UPDATED - Harmonic Implications of the Overtone Series, Part I
This is a re-visit to my earlier Musical Philosophy post, which generated some hilarious responses, all from proponents of adding atonality to the definition of music, and all rather infantile. If I think a comment is worth responding to or worth a viewer's read, I'll post it; if not I won't. So, there were none worth responding to, obviously. And here's a hint: If you use dummy e-mails and obviously idiotic names, I won't even read the comment when it comes in. This is primarily directed at a fan of mine named, aptly, U.R.N. Idiot (I did get a laugh out of the name though. I mean, "Dear Mr. Urine Idiot, Sorry about the bad luck with that name of yours, but as they say, if the shoe fits...").
There are strange people on the internet.
*****
Anyway, I've revised my terminology a tad to tighten up the consistency, and I think all lovers of music theory will be able to derive some use out of this post.
*****
UPDATE 09/17: I suppose I ought to take a moment to describe the logic behind my analysis symbols, as they are slightly different than "normal," if there is such a thing in music analysis. I developed these with the idea that they would be 1) Consistent, 2) Unambiguous, and 3) Executable on a QWERTY keyboard.
A bold capital Roman numeral denotes that the fifth has a major third, while a bold small case Roman numeral indicates that the fifth has a minor third. Fifths are perfect by implication. In the case that the fifth is diminished, a small case "d" will appear in parenthesis after the analysis symbol, and if the fifth is augmented, a capital "A" will appear in the parentheses. To be extra-clear on major thirds with diminished and augmented fifths, the numeral "5" will appear after the modified fifth designator.
Senenths (and ninths &c.) will be described with the M, m, A, d symbology as well, with the numeral following the modifier.
*****
The first section will be using the following example:
As always, every discussion of music must begin with the harmonic series, which has been true since the first music theorists investigated it with their monochords. The entire history of Western Art Music has been the ongoing investigation into the implications of this series.
Today I want to discuss the harmonic implications of the series. Obviously, the series makes a major minor-seventh chord, also called a dominant seventh chord, and it generates the gravitational force in music. The first and most obvious implication of the series is the primordial progressive root motion, as seen on the second system. This decending fifth or rising forth progression is statistically the most common type of progression in Western Art Music, and popular forms as well.
The technologically correct voice leading for this progression is a crosswise transformation in which the root of the first chord becomes the fifth of the target chord, the third of the first chord becomes the seventh of the target chord, and vice versa, in a purely tetradic environment. Traditionally, the target has been a triad momentarily, as I'll demonstrate shortly.
The implications of this progressive root motion lead to major tonality, which is the most perfect in accordance with the implications of the series for a triadic system. The root is the point at which arrivals or departures can be made in a progressive root motion to and from major triads, as you'll see on the third set of staves. Note that where the texture is triadic, there can only be clockwise or counterclockwise circular transformations between chords. In progressive root motions in a triadic texture, the transformations are counterclockwise: Root becomes fifth, fifth becomes third, and third becomes root.
If you simply read the third system backwards, you get the opposite of progressive motion, which is regressive. Regressive root motion types go against the dominant resolution impulse of the harmonic series. This in no way implies that these root motion types are inferior, rather they are just a resource and are necessary for variety and balance, as you'll soon see.
Note also that I not only analyze the degree and gender of the chords, but the root motion and transformation types as well. I have found this to be of great assistance in composing harmonic continuities, because I can visually check the paterns I'm creating.
*****
UPDATE 09/17: Again, I developed these symbols to be easily executable on a QWERTY keyboard, so ---> = clockwise transformation, <--- = counterclockwise transformation, and <-^-> = crosswise transformation. For the root motion types, P = progressive root motion, R = regressive root motion, .5P = a half-progression, .5R = a half-regression, SP = a super-progression, SR= a super-regression, Ptt = a progressive tri-tone root motion, and Rtt = a regressive tri-tone root motion.
