Harmony and the Harmonic Series: I-V to Rule Them All

Preamble

Since writing my article on Music and Evolution, I have received many emails asking the same question: "How does harmony and the movement of chords function in conjunction with the harmonic series and our ability to predictively 'hear' simple resolutions to complex wave relationships?" This article will try to answer that question by delving into the relationship between the three most important chords in tonal music—I, IV, and V. These three chords and their relationship with each other are at the foundation of all types of tonal music—classical, jazz, folk, country, rock, pop, etc. As such, an exploration into how the harmonic series governs these relationships provides, by extension, an explanation for how all of the chordal relationships in tonal music function.

I. Background: The Ubiquitous Tonic (I), Subdominant (IV) and Dominant (V)

These chords are created by building triads on the first, fourth and fifth scale degrees of any major or harmonic minor scale. Here are the triads built on each scale degree from the key of C major:



Note that I, IV, and V are the three major chords (indicated by upper case Roman numerals), ii, iii, and iv are minor chords (as indicated by lower case Roman numerals), and vii is a diminished triad (indicated by the "o,") which has a minor third and a diminished fifth.

The proper names of the chords require some explanation, particularly the names of I, IV, and V. The chord on I is the tonic (tonicus in Latin), signifying that this pitch is the tonal center of the scale, which therefore also identifies the key. The chord on V is named "dominant" because it is the most powerful of all—it elicits the strongest response in listeners' expectations of what "should" follow in the next chord. Our remarkable aural faculty automatically calculates the simplest solution to the complex wave relationships, which makes us "hear" the tonic after the dominant as a "resolution." ¹

The subdominant chord is on scale degree 4. It is named "subdominant" ("the under dominant") because it is the same distance—a perfect 5th—from the tonic as the dominant but in the opposite direction. Its name is thus derived from the all-important tonic/dominant relationship²—it is the dominant's inverted mirror image.


These chords, and their minor key versions, are ubiquitous in tonal music of all kinds. In particular, we find that virtually every piece of tonal music, in all styles, ends with the movement of IV-I³ or V-I. These are so common that they have proper names—IV-I (or iv-i) is called a "plagal cadence," and V-I (or V-i) is called an "authentic cadence." In most styles of music, however, the authentic cadence is found much more often as the definitive closing cadence than the plagal cadence. (In classical music, the plagal cadence is more likely to be found after the authentic cadence as a palliative, soporific "Amen" to end the piece.

The number of songs, particularly in popular music of all kinds, that rely on I-IV-V (along with a few minor variations) is likely in the hundreds of millions. Here are a few examples from popular music, classical music, and blues/jazz, along with the chords used:



In "Louie Louie," the three chords are ordered as follows: I-IV-V-IV, which repeats over and over again.



In "Twist and Shout," the three chords are ordered as follows: I-IV-V, which also repeats over and over again.



The anthemic chords in "Baba O'Riley" that enter after the arpeggiated synth introduction are I-V-IV, which, of course, repeats over and over again. The "bridge" (2'18") provides some textural and melodic variation, but it does not provide any harmonic variation—that entire section is on V.

Here is an example from classical music, from the first movement of one of Mozart's most famous pieces, "Eine Kleine Nachtmusik."



The first 20 seconds are I-V-I-V-(over tonic pedal)-I-V-(over tonic pedal)-I. (Note that the repetition found in the other pieces is missing here.)

And finally, one cannot talk about these chords without acknowledging that the blues, upon which hundreds of thousands of tunes are based, is I-IV-I-I-IV-IV-I-I-V-VI-I-I (12-bar blues).

II. The Birth of the Chord

To understand why chords move to other chords and how that is related to the harmonic series, we first need to examine the concept at the heart of this discussion—the idea of a "chord"—from a broad historical perspective.

Classical musicians, composers and music theorists before the early 18th Century were undoubtedly aware of what we now call "chords"—i.e., triads built on the seven scale degrees of every major or minor key, with notable hierarchies in terms of their power to invoke the tonic.

The idea of "chords" as we know them today, came to the forefront when composer and theorist Jean-Philippe Rameau introduced his theory of "chord inversion" in his groundbreaking work from 1722 entitled "Treatise on Harmony." Before Rameau, we can be sure that composers and musicians were aware that each inversion of a chord contained the same three pitches as the root position chord. They saw that the inverted chords have different harmonic functions and different levels of stability. They did not, however, consider them as primarily vertical phenomena but rather as horizontal ones—i.e., sonorities created by linear movement. As shown in the example, before Rameau's theories took hold, these sonorities were built using the interval distances above the bass note, a notation system known as "Figured Bass."

Rameau recognized that the various ways a chord could be found— root position, first inversion, second inversion—could be conceptualized as being three versions of the same triad named after its root (C major or C minor, for example). After Rameau, the root position and its two inversions were seen as the same chord, albeit in different inversions. Theorists and musicians adopted Rameau's theory, and we use it to this day.

It is a useful tool that gives musicians an effective short-hand code, which jazz musicians, with the sophisticated chord symbols found in lead sheets, have perfected as what could legitimately be called "Figured Soprano"—a modern-day version of the Baroque era's numerical nomenclature. While they consist of the same three notes, their harmonic tendencies are entirely different, to the point where the second inversion is a doppelganger of sorts that looks like a C major chord, but it functions most often as a precursor to the dominant chord, making it a double suspension over the dominant. It is also, along with the first inversion, used as a passing chord, which speaks to the lack of ability of both first and second inversions to assert themselves as the tonic.

So, like all theories, Rameau's has strengths and weaknesses. It is undoubtedly a convenient and helpful way to quickly and easily identify chords in this manner, but it masks the differences in how the inversions function and how they are used. Every budding jazz musician finds this out very quickly—it is easy to play the chords, but without proper voice leading, they do not sound very good. This vertical approach does not, for the most part, show how to connect the chords in a way that acknowledges the linear/horizontal movements needed for a melodically pleasing unfolding of the harmony. In other words, chords need to move from one to the other almost like a multi-voiced choir would sing them—largely stepwise movement, leaps followed by stepwise motion in the opposite direction, and with some measure of independence between each "voice" in the chord "choir."

The horizontal element is not present in the vertically focused chord approach, which is why a beginner jazz musician can play from the same lead sheet as a professional, but the resulting chords or improvisations are starkly contrasting—the professional knows how to voice the vertical structures using melodic principles and good voice leading, which is difficult for a beginner or amateur to do. While the horizontal element is essential in classical music and jazz, it is less so in popular music, where barre chords, which move in parallel motion without independent voice leading, are regularly utilized; indeed, the lack of voice leading is part of what defines and demarcates the genre's sound.

This very same voice-leading makes the relationship between these three chords so powerful, but, as described above, that voice-leading is not seen in the chordal "foreground."

This now brings us to the original question regarding how harmony and the harmonic series are related. To that end, we will first look at the 11th partial (F#) that creates the Lydian mode and implies that the Lydian mode is, perhaps, the "natural major scale." We will then look at the voice-leading relationships between the dominant and the tonic, the subdominant and the tonic, and how they function in regards to the harmonic series.

III. The Missing Link?: The Subdominant in the Harmonic Series

Given the importance of the subdominant, we might expect it to have a prominent position in the harmonic series, but it does not appear in the first 16 partials. ⁴



Curiously, the subdominant does occur as the 11th partial, but it is raised by a half-step (and is somewhat out of tune). This has been a matter of discussion by musicians, composers, and music theorists. Since the raised subdominant is found naturally in the harmonic series, why is music not based on the scale with the raised subdominant, which is the Lydian mode?⁵ The answer is that the Lydian mode is less stable than the major/minor scale, and less stability means less potential for powerful resolutions and modulations to other keys.

Why is it less stable? It is less stable because the subdominant in Lydian mode is a diminished triad, with the most unstable interval, the tritone, built on the leading tone of the dominant, making it a secondary dominant chord. The supertonic in Lydian is a major chord, the dominant of the dominant (V/V), which destabilizes the tonic. Additionally, in Lydian, the leading tone chord is a minor triad, stripped of the tritone that so powerfully begs for resolution to the tonic. Thus, in the Lydian mode, there is a power struggle between the dominant and the tonic. At the same time, in the major-minor scale with the diatonic subdominant, there is no such struggle—the tonic reigns supreme and unchallenged. This is why the major-minor scale system is much more stable, robust, and imbued with greater harmonic potential than the Lydian mode, or any of the other modes.⁶ Compare the chords in Lydian with the chords in the major scale:



To answer the question regarding the harmonic series and harmonic movement, we must first explain why the subdominant is such a powerful chord in terms of its ability to summon the tonic. This presents us with somewhat of a mystery—the second most powerful chord in any key is not found in the first 16 partials of the harmonic series. How can that be? I believe the answer is to be found in the harmonic series and the voice leading that exists between the dominant and the tonic chord.

First, why is the dominant chord the most powerful in terms of its pull to the tonic? If we look at it from a voice-leading perspective, instead of as a triad/chord, we see the linear relationship between the tonic and dominant.



Here we see the following: ⁷
1. The fifth of the tonic chord (G) is retained in all of the chords.
2. The tonic (C) moves to the supertonic (D) and leading tone (B) by step in opposite directions.
3. The supertonic (D) and leading tone (B) converge back to the tonic, creating a powerful resolution.

The fifth is a common pitch, which functions linearly as a pedal, but the other two pitches introduce strong dissonances with the tonic. Specifically, the two are tonic-supertonic (1:9, C-D) and tonic-leading tone (1:7, C-B), which occur simultaneously, creating even greater complexity as the two interact. The fifth of the tonic, G, becomes the root of the dominant chord, which is the 3rd partial of C and is thus a consonance. We should note that these are "background" or "hidden" dissonances that occur only in the listener's mind, who has been acclimated to the key and hears C major as the tonic, even when it is not being played. Thus, the harmonic series governs the harmonic movement. When the complex waves are created, the aural faculty recognizes the complexity—it is a dissonance that requires resolution. Our aural faculty then supplies the listener with the solution, which the listener yearns for automatically. That solution is the nearest factor of two, which is what the tonic (C) provides when the tonic-supertonic (1:9—1:8, D-C) and tonic-leading tone (1:7—1:8, B-C) resolve downwards and upwards respectively.

This remarkable phenomenon is masked by the fact that the dissonant pitches, the supertonic and leading tone, are consonant with the G major chord that is sounding but dissonant with the tonic that exists in the listener's mind.

IV. The Return of the Subdominant

With that in mind, we can now revisit the mystery of the subdominant as the second most powerful chord. As mentioned, we can find the subdominant by taking the distance between the tonic (C) and its 3rd partial (G), a perfect fifth, and inverting it. Thus, we arrive at the subdominant (F), which is a perfect fifth down from the tonic.

As a mirror image of the dominant, it also mirrors the tonic-dominant-tonic's voice-leading characteristics. Visually, it is the literