I noticed that if the root note of the dominant 7 chord is omitted, it makes it a diminished chord. Here is an example: if the G note in the G dominant 7 chord is omitted, it becomes a B diminished chord. So, is the diminished chord even considered an incomplete dominant 7 chord?
Certainly in some circumstances it functions as a V7 chord. Interestingly/ambiguously, depending on the inversion/voicing, it can pretend to be an incomplete V7 in 4 different (major) keys. :)
I recall, as a kid, thinking that the diminished chords in classical music "really needed to be filled out to V7 chords"... but I've since recovered. :)
If the question is about whether or not various specific practice traditions would consider it so... well, that depends on the tradition. But, still, I'd agree that its sound/function often is V7-ish.
Not always, though! Already in Brahms, and in jazz-standard-type stuff, the voicings often are stretched-out diminished chords, whose function is as much voice-leading as V7. Not going to the tonic or even secondary tonics, etc. Further, as in George Shearings "locked hands" voicings, diminished chords are simply not-quite-in-the-scale passing chords between inversions of something like I69 or other pentatonic-ish chords. It would be a stretch to declare all of them V7-ish, unless one had a very limited list of allowed labels. :)
There are several theorists (Walter Piston for one) who have suggested that the vii0 is equivalent to a rootless V7. In practice, it's sometimes used that way; the voice leading is identical to the V7-I or V7-i (except for step 5) in that scale step 7 moves to the tonic.
There is another usage of vii0 that is quite different and not equivalent harmonically to a rootless V7. In a sequence (such as those using the Circle of Fifths). In major keys, the chord sequence I-IV-vii0-iii-vi-ii-V7-I is often used with the vii0 in root position where the pattern of fifths in the bass dominates the usual diminished chord voicing.
Do you mean "...is the diminished chord EVER considered an incomplete dominant 7 chord?"
That simple question deserves a simple answer. Yes, it sometimes is. A diminished triad isn't JUST an incomplete dom7, it can do other things as well. But Bdim and G7 (B, D, F and G, B, D, F) both contain the tritone B - F, and when it resolves to C - E there's certainly a dominant - tonic thing going on!
Bdim can also resolve to Bmajor, which is acting quite differently to G7. Or to F♯7(sus4)... So it isn't merely an incomplete G7. But it can do that.
The short answer is no. Your observations are correct but a diminished triad and a dominant 7th chord are two different things. Following your logic you might ask if an Em chord is an incomplete Cmaj7 chord or a host of other scenarios.
A diminished chord sometimes but not always functions much like a dominant chord though. Take your G7 and Bdim. Both are diatonic to the key of C. Both contain a tritone interval that resolves inward to a major 3rd on a C chord. However both are not 7th chords and both do not contain a major triad. The diminished triad on its own doesn’t imply a dominant seventh chord with no root.
Yes, but pay attention to the context of the music and whatever you are analyzing.
For example, don't try labeling a perfect cadence in a passage with
viio I, justified by
viio = V7. Certain passing diminished chords aren't necessarily incomplete dominant seventh chords. There are times when the specifics, not equivalencies, are most important.
A dominant seventh chord (V7) generally resolves to i, unless we're in Blues territory. Component-wise, a diminished is an incomplete dominant seventh chord - however, one main component is missing - its root. So we can't call V7 - G7 in key C, without that root, an incomplete dominant seventh, as it can't even be named G7 - that G isn't there.
That apart, any diminished triad could and often does (it's one of its useful positions)lead to other places. True, BDF could lead to CEG, actually making iio>I, but could just as easily move to E♭G B♭, or F♯A♯C♯.
So, no is the answer. Unless a different instrument (bass, maybe) was providing that root (of G here).
A diminished 7th chord is a V7b9 chord without the root. Every diminished 7th can substitute for a V7b9 and vice versa. The issue is that a diminished 7th chord has 4 V7b9 chords and not every one will work in the global context and they do have different sounds in some cases[In the sense that it might not be what you want due to subtle implications due to the larger context. Sorta like how in English(or any language) some words have multiple means].
You want to always think of a dim7th as also being 4 V7b9 along with variations such as #ivo7, #vio7, io7, etc.
