Why does the dominant lead to the tonic?

Question

Going from a dominant to the chord to which it is a dominant always sounds good to the ear. It feels like the chord leads to the next one. And if the composition or movement is written in a specific key, then when the harmony reaches a dominant chord, you feel a sense of unease and instability. Moreover, if the 7th key is added to the chord, the effect is increased.

I've heard that the reason for this has to do with half steps. For instance, if we consider the key to be C-major, then in order to form a C-major chord from a G-major chord, you only need to change the B to a C (minor second difference) and the D to an E (major second up). However, this also applies to the subdominant F-major, but the effect is completely different. And of course, you can go the other way - from the dominant to the tonic - and the steps are the same but reversed. So, it seems to me that just saying that the steps cause this effect isn't the full truth.

Basically, is there any known explanation as to why the dominant sounds like it leads to the tonic? When we listen to music, why does it sound so natural when chords follow in sequence through different dominants?

Answer 1

You've done a nice job of describing the V-I progression, and you've identified the resolution through half steps and whole steps. All of this is important to understanding why the V-I sounds so good, but it's not the whole story. There is something else that a V-I progression possesses which is not present in a I-V progression or a IV-I progression. In a V-I progression, the V chord contains a chord tone that resolves to the root of the I chord. This is a crucial distinction which allows the V chord to resolve strongly to the I chord.

For example, here's a V⁷-I resolution in C maj, taken from Beethoven's Fifth Symphony:

The V⁷ chord contains a B which resolves to the root of the I chord, C.

By contrast, playing | F maj | C maj | (a IV-I progression) would not be the same. The notes of the F maj chord are F A C, and the notes of the C maj chord are C E G. We can set up our chords so that:

• the F resolves down a half step to the E
• the A resolves down a whole step to the G
• the C remains a C

As this shows, there are no notes from the IV chord (F maj) which resolve up or down a step to the root of the I chord (C maj). Rather, in this progression (with this particular voice leading), the tones of the IV chord only lead in to the 3rd and 5th tones of the I chord. To our ear, this IV-I resolution is weaker as the IV's chord tones provide no movement toward the root of the I chord. That movement toward the root of the I is present, though, in a V-I progression; in fact, our ear anticipates that movement/resolution of the chord tones, making the V-I progression even more natural sounding.

The instability of the V⁷ chord, as you've described, is largely due to the presence of a tritone (diminished fifth) interval in the V⁷ chord. In particular, the V⁷ chord contains both an F (the seventh of the V⁷ chord) and a B (the third of the V⁷ chord), and these two notes are 6 half notes apart. That particular interval has a quality of instability and dissonance that further drives the ear to desire resolution. This is part of why a I-V progression would not sound like a resolution.

When analyzing a progression, a key thing to look for is the particular chord tones involved in the resolution. For example, resolving to the root of the tonic creates stronger movement to the tonic than resolving to the fifth of the tonic.

Answer 2

Take a look at this link here : https://www.historicaltuning.com/Harmonics.html It gives an example of particular keys on the keyboard, and discusses harmonics. One of the G keys shown happens to be the third harmonic relative to a 'fundamental' frequency note that happens to be one of the 'C' keys. And there are information sources out there (such as physics and applied science) that discuss adding particular harmonics together -- eg. if you add the 'fundamental' (aka first harmonic) and the third harmonic together. When they do it, some sort of 'constructive' (non-clashing) effect occurs when we look at the resulting 'summed' waveform (combined result). But when we focus on the third harmonic ----- this one, and the first harmonic ----- the transition from third to first harmonic (aka fundamental) appears very natural - without clashing ----- due to their particular constructive, non-clashing nature.

Also, when we're talking about a harmonic, it refers to a sinusoidal behaviour. And a sinusoid can be defined by a frequency, and an amplitude, and also a reference point in time. So if we compare a 'fundamental' sinusoid with a third-harmonic sinusoid, then we would be able to see that the peaks of both the fundamental and the third harmonic sinusoid are "symmetrical" (in a time plot) ----- symmetry around the point in time where the fundamental has a peak (highest point). Eg. see this link here : https://electrical-engineering-portal.com/harmonics-what-are-they-what-do-they-do

However - unlike music notes, the information in the above link does NOT have the third harmonic being at the same 'amplitude' as the fundamental. But that's ok, as that information is under a slightly different context. We're just borrowing their diagram only.

And the third harmonic just so-happens to have its highest point there too - at the same time. So this relates to the strongly additive nature of fundamental components and third harmonic components. They're sort of (or strongly) on the 'same page' ---- or a nice match-up. So hopefully these details can help go toward getting ideas about why 'dominant' notes transition nicely (resolve) to fundamental/root/tonic notes.

Although ----- when we strike a particular note on the piano, such as hit a 'G' key, then the sinusoidal waveform won't necessarily be (at first) lined up (in time) with the 'fundamental' waveform. But - maybe - as seen in some science experiments, some physics mechanism occurs, in which the waveforms become aligned. And maybe electronic tone generators may need to have this sort of alignment as well. Or --- maybe human ears aren't sensitive to timing differences when we transition from a third harmonic note to a fundamental note. I thought I'd just mention it - just in case!

Edit (addition) - a contributor in the comments section made an excellent point - in that - for example, when G resolves to C, the two notes are not sounded simultaneously. There is certainly an interesting link between third harmonic frequency and fundamental frequency.

