Diminished Sevenths and Secondary Diminished Sevenths
The diminished seventh chord is one of only two symmetric harmonic structures, the other being the augmented triad. By this I mean that all the intervals are all the same and the structure repeats at the octave without interruption, unlike diatonic symmetries like major and minor seventh chords which have whole tones or semitones interrupting the octave repititions of their symmetries. For the diminished seventh it is all minor thirds, and for the augmented triad it is all major thirds.
In the example above on the top stave I have demonstrated the theoretical origin of this sonority and it's traditional analysis. If the root of the dominant seventh chord found on the bVII degree of the minor key is raised to obtain a leading tone, the result is a fully diminished seventh chord. In the traditional analysis, the "root" of the diminished seventh progresses up by a semitone to resolve to the tonic, while the rest of the tones move down by whole step or half step to complete the tonic triad. This would have to be analyzed as a strong ascending root progression in which the tones transform in a counterclockwise direction with 3 becoming 1, 5 becoming 3, 7 becoming 5, and the "root" being constant between the two chords. I have indicated this with a perenthetical (1) to the right of the counterclockwise transformation symbol underneath the "S+" root progression indicator. The first two measures show this traditional explaination in the minor, while the second two measures show the traditional resolution in the major.
For the major key, the vii is already a diminished triad, so the minor seventh of that half-diminished seventh needs to be lowered by a semitone to get the fully diminished sonority. Note that the minor key and major key diminished sevenths are enharmonically the very same chord, since the G-sharp and the A-flat are the same tone notated differently. So, you could easily be cruising along in A minor and modulate to C major by introducing the diminished seventh for C major and then resolving it to that relative major tonic: The ear wouldn't know you weren't going to target the minor tonic until the modulation had been made. This effect has been used since before Bach's time.
Since one version of the diminished seventh was created by raising the root of a dominant seventh chord by a half step, it follows that any degree of a diminished seventh chord can be lowered a semitone to get any one of four different dominant seventh chords since it is a perfectly symmetrical harmonic structure. I have demonstrated this on the second staff of the example. Composers from Mozart on used this with increasing frequency to achieve startling modulations, and Romantic era composers used it constantly. Schubert was probably the best at reinterpreting symmetric structures, and his pieces often sound like clouds have obscured the view until the sun comes back out, and the listener emerges into a whole different world. Quite effective.
More than this though, since two of the tones of the diminished seventh can be interpreted to be the leading tone in order to target the tonic minor or it's relative major (Or, vice versa), it follows that any one of the four degrees of the sonority can be the leading tone. I have demonstrated this on the third staff of the example.
As you can see, there are eight possible modulations for this chord if you consider both major and minor tonics: That includes fully one third of all of the available keys in the twelve tone system. Since there are only three diminished seventh chords in the twelve tone system, and diminished sevenths can be created between any two degrees seperated by a whole tone by raising the root of the chord on that degree along with either also raising the third (for minor seventh chord degrees), or lowering the seventh (for major seventh chord degrees), and diminished sevenths can easily be created on any actual degree of the key through similar processes, modulations to any one of the twenty four regions of the home key are always only a couple of chords away.
Previously I mentioned the traditional analysis method for this sonority in terms that may have been interpreted as disparaging. There is a reason for this. Since the diminished seventh is a symmetric structure that can be interpreted in any one of four ways, in actual point of fact, it has no root out of context. In other words, there is no way to know what the "root" of a diminished seventh chord is until it resolves. In this respect, the diminished seventh momentarily suspends toinality, and it has been used to achieve this effect since the early Baroque era (Or even earlier).
The other problem with the diminished seventh is that all four of it's tones are active, and require resolution no matter how you interpret it. That means that in the method of writing trasformations over a constant root bass, parallel octaves are unaviodable unless a voice is sustained through the diminished seventh that is not really a member of that structure (As I did in the penultimate measure of the Theorem of Pythagoras sketch from the previous post). While this is "doable", and some interesting sonic effects can be achieved in this way, it is not the ideal solution.
