Implied notes in a chord

Question

Often, especially in jazz, and on guitar, the P5 of a chord can be omitted. In jazz, it's because there are sometimes quite a few other notes attached to particuar chords, on guitar it's because of the same reason, and/or impossible fingerings.

It's said that the P5 can go first due to it still being audible as a harmonic of the root - it's the second harmonic.

However, coming very close after is M3 in the harmonic series. That could mean M3 ought to be dispensible (in major chords, obviously!). How true is that?

And as a follow-up, how come that same M3 harmonic doesn't affect the 'minorness' of minor chords?

Answer 1

Everyone knows what you mean, but really the fifth is the third harmonic. It IS the second overtone though.

That fifth harmonic, the major third one, probably does contribute to the instability of a minor chord, and this is no doubt why minor chords weren't seen as completely consonant until hmmm... the end of the C17th, and not everyone agreed even then. Despite what it says about them in the Wikipedia article, Picardy thirds weren't used to provide a happy ending, but a consonant one.

About the 5th being the first to go in jazz, I think it IS a sort of general rule, but some chords call for certain notes to be left out, and others others. In 11th chords the third is usually ditched: not because it's present in the harmonics but because it's not wanted! The main characteristic of the 13th chord is the clash of the 7th with the 13th: the 11th and the 9th are commonly left out.

Answer 2

...It's said that the P5 can go first due to it still being audible as a harmonic of the root

Invert the chord with fifth omitted (E C of a C major chord). The overtone from the bass should be a B, a major seventh relative to the root. This provides no explanation about why the fifth is omittable (although it does provide some explanation of the instability of the inversion.)

However, coming very close after is M3 in the harmonic series. That could mean M3 ought to be dispensible (in major chords, obviously!). How true is that?

Depending on style I think you can omit the third and still have a clear major tonality.

If we are really talking about implied harmony and omission of chord tones, just cut to the chase and look at two part harmony. Of course there are countless examples of two part harmony which end on an octave - third and fifth "omitted" - where the major or minor tonality of the music is perfectly clear. The third can be omitted from an incomplete dominant, more about that below.

...how come that same M3 harmonic doesn't affect the 'minorness' of minor chords?

Maybe it does. Maybe the special quality of minor comes from the clash with that harmonic?

But, I think the obvious answer dissatisfies many: the overtone series doesn't provide much explanation about harmony.

There is no grand unifying theory of harmony based on the overtone series.

---

My thinking on this topic has almost nothing to do with the harmonic series except a possible connection with the chord of nature. The following is based on simple common practice harmony.

It seems a basic convention that a chord is in root position until something concretely inverts it. (Perhaps you could back that notion up further to a single tone implies a root position triad until something concretely defines the chord differently.)

C E by default is root position. Why root position by default? As a tertian chord there is nothing necessitating it to be an inverted chord so the simplest explanation is it is not inverted. The implied triad is C E G. The C is essential because it is the root. The E is essential to give the quality in the major/minor system. When a G is added nothing about the root or quality changes. The fifth can be omitted simply because the only thing it's presence does is confirm the chord is a root position chord.

C A a simple sixth immediately defines an inverted chord. There is no tertian stack going C up to A, except a 13th chord which is not part of this style, the inversion of C A is tertian A C therefore it's an inversion. The root is A. The C is the minor third. The chord's fifth E can be omitted with no loss of essential information about the chord's root and quality. (Possibly an F could be added to make a second inversion chord, but that is a special case listed below.)

So while a perfect fifth above a bass is not necessary to make clear the root and inversion type of a root position chord, the sixth above a bass is the essential thing defining a first inversion chord and is therefore necessary.

Changing C E to minor C Eb obviously requires the chord's third so that essential tone cannot be omitted, at least not without careful handling.

A seventh chord can be effectively implied with only a minor seventh over a root C Bb. This is one common case where both the third and fifth can be omitted, because the seventh is the essential quality. The fifth can be ommitted for the same reasons it can be omitted in triads as explained above. The seventh typically resolves to a third in the next chord. That third in the chord of resolution seems to mitigate the absence of a third in a dominant seventh chord. Bb C is understood as an inversion similar to the triad case above, you don't need the fifth just the inversion of the tertian stack.

So, you can't really talk about chords if you can't identify roots and you don't need a chord's fifth present to identify the root. You don't need the fifth to define major/minor quality. You don't need the fifth to define a seventh chord. You really don't need it to define anything about tertian chords in the major/minor system. That's why you can omit the fifth.

There are special cases where the fifth of the chord actually defines something about the chord and then it is essential:

6/4 inversions require the fifth in the bass by definition and those chords get special treatment.

Diminished chords are a special case too. Without the diminished fifth actually present chords like viio or iiø7 can sound like other chords. So tones forming the diminished fifth are essential.

Answer 3

I really don't know the answer and this is largely speculation, but it won't fit in a comment.

What are chords? What are triads? IMO, they are commonly known and used building blocks for shaping harmony quickly and effectively. An abstraction like Michael Curtis says in a comment. But you can define and shape harmony even with sequences of single notes. Who said you need triads to define and shape harmony? Is there a law of nature that there needs to be at least three notes to do something, and that when you have those three notes or a stack of thirds, then it becomes some kind of physical entity that it isn't if there are only two notes? I think that chords being atomic or monolithic physical entities is a myth implied by our culture where chords are used as if they were physical entities. Our musical culture gives chords that status.

Here's something I'd call almost a C - G7 - C cadence, even though it only has at most two simultaneous notes playing. Maybe there's an actual name for this:

I think that works quite well as a harmony definition. Even the topmost note alone would work to some extent.

I guess my answer to the "why can you leave out the fifth but not the third" question is: because the fifth doesn't define harmony as well as the third does. Maybe the overtone series has something to do with it, but to me that doesn't feel like a practical explanation I could use for anything. There may even be an actual biological reason for the overtones/harmonics thing, but I still cannot utilize that for anything - it could just as well be because of magic fairies. But defining harmony - that's just intuitively so, I can check if that's true by trying to leave out the third - and then I realize that my harmony became less defined! :)

Answer 4

One way to look into this is to consider the actual harmonic frequencies that would be present.

If we have a note at 100Hz, and assume it has perfectly harmonic overtones, then it will have harmonics at:

100, 200, 300, 400, 500, 600, 700, 800, 900.... Hz.

The (justly-intoned) perfect fifth up would be at 150Hz. So it would have harmonics at

150, 300, 450, 600, 750, 900.... Hz.

From that, we can see that

• The fundamental of the fifth (150 Hz) is not present in the harmonic series of the root
• none of the odd harmonics are present in the harmonic series of the root. Only the even harmonics are present.

However,

If we consider the perfect fifth in the next octave up - 300Hz, its perfect harmonic series is

300, 600, 900... Hz

All those frequencies are contained in the series of the root. Having said that, the strengths and decay lengths are unlikely to be the same as if you were to really play a note with the fundamental of 900Hz.

We can say that within the harmonic series of the root is a weak version of the perfect fifth in the next octave up.

Following that logic, there's also a (probably even weaker) version of the major third one more octave up - with harmonics at 500, 1000, 1500... Hz. And other, even weaker versions of other notes.

It's said that the P5 can go first due to it still being audible as a harmonic of the root - it's the second harmonic.

To say the P5 can go first is probably a fair statement, because that note (if we ignore higher octave repetitions of the same note) has the strongest overlap with the harmonics in the root.

However, we have to be careful of more casual statements like "The P5 is present in the root note". Playing the root note will not sound the same as playing the root and the fifth, or even the same as playing the root and actually playing the fifth in the octave up.

Equally, we can't omit the minor seventh because it's "already in" the minor third, or the major seventh because it's "already in" the major third. If we could, we wouldn't need extended chords, and we might not even have jazz at all...

Answer 5

When we talk about chords, we're typically talking in a tonal context with certain assumptions. One of assumptions is what intervals affect the overall harmony.

If we focus on simple chords for now triads and 7ths, the major flavors of the chord tend to come from the 3rd and the 7th. It's viewed from this perspective that the 5th dose not bring out the quality of a chord that much so when strapped for voices, leaving out the 5th itself. It most likely developed this way due to the strength of the 5th in the harmonic series, but it's more of a cultural reason than anything else.

The 3rd being a very strong indicator of quality typically cannot be dropped without making the harmony more ambiguous. This is not a hard rule and there are some exceptions one really common example is the fully diminished 7th which sometimes in jazz the 3rd gets dropped to better inflect the fully diminished flavor of the chord.

---

As a footnote this does not always hold true especially as you deviate from tertian harmony. For example if you were working in quartal or quintal harmony, dropping a 5th would change the overall harmony .

Answer 6

However, coming very close after is M3 in the harmonic series. That could mean M3 ought to be dispensible (in major chords, obviously!). How true is that?

I‘ve found this source that confirms your assumption reasoning exactly the same way like you.

A. B. Marx: p.193

https://books.google.com/books/about/Allgemeine_Musiklehre_Ein_H%C3%BClfsbuch_f%C3%BC.html?hl=de&id=PIUpBDV9Pl4C

Translation by google:

2. A u s l a s s u n g. One or a few intervals of a chord can be „left out“ (omitted). As a rule, the omission is the least likely to hit those intervals that belong to the excellent characteristics of the chord. If you wanted to omit the third from a triad, you would no longer be able to tell whether it was a major or minor, major or minor. If one wanted to omit the seventh chord from a seventh chord, or the none from a ninth chord, the former would again become a dri chord, the latter again a seventh chord. In the same way, the omission of the uppermost or the fundamental in a triad can make it doubtful which of two chords is meant: whether one without a fundamental or one without a fifth. 235 With the dominant chords and the ninth chords, the omission of the root note takes place with a special effect. The dominant chord becomes a new one (already p. 189 13

In a quintfall sequence of 7th and 9th chords the 3rd or 5th can be omitted as we are in a diatonic progression. Our mind and cognition will fill the lacking tones.

Btw. in early music the the final chord had to be built only by perfect intervals.

Answer 7

It's said that the P5 can go first due to it still being audible as a harmonic of the root - it's the second harmonic.

This harmonic (actually the third harmonic) is not audible. The ear-brain system takes any periodic wave and fuses it into a single sensation of tone. This tendency is so strong that often multiple notes will sound like a single note if they have a common period. So in fact, even if you play the P5 in the voicing, it may be difficult to "hear out," depending on various factors. This is the reason for the traditional prohibition on parallel and direct fifths and octaves in classical voice leading. So a better explanation would be almost the opposite of the one you propose: we don't leave out the 5th because we hear it whether it's present or not -- we leave it out because we may not hear it whether it's present or not.

We typically try to include the 3 and 7 in any voicing, and we tend to leave out the 1 and 5. The reason for this is mainly that we're talking about functional harmony, and the function of the chord depends completely on the 3 and the 7. The chords A7, Am7, and Amaj7 all have completely different functions. For example, the A7 typically acts as a dominant leading to a D or Dm chord, whereas an Am7 is not capable of doing this job.