Just giving an example - to be clear about third harmonic frequencies. Take the dominant chord (G-B-D) from C major. Then consider the root chord of C major, which is C-E-G. In the frequency chart --- you'll find that G (ie. first note of the dominant chord) is the third harmonic of C (ie. first note of the root chord). And B (2nd note of the dominant chord) is the third harmonic of E (2nd note of the root chord). And D (3rd note of dominant chord) is the third harmonic of G (3rd note of the root chord). So - when comparing counterpart individual notes between the dominant chord and the root chord --- we have three 3rd harmonic notes transitioning to three fundamental notes (when we transition from a dominant chord to a root chord). Quite interesting.

Just to be extra clear ----- let's take the note C2 on the keyboard, which has a frequency of 65.406 Hz. The third harmonic frequency is three times that frequency, which is 196.218 Hz, which is close enough to 196 Hz, which is the G3 key. So that takes the G to C transition into account.

And let's take the highest note of the root chord, which is G2 (97.999 Hz). Then the third harmonic frequency will be three times that ---- ie. 293.997 Hz, which is close enough to D4 --- which happens to be the highest note of the dominant chord (ie. highest note of G-B-D is 'D').

And one extra observation that could be important too is ------- note that the highest note in the dominant chord (for our example) is a 'D'. So - when we play the notes of the dominant chord one note at a time, we get G, then B, then D. And if we immediately follow on by playing all three notes of the root chord (all at once ----- C-E-G simultaneously) ....... then the overall result - to our ears, will actually sound like playing these four notes (individually) in sequence ---- G-B-D-C. The step-down from D to C sounds natural ---- a stepping down of the 'major 2nd' from D to C. And - the 'C' note (being the root/tonic note of the root chord) will have its side-kicks E and G (as part the root chord). And those two side-kick notes will just be secondary ------ as 'C' will have the biggest influence in the root chord. That is, even though the root chord is played (with C, E and G played simultaneously), it really sounds more like an overall 'C'. And we just feel that this sequence --- G-B-D-C ----- like a 'done deal' when we play it. Or, alternatively, if we play the dominant chord notes (simultaneously G-B-D), followed by playing the root chord (simultaneously), then --- to our ears, it just sounds overall like playing a 'D' followed by a 'C', which also sounds like a 'done deal'. This sort of natural terminating pattern certainly occurs when the two chords of interest happen to be the dominant chord and the root chord.

Also - there is also the situation about what happens in our mind (or brains) - in the way we process sound information. For example - somebody or some animal that hasn't been conditioned to recognising and accepting audible frequency sound sequences - such as a transition from a 'dominant' to root - probably wouldn't make much of this interesting pattern. Maybe getting into realm of 'psycho-acoustics'.

Answer 3

Yes, it's about the minor second ("half step") movement.

The progression can be generalized in terms of root progression and voice leading: root progression by descending perfect fifth, and voice leading of one voice held and two voices moving by step.

In the diatonic gamut that only happens in two places: Vb I and Ib IV. (b means first inversion.) There is a sort of equivalence in those two progressions in terms of root progression and voice leading. In a purely diatonic context the two can be distinguished by making V a dominant seventh chord, like V7b I. You cannot do the same diatonically moving to IV and the added seventh in V7b provides two minor-second motions resolving to I in TI DO and FA MI.

It's interesting to compare these progressions with some other progressions that also have the voice leading of one voice held and two moving by step:

• ivc I the borrowed minor iv
• IIIc I a chromatic mediant relationship
• Fr+6 V an imperfect cadence progression (Fr means the French sixth)

While those chords are all chromatic it's important to note the chord roots are diatonic. So while they are chromatic, they are also quite "at home" in a key.

Technically Fr+6 V is root progression by descending P5 and it has two minor-second resolutions, with one of those moving to the chord root of V. That is quite strong.

ivc I is root progression by descending P4 a bit "less strong" than descending P5, and the non-movement to the root of I as a held note provides some theoretical explanation of why it's "less strong."

IIIc I is interesting, because while the root progression would be considered a "weaker" roots by descending third it does have a minor-second motion to the root of I. This is a chromatic mediant relationship. And while the root progression may be "weak" the chromatic mediant is unique and very identifiable. They are "strong" in chromatic color and expressive potential.

To some degree I'm simply saying an important, unique distinction for V7 I, in addition to the minor-second movements, is that it's a diatonic progression. But some other chromatic progressions share similar characteristics. It's good to be aware of those progressions. And they also reinforce the notion that minor-second motions do indeed create "strong" progressions.

Finally, I think it needs to be stated that the "strength" of the dominant leading to the tonic, the strength of the leading-note moving up to the tonic, is a matter of convention. It's perceived as strong, because that is how the major/minor system treats it. However, in modal style, in the Phrygian mode, this convention is literally turned upside down. In Phrygian the "leading note" is a minor-second above the tonic. Final cadences might look like vii I or viib I, a minor subtonic chord moving to a tonic with a tierce de Picardie (two voices moving by a minor second, but root progression by a major second). (In the Phrygian mode, the seventh degree --- the root of the chord vii --- is a major second below the tonic.) That progression totally reinforces the theory that minor-second movements make strong progressions, but allows for tonalities other than only major and minor keys.