Since the diminished seventh chord has a dominant function it also has any one of four dominant roots available a major third below any one of the chord tones. Adding any one of these potential real roots will create the sonority of a dominant seventh with a minor ninth. This method allows for the constant root bass line to be employed to write transformations over, and at the same time for parallel octaves to be avoided with the virtual five voice texture this technique requires in an all seventh chord environment. I have given a couple of examples of this on the bottom staff of the above example.
In this context, the diminished seventh has a real percievable root, and all that "real root versus theoretical root" nonesense can be avioded entirely. Notice that in those examples the resolution is the proper delayed crosswise transformation that dominants and secondary dominants normally participate in, and that the minor ninth takes the place of the root momentarily in the normative seventh chord transformations. This is indicated with the numeral "9" in place of the "up arrow" to "1" in the transformation analysis symbol. This method is technologically far superior to the traditional analysis method, especially if the goal is to employ these sonorities in the composition of harmonic canons, as I have given an example of below.
As you can see, treating secondary diminished sevenths as upper structures of a real, actual dominant seventh chord is the most logical solution for the composer, if not the theorist. They become just another special type of secondary dominant sonority like the so-called French Sixth is (Which I will refer to as a V(4/2/b) from now on).
Since there are transformation types other than crosswise in this canon, all four of the voices get all four of the canonic parts in turn. That makes an eight measure canonic voice, and since the root progression pattern that repeats is two measures long, the voices follow at that distance. I composed the canon in the first twelve bars and extracted it starting in measure 13. Since I have delayed the second voice in the first pair until the second measure, it takes eleven measures before all four of the voices exchange all of the parts and participate fully in the canon. That is why I made this canon twelve measures long. I didn't bother analyzing the whole thing since it is a sequentially modulating two bar phrase, and the analysis simply follows it's due course.
In the example above on the top stave I have demonstrated the theoretical origin of this sonority and it's traditional analysis. If the root of the dominant seventh chord found on the bVII degree of the minor key is raised to obtain a leading tone, the result is a fully diminished seventh chord. In the traditional analysis, the "root" of the diminished seventh progresses up by a semitone to resolve to the tonic, while the rest of the tones move down by whole step or half step to complete the tonic triad. This would have to be analyzed as a strong ascending root progression in which the tones transform in a counterclockwise direction with 3 becoming 1, 5 becoming 3, 7 becoming 5, and the "root" being constant between the two chords. I have indicated this with a perenthetical (1) to the right of the counterclockwise transformation symbol underneath the "S+" root progression indicator. The first two measures show this traditional explaination in the minor, while the second two measures show the traditional resolution in the major.
For the major key, the vii is already a diminished triad, so the minor seventh of that half-diminished seventh needs to be lowered by a semitone to get the fully diminished sonority. Note that the minor key and major key diminished sevenths are enharmonically the very same chord, since the G-sharp and the A-flat are the same tone notated differently. So, you could easily be cruising along in A minor and modulate to C major by introducing the diminished seventh for C major and then resolving it to that relative major tonic: The ear wouldn't know you weren't going to target the minor tonic until the modulation had been made. This effect has been used since before Bach's time.
Since one version of the diminished seventh was created by raising the root of a dominant seventh chord by a half step, it follows that any degree of a diminished seventh chord can be lowered a semitone to get any one of four different dominant seventh chords since it is a perfectly symmetrical harmonic structure. I have demonstrated this on the second staff of the example. Composers from Mozart on used this with increasing frequency to achieve startling modulations, and Romantic era composers used it constantly. Schubert was probably the best at reinterpreting symmetric structures, and his pieces often sound like clouds have obscured the view until the sun comes back out, and the listener emerges into a whole different world. Quite effective.
More than this though, since two of the tones of the diminished seventh can be interpreted to be the leading tone in order to target the tonic minor or it's relative major (Or, vice versa), it follows that any one of the four degrees of the sonority can be the leading tone. I have demonstrated this on the third staff of the example.
As you can see, there are eight possible modulations for this chord if you consider both major and minor tonics: That includes fully one third of all of the available keys in the twelve tone system. Since there are only three diminished seventh chords in the twelve tone system, and diminished sevenths can be created between any two degrees seperated by a whole tone by raising the root of the chord on that degree along with either also raising the third (for minor seventh chord degrees), or lowering the seventh (for major seventh chord degrees), and diminished sevenths can easily be created on any actual degree of the key through similar processes, modulations to any one of the twenty four regions of the home key are always only a couple of chords away.
Previously I mentioned the traditional analysis method for this sonority in terms that may have been interpreted as disparaging. There is a reason for this. Since the diminished seventh is a symmetric structure that can be interpreted in any one of four ways, in actual point of fact, it has no root out of context. In other words, there is no way to know what the "root" of a diminished seventh chord is until it resolves. In this respect, the diminished seventh momentarily suspends toinality, and it has been used to achieve this effect since the early Baroque era (Or even earlier).
The other problem with the diminished seventh is that all four of it's tones are active, and require resolution no matter how you interpret it. That means that in the method of writing trasformations over a constant root bass, parallel octaves are unaviodable unless a voice is sustained through the diminished seventh that is not really a member of that structure (As I did in the penultimate measure of the Theorem of Pythagoras sketch from the previous post). While this is "doable", and some interesting sonic effects can be achieved in this way, it is not the ideal solution.
Since the diminished seventh chord has a dominant function it also has any one of four dominant roots available a major third below any one of the chord tones. Adding any one of these potential real roots will create the sonority of a dominant seventh with a minor ninth. This method allows for the constant root bass line to be employed to write transformations over, and at the same time for parallel octaves to be avoided with the virtual five voice texture this technique requires in an all seventh chord environment. I have given a couple of examples of this on the bottom staff of the above example.
In this context, the diminished seventh has a real percievable root, and all that "real root versus theoretical root" nonesense can be avioded entirely. Notice that in those examples the resolution is the proper delayed crosswise transformation that dominants and secondary dominants normally participate in, and that the minor ninth takes the place of the root momentarily in the normative seventh chord transformations. This is indicated with the numeral "9" in place of the "up arrow" to "1" in the transformation analysis symbol. This method is technologically far superior to the traditional analysis method, especially if the goal is to employ these sonorities in the composition of harmonic canons, as I have given an example of below.
As you can see, treating secondary diminished sevenths as upper structures of a real, actual dominant seventh chord is the most logical solution for the composer, if not the theorist. They become just another special type of secondary dominant sonority like the so-called French Sixth is (Which I will refer to as a V(4/2/b) from now on).
Since there are transformation types other than crosswise in this canon, all four of the voices get all four of the canonic parts in turn. That makes an eight measure canonic voice, and since the root progression pattern that repeats is two measures long, the voices follow at that distance. I composed the canon in the first twelve bars and extracted it starting in measure 13. Since I have delayed the second voice in the first pair until the second measure, it takes eleven measures before all four of the voices exchange all of the parts and participate fully in the canon. That is why I made this canon twelve measures long. I didn't bother analyzing the whole thing since it is a sequentially modulating two bar phrase, and the analysis simply follows it's due course.
posted by Hucbald at 3:50 PM
2 Comments:
Ah, so you are a Rameau-ian! He believed that vii0 chords were V9s with the missing root, and also believed that IV chords were ii7 chords with the missing root. In both cases, it was to support his contention that all harmonic progressions are by fifth motion. So viio7 - I is really V9 - I, and IV - V is really ii7 - V. If you haven't read his Traité de l'Harmonie before, you should check it out.
Hi Scott,
Yes, I am familiar with Rameau's "Traite" (Sorry, I don't know how to get the accent mark), as it is a part of my music theory text library and I have studied my way through it a couple of times over the years (Many college music libraries would be shamed by my collection). I only follow him so far though. I think he has it exactly right about the vii(d7) chord, but not about the IV=ii or about all root progressions being by fifth. Considering his obvious placement as the FIRST harmonic theorist, I think that is nothing short of stupifyingly brilliant.
Bach did not agree with Rameau either (Bach was a voracious reader of theory texts), but I think Bach was arguing over the same details I am.
Unfortunately, I found the English translation of Rameau to be very frustrating to study. I'm not sure if it was the translation, or Rameau's lack of consision in his terminology, but my beaten-to-death copy is full of copious exclamations from my younger days as a musical hot-head know-it-all. :^)
Cheers.
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