It is not about it being a specific thing. It's a color, a sound. You simply want to learn how it works by usage rather than try to peg it to some ideal or specific thing that it must be. It can be anything. You can make a diminished chord be a minor chord, a major 7th chord, or whatever. It is harder to do but if you've established Cmaj7 as the quality for the C chord and then play Cdim7 you are essentially combining Cmaj7 and Cdim7 and it will color that Cdim7 as a Cmaj7 sound. E.g., if you play Cmaj7 Cdim7 Cmaj7 and Cmaj7 is clearly the quality for the C chord then that Cdim7 is functioning as an alteration of Cmaj7 and it will sound different than if you did G7 Cdim7 G7 in which case the Cdim7 will be functioning in terms of G7 giving a sort of G7 Lydian augmented if the context is appropriate. Of course it might be clear it is part of something else, e.g., G7 Ab7b9 G7 if that is more natural. It all depends. Also different people will have different "expectations". It's not about tying things down and saying it has to be X. We get in to the trap because "basic" harmonies are generally not ambiguous(but they can be) and we think all other harmonies sorta have the same simple interpretation. Ultimately one just needs to familiarize themselves with how harmony works and how to name chords/colors and then everything else takes care of itself. Once you understand it through experience then you realize it is not something that can be explained using worse because the words are not the thing. You just know how to communicate the basic color and then use your intuition(experience) to interpret it and understand it.
For example, in moonlight sonata there is a section of 3 dim7th arpeggios(2 out of the 4 possible) over a G# bass. The G# acts as the V7b9 chord for the C#min tonic but the diminished sound is expressed and not the dominant sound due to the arpeggiation of D#o7(a V7b9), then C#o7(a io7 or a #ivo7 etc), then F#o7(V7b9). All these though are essentially a G#7b9 sound as Beethoven peddles the G# to reinforce this. In each case though they are variations of coloration of the V7b9 chord. In the case of C#o7 over the G# bass, it creates a Lydian augmented sound: G# C# E(D##) G Bb(A#). The G and A# are the maj7th and maj9th and act as dissonances but work because they make sense locally(It's just a Dim7 chord) and make sense globally as just dissonances that resolve to something that works/make sense next(A true G7b9). The point here is that it doesn't matter how you think about it. Maybe the lydian explanation I gave is nonsense because it makes more sense to think of it just as a C#o7 chord over a G# pedal. Ultimately it depends on context and what one is trying to do with it. (e.g., if you are improvising you might want to think of it as sort of Lydian augmented so you can come up with new material that still relates in some way)
The the entire point of a dim7th chord is that it has many different interpretations and this can help one reinterpret it different to lead to different harmonic consequences. Not everything will work, one has to learn how it works through experience.
Diminished 7th chords have unique tense sounds and so the point is that they can act as drivers of momentum and interest and you won't learn how to understand that unless you learn to use them through improvisation which requires you to master the technical aspects of them such as learning their arpeggiations and chord forms along with the scales that fit them.
Again, the idea is that if it fits then it fits. It may not be the best fit but the idea is not to try to narrow down but to expand. You want to understand all the ways, not just one way. There is no best way(although there is, generally, a best way in terms of memorization and recall because our brains will latch on to what is the most "obvious" understanding but that depends on our experiences).
E.g., if you can see a chord/melody/scale/theme/etc as X, Y, Z, A, B, C, Q, DFS, AIKDFJKSJF, ETEREA, etc then it is better than to see it just as X or A.
Most of the time in musical analysis we simply choose the most direct understanding because most music is pretty simple. For example, a i chord in our traditional CPP understanding generally is pretty simple and most people use it in a simple way. This does not mean it doesn't have far more complexity but that complexity has yet to be understood by modern music and modern ears. Over time things develop as people "discover" new things and they become popular. With 12-TET though there is a limitation because there has to be some separation between colors or you just have noise. So really what we do as humans is have a sort of basic palette of colors then a few colors that we can use that are very special and complex and this offers us a nice balance and set of building blocks to do interesting things. One should learn basic harmony and theory in this nature but realize it's just a topological analysis of music... it isn't music. Ultimately if one wants to be a true musician they have to develop their own understanding that transcends words and this can only come through "practice"/experience and this is what ultimately separates the "masters" from everyone else.