*****
If we take the notes of the major triads of the cardinal degrees and put them into a scale, we get the Ionian or Major mode. It is important to understand that it is the harmonic implications of the series that generate the scale, and not the other way around (Technically, the series itself creates a Mixolydian Augmented Fourth scale). Historically, the melodic and contrapuntal implications of the series were worked out first, but this was because the original medium was vocal, I believe. The archaic Church Mode system also held back musical development for centuries, and this lineal thought lead to a poly-lineal musical conception, or counterpoint. As early as the 1300's though, the common musicians were using the Ionian mode in many of their pieces. In any case, what I'm discussing here today was not initially intuited until just before the time of J.S. Bach, and it was Rameau who first attempted (poorly) to explain it theoretically.
Not until Schoenberg's "Structural Functions of Harmony" were root motion types first categorized (Again, poorly: He obviously hadn't a clue about the implications of the harmonic series), and it was Joseph Shillinger who first correctly worked out the transformational nature of harmonic voice leading. Until Schillinger (And still among most traditional tonal composers of today) harmonic voice leading was a watered down form of counterpoint, gleaned primarily from the chorale harmonizations of Bach (Which explains the situation perfectly).
On the bottom staves you see the resulting harmonized scale, and unfortunately, this is what students are often given first. Without the preceeding understanding, confusion is inevitable: For years I thought scales generated harmony. They don't. It's exactly the other way around.
*****
Instead of presenting the root progression types in sterile isolation, I decided to use some phrased continuities. In the first example, after a super-regression from I to vii(d), the phrase has all of the chords in the diatonic system arranged in progressive order.
It is important to note that the constant root bass part is just a checking tool: The triadic continuity exists solely on the upper staff of each system. After the super-regression's clockwise circular transformation, all of the progressive root motions generate counterclockwise transformations. I turned the phrase around in the last measure with a super-progression from IV to V.
The reason for the terminology of super-progression and super-regression is simple: What is actually happening in these cases is that there is an implied root not present a third below the "root" of the first chord, which progresses or regresses to the target chord. By comparing every root progression type to the primordial natural progression, their various effects can be well understood. In part two, I'll clarify this further with secondary diminished seventh chords, which are actually major minor-seventh chords with a minor ninth and a missing root (If you want to prove this to yourself, try to write a four voice continuity employing these chords over a constant root bass: You'll end up with parallel octaves unles you put the REAL root in the bass line).
Note also that in a triadic texture, the voice leading is not totally smooth where super-progressions and super-regressions are used: The series actually implies a four part texture.
In the example on the second system, I have arranged all the chords in the system in half-progressive order. In a half-progression the two chords only go half way to the fifth the series desires to go to. As you can see, If you were to go directly from the I chord to the IV chord, not only would the root fall the required fifth, but the triadic continuity would make the progressive transformation all at once as well. Again, a half-regression would simply be reading that section backwards.
So, there are only seven types of root motions:
1) Progressive root motion (Falling fifth/rising fourth)
2) Regressive root motion (Rising fifth/falling fourth)
3) Half-Progressive root motion (Falling third/rising sixth)
4) Half-Regressive root motion (Rising third/falling sixth)
5) Super-Progressive root motion (Rising second/falling seventh)
6) Super-Regressive root motion (Falling second/rising seventh)
7) Tri-Tone root motion (Which is Ptt if it is a falling diminished fifth, or Rtt if it is a falling augmented fourth).
In example III, I used all of the root motion types in a single phrase with the exception of the Rtt (Regressive tri-tone). Note that the Ptt breaks the phrase up into two four measure sub-phrases, and that these sub-phrases have mirrored root motion types: The initial regression from measure one to measure two is answered by a progression from measure four to measure five; then the super-progression from measure two into measure three is answered by a super-regression from measure five into measure six, &c. Well ordered root motion patterns separate good progressions from bad ones, for the most part, and using the extended analysis symbol sets that I do makes the patterns easier to create and to see.
As I said, the series really implies a four-part texture, so the first tetradic example is the same as the first triadic example with the exception of the number of voices. Note that in a purely tetradic texture even the super-progressions and super-regressions have a common tone. Progressions and regressions have two common tones, and half-progressions and half-regressions have three common tones. The number of common tones also has a huge effect on the nature and effect that the various root motions create, obviously.
Now for the (Insert horn fanfare here) parallel perfect fifths in the super-progressions and super-regressions: They are perfectly natural and are not an issue in harmony at all, unless the chordal functions of the tones do not transform! In other words, parallel perfect fifths only sound "crude" when the root remains the root and the fifth remains the fifth (Or the same relationships involving the thirds and sevenths).
If you get the impression that I'm calling jazz vioce leading crude, well, yeah, on a certain level, but not transforming the chord tone functions is simply an aspect of that style. No doubt in my mind that the "proper" transformations sound "cooler" though, which is why I use them (Some of the "style" listeners percieve in the late music of George Gershwin is due to his being a student of Schillinger and employing some of these techniques).
*****
UPDATE 09/17: But, one thing the jazz musicians did properly intuit is that the overtone series implies a four-part texture. This actually began with blues tonality, which is simply a musical system built upon overtone chords on all of the cardinal degrees (I7, IV7, V7 in traditional parlance, I(m7), IV(m7), V(m7) as I designate them).
*****
Finally, as with the primordial progression implied by the series, progressive root motion creates crosswise transformations (As does regressive tetradic motion).
The second tetradic example at the bottom of the page is just a four voice version of the second triadic example. Note all the common tones: This series of falling thirds and one-at-a-time note movements in the transformations creates a very mild effect.
If you want to make a four voice version of the third triadic continuity, that would be good exercise, but I want to go back to the first example for my last point today.
*****
UPDATE 09/17: Also notice that in triadic textures the series of progressive and half-progressive root motions cause the voices to rise in pitch through time (And, regressive/half-regressive root motions in triadic textures would cause them to fall), but in tetradic environments, the same root motion types have the opposite effect. Imbalanced progressions, like these examples, can get you in "trouble" with ranges if they are not balanced out by alternating three and four voice episodes. Noticing all of these details enables the composer to gain total control over his work, and freedom is control.
*****
The pull of the dominant resolution/progressive root motion force in music - well, in sound, actually - lead to composers intuiting overtone chords on every degree of the diatonic system over time. The original continuity was written with this demonstration in mind.
After the super-regression from measure one, the thirds of the chords are raised and/or the sevenths are lowered to get an overtone chord. I did not raise the fifth of the V(d5m7)/iii to show you the harmonic origin of the so-called French Augmented Sixth Chord: In the second inversion, you get the Augmented Sixth interval, but in reality there is nothing "French" about this chord, and that description is useless insofar as explaining what it is. What it is, is merely a secondary dominant seventh chord with a diminished fifth in a particular inversion. All inversions of it are available, however, and on any degree, I might add.
Note that the secondary leading tones are allowed to resolve to the new roots-of-the-moment in this example. This is the traditional way to deal with these chords, but it is not the only way: The secondary leading tone can come down chromatically to give seventh chords as targets (Another thing that sounds vaguely Gershwinian in abundance, but composers have done this as far back as Bach's time in isolation). What this more traditional example's resolutions do is to demonstrate that inserting triads as target chords only delays the transformation type (crosswise), it does not negate it entirely (Or rather, that's the way that the series implies it ought to go).
*****
UPDATE 09/17: Yet another thing to note is that if the continuity started out with a minor seventh chord instead of a diminished minor seventh chord, and the remaining targets were also all made into minor seventh chords (On the dominant, tonic, and subdominant degrees), a very simple harmonic double canon would be created. I have covered that subject in depth previously, but I guess I'll hit it again in the next post (Because it really is a super-cool musical technology).
*****
Please understand that this is something I have been pondering recently, and I wrote this post in a stream-of-consciousness kind of way to put my own mind in the driver's seat with these concepts. I will study and re-read this a few times and try to clear up and points that might be confusing with a summary at the beginning of the second part of this subject.
There are strange people on the internet.
*****
Anyway, I've revised my terminology a tad to tighten up the consistency, and I think all lovers of music theory will be able to derive some use out of this post.
*****
UPDATE 09/17: I suppose I ought to take a moment to describe the logic behind my analysis symbols, as they are slightly different than "normal," if there is such a thing in music analysis. I developed these with the idea that they would be 1) Consistent, 2) Unambiguous, and 3) Executable on a QWERTY keyboard.
A bold capital Roman numeral denotes that the fifth has a major third, while a bold small case Roman numeral indicates that the fifth has a minor third. Fifths are perfect by implication. In the case that the fifth is diminished, a small case "d" will appear in parenthesis after the analysis symbol, and if the fifth is augmented, a capital "A" will appear in the parentheses. To be extra-clear on major thirds with diminished and augmented fifths, the numeral "5" will appear after the modified fifth designator.
Senenths (and ninths &c.) will be described with the M, m, A, d symbology as well, with the numeral following the modifier.
*****
The first section will be using the following example:
As always, every discussion of music must begin with the harmonic series, which has been true since the first music theorists investigated it with their monochords. The entire history of Western Art Music has been the ongoing investigation into the implications of this series.
Today I want to discuss the harmonic implications of the series. Obviously, the series makes a major minor-seventh chord, also called a dominant seventh chord, and it generates the gravitational force in music. The first and most obvious implication of the series is the primordial progressive root motion, as seen on the second system. This decending fifth or rising forth progression is statistically the most common type of progression in Western Art Music, and popular forms as well.
The technologically correct voice leading for this progression is a crosswise transformation in which the root of the first chord becomes the fifth of the target chord, the third of the first chord becomes the seventh of the target chord, and vice versa, in a purely tetradic environment. Traditionally, the target has been a triad momentarily, as I'll demonstrate shortly.
The implications of this progressive root motion lead to major tonality, which is the most perfect in accordance with the implications of the series for a triadic system. The root is the point at which arrivals or departures can be made in a progressive root motion to and from major triads, as you'll see on the third set of staves. Note that where the texture is triadic, there can only be clockwise or counterclockwise circular transformations between chords. In progressive root motions in a triadic texture, the transformations are counterclockwise: Root becomes fifth, fifth becomes third, and third becomes root.
If you simply read the third system backwards, you get the opposite of progressive motion, which is regressive. Regressive root motion types go against the dominant resolution impulse of the harmonic series. This in no way implies that these root motion types are inferior, rather they are just a resource and are necessary for variety and balance, as you'll soon see.
Note also that I not only analyze the degree and gender of the chords, but the root motion and transformation types as well. I have found this to be of great assistance in composing harmonic continuities, because I can visually check the paterns I'm creating.
*****
UPDATE 09/17: Again, I developed these symbols to be easily executable on a QWERTY keyboard, so ---> = clockwise transformation, <--- = counterclockwise transformation, and <-^-> = crosswise transformation. For the root motion types, P = progressive root motion, R = regressive root motion, .5P = a half-progression, .5R = a half-regression, SP = a super-progression, SR= a super-regression, Ptt = a progressive tri-tone root motion, and Rtt = a regressive tri-tone root motion.
*****
If we take the notes of the major triads of the cardinal degrees and put them into a scale, we get the Ionian or Major mode. It is important to understand that it is the harmonic implications of the series that generate the scale, and not the other way around (Technically, the series itself creates a Mixolydian Augmented Fourth scale). Historically, the melodic and contrapuntal implications of the series were worked out first, but this was because the original medium was vocal, I believe. The archaic Church Mode system also held back musical development for centuries, and this lineal thought lead to a poly-lineal musical conception, or counterpoint. As early as the 1300's though, the common musicians were using the Ionian mode in many of their pieces. In any case, what I'm discussing here today was not initially intuited until just before the time of J.S. Bach, and it was Rameau who first attempted (poorly) to explain it theoretically.
Not until Schoenberg's "Structural Functions of Harmony" were root motion types first categorized (Again, poorly: He obviously hadn't a clue about the implications of the harmonic series), and it was Joseph Shillinger who first correctly worked out the transformational nature of harmonic voice leading. Until Schillinger (And still among most traditional tonal composers of today) harmonic voice leading was a watered down form of counterpoint, gleaned primarily from the chorale harmonizations of Bach (Which explains the situation perfectly).
On the bottom staves you see the resulting harmonized scale, and unfortunately, this is what students are often given first. Without the preceeding understanding, confusion is inevitable: For years I thought scales generated harmony. They don't. It's exactly the other way around.
*****
Instead of presenting the root progression types in sterile isolation, I decided to use some phrased continuities. In the first example, after a super-regression from I to vii(d), the phrase has all of the chords in the diatonic system arranged in progressive order.
It is important to note that the constant root bass part is just a checking tool: The triadic continuity exists solely on the upper staff of each system. After the super-regression's clockwise circular transformation, all of the progressive root motions generate counterclockwise transformations. I turned the phrase around in the last measure with a super-progression from IV to V.
The reason for the terminology of super-progression and super-regression is simple: What is actually happening in these cases is that there is an implied root not present a third below the "root" of the first chord, which progresses or regresses to the target chord. By comparing every root progression type to the primordial natural progression, their various effects can be well understood. In part two, I'll clarify this further with secondary diminished seventh chords, which are actually major minor-seventh chords with a minor ninth and a missing root (If you want to prove this to yourself, try to write a four voice continuity employing these chords over a constant root bass: You'll end up with parallel octaves unles you put the REAL root in the bass line).
Note also that in a triadic texture, the voice leading is not totally smooth where super-progressions and super-regressions are used: The series actually implies a four part texture.
In the example on the second system, I have arranged all the chords in the system in half-progressive order. In a half-progression the two chords only go half way to the fifth the series desires to go to. As you can see, If you were to go directly from the I chord to the IV chord, not only would the root fall the required fifth, but the triadic continuity would make the progressive transformation all at once as well. Again, a half-regression would simply be reading that section backwards.
So, there are only seven types of root motions:
1) Progressive root motion (Falling fifth/rising fourth)
2) Regressive root motion (Rising fifth/falling fourth)
3) Half-Progressive root motion (Falling third/rising sixth)
4) Half-Regressive root motion (Rising third/falling sixth)
5) Super-Progressive root motion (Rising second/falling seventh)
6) Super-Regressive root motion (Falling second/rising seventh)
7) Tri-Tone root motion (Which is Ptt if it is a falling diminished fifth, or Rtt if it is a falling augmented fourth).
In example III, I used all of the root motion types in a single phrase with the exception of the Rtt (Regressive tri-tone). Note that the Ptt breaks the phrase up into two four measure sub-phrases, and that these sub-phrases have mirrored root motion types: The initial regression from measure one to measure two is answered by a progression from measure four to measure five; then the super-progression from measure two into measure three is answered by a super-regression from measure five into measure six, &c. Well ordered root motion patterns separate good progressions from bad ones, for the most part, and using the extended analysis symbol sets that I do makes the patterns easier to create and to see.
As I said, the series really implies a four-part texture, so the first tetradic example is the same as the first triadic example with the exception of the number of voices. Note that in a purely tetradic texture even the super-progressions and super-regressions have a common tone. Progressions and regressions have two common tones, and half-progressions and half-regressions have three common tones. The number of common tones also has a huge effect on the nature and effect that the various root motions create, obviously.
Now for the (Insert horn fanfare here) parallel perfect fifths in the super-progressions and super-regressions: They are perfectly natural and are not an issue in harmony at all, unless the chordal functions of the tones do not transform! In other words, parallel perfect fifths only sound "crude" when the root remains the root and the fifth remains the fifth (Or the same relationships involving the thirds and sevenths).
If you get the impression that I'm calling jazz vioce leading crude, well, yeah, on a certain level, but not transforming the chord tone functions is simply an aspect of that style. No doubt in my mind that the "proper" transformations sound "cooler" though, which is why I use them (Some of the "style" listeners percieve in the late music of George Gershwin is due to his being a student of Schillinger and employing some of these techniques).
*****
UPDATE 09/17: But, one thing the jazz musicians did properly intuit is that the overtone series implies a four-part texture. This actually began with blues tonality, which is simply a musical system built upon overtone chords on all of the cardinal degrees (I7, IV7, V7 in traditional parlance, I(m7), IV(m7), V(m7) as I designate them).
*****
Finally, as with the primordial progression implied by the series, progressive root motion creates crosswise transformations (As does regressive tetradic motion).
The second tetradic example at the bottom of the page is just a four voice version of the second triadic example. Note all the common tones: This series of falling thirds and one-at-a-time note movements in the transformations creates a very mild effect.
If you want to make a four voice version of the third triadic continuity, that would be good exercise, but I want to go back to the first example for my last point today.
*****
UPDATE 09/17: Also notice that in triadic textures the series of progressive and half-progressive root motions cause the voices to rise in pitch through time (And, regressive/half-regressive root motions in triadic textures would cause them to fall), but in tetradic environments, the same root motion types have the opposite effect. Imbalanced progressions, like these examples, can get you in "trouble" with ranges if they are not balanced out by alternating three and four voice episodes. Noticing all of these details enables the composer to gain total control over his work, and freedom is control.
*****
The pull of the dominant resolution/progressive root motion force in music - well, in sound, actually - lead to composers intuiting overtone chords on every degree of the diatonic system over time. The original continuity was written with this demonstration in mind.
After the super-regression from measure one, the thirds of the chords are raised and/or the sevenths are lowered to get an overtone chord. I did not raise the fifth of the V(d5m7)/iii to show you the harmonic origin of the so-called French Augmented Sixth Chord: In the second inversion, you get the Augmented Sixth interval, but in reality there is nothing "French" about this chord, and that description is useless insofar as explaining what it is. What it is, is merely a secondary dominant seventh chord with a diminished fifth in a particular inversion. All inversions of it are available, however, and on any degree, I might add.
Note that the secondary leading tones are allowed to resolve to the new roots-of-the-moment in this example. This is the traditional way to deal with these chords, but it is not the only way: The secondary leading tone can come down chromatically to give seventh chords as targets (Another thing that sounds vaguely Gershwinian in abundance, but composers have done this as far back as Bach's time in isolation). What this more traditional example's resolutions do is to demonstrate that inserting triads as target chords only delays the transformation type (crosswise), it does not negate it entirely (Or rather, that's the way that the series implies it ought to go).
*****
UPDATE 09/17: Yet another thing to note is that if the continuity started out with a minor seventh chord instead of a diminished minor seventh chord, and the remaining targets were also all made into minor seventh chords (On the dominant, tonic, and subdominant degrees), a very simple harmonic double canon would be created. I have covered that subject in depth previously, but I guess I'll hit it again in the next post (Because it really is a super-cool musical technology).
*****
Please understand that this is something I have been pondering recently, and I wrote this post in a stream-of-consciousness kind of way to put my own mind in the driver's seat with these concepts. I will study and re-read this a few times and try to clear up and points that might be confusing with a summary at the beginning of the second part of this subject.
posted by Hucbald at 6:27 PM
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