtualy any science, these rule-sets were clumsy and sometimes even contradictory. For a time, thirds were considered consonances, but sixths - their octave displacement inversions - were not, for instance. Also, perfect fifths were not allowed to move in parallel, but perfect fourths - their octave inversions - were.
As with any true solution, the answer to both interval classification and parallel prohibitions is simple, and is explained by the implications of the series.
1) All adjacent intervals contained within the first seven partials of the series - plus their compounds and inversions - are consonances:
A) The Perfect Octave at 2:1
B) The Perfect Fifth at 3:2
C) The Perfect Fourth at 4:3
D) The Major Third at 5:4
E) The Minor Third at 6:5 (And 7:6)
F) The Major Sixth at 5:3
G) The Minor Sixth at 8:5
2) All adjacent intervals after the seventh partial - plus their compounds and inversions - are dissonances:
A) Major and Minor Seconds
B) Major and Minor Sevenths
C) Major and Minor Ninths
Note: I didn't bother with the ratios for the dissonances, as the laws that govern them are so simple that they are not required.
3) Consonances which are superparticular ratios* (Or even particulars) in both octave inversions are Perfect Consonances:
A) Perfect Octave/Perfect Unison
B) Perfect Fifth/Perfect Fourth
* In superparticular ratios, the difference between the two terms is one, as in 2:1 (2- 1= 1).
4) Consonances which are superparticular ratios in only one position are Imperfect Consonances:
A) Major Third/Minor Sixth= 8:5
B) Minor Third/Major Sixth= 5:3
The First Law of Contrapuntal Motion: Parallel Perfect Consonances are Prohibited.
The reason for this is twofold: 1) The two sets of overtones for pitches in perfect relationships interweave so well as to cancel out any sense of melodic independence - the two pitches blend so well as to become one in the ear of the listener - and, 2) in the diatonic system all octaves and unsons are perfect, and only the tri-tones are different among the fifths and fourths: There is almost no oportunity for variety with parallel perfect intervals.
In so-called simple counterpoint, it has been taught that parallel perfect fourths are OK in three or more voices in situations where the voices will not be inverted at the octave (Which would produce parallel perfect fifths), but this is really wrong. Nonetheless, it's a convention, and I'm sure I'll never be able to stamp the notion out. I would only point out though, that if you take octave inversion as being a natural feature of all correctly written counterpoint then parallel perfect fourths must be proscribed. I hold to the view that all correctly written counterpoint inverts at the octave, naturally, but I too write in the relaxed "simple" style as well.
Interestingly - to me anyway - the allowance for parallel perfect fourths in traditional simple counterpoint and traditional harmonic voice leading (Which is just a simplified form of counterpoint) comes from the harmonic implications of the series.
Remember those parallel perfect fifths in the transformations between diatonic tetrads involved in super-progressions and super-regressions? There you have it: In another inversion, those would be parallel perfect fourths. Composers just intuitively "borrowed" a feature of harmonic transformation to relax the laws of contrapuntal motion.
The Second Law of Contrapuntal Motion: Parallel Imperfect Consonances are Allowed.
The only thing limiting the number of parallel thirds and sixths you can string together is taste. There are never more than two major thirds/minor sixths in succession, or three minor thirds/major sixths succession in the diatonic system, so there is plenty of variety in the genders one hears, and the overtones of two pitches in imperfect relationships do not blend so well as to allow the tones to completely unify in the ear.
The Third Law of Contrapuntal Motion: Parallel Dissonances are Proscribed.
Intervals in imperfect but consonant relationships are sonorous in quality, and so mediate between the perfect consonances and the dissonances. To modern ears, minor sevenths and major ninths may sound "pretty" in a harmonic context, but in a contrapuntal context, they are not allowed by the implications of the series. Personally, I hate the details of musical acoustics - I find the topic mind-numbingly boring and I can never retain any of it - but I have studied the subject enough to realize how the implications of the series make these three laws sensical from an acoustic standpoint.
*****
From these three simple laws of musical motion everything else about counterpoint can be extrapolated: Any intervallic sequence may be justified by oblique or contrary stepwise motion; Parallel unequal fourths and fifths are allowed; Parallel name-only dissonances are allowed if the sonority is the acoustical equivalent of an imperfect consonance, &c.
The idea that dissonances must be resolved is really a stylistic affectation, as are things like prohibitions against "direct fifths" (But, "horn fifths" are OK (?!)) &c. I have written counterpoint in the jazz swing style, and I can attest to this fact.
What is happening in counterpoint, as I mentioned earlier, is that the composer is playing with the active tone/passive tone dichotomy inherent in the series. In harmony the root progressions mandate which transformation is most logical (But, composers have traditionally used contrapuntal "surprises" in that contex too, obviously), but in counterpoint it is the less certain transformations which are exploited to achieve independence between the melodic trajectories.
By dispensing with the ponderous rule-sets that are usually taught, counterpoint can be presented in a simplified form that allows for the student's own style to develop. After all, the rule-set of so-called modal counterpoint merely explains aspects of the personal style of Palestrina, and the rule-set of so-called tonal-counterpoint only explains the personal style of Bach. If the student wishes to learn these styles (As I did), he is certainly free to do so, but the underlying laws governing musical motion in counterpoint as implied by the overtone series are elementally simple.
*****
It would not be a good idea to allow me to get ahold of that quill pen.
posted by Hucbald at 2:27 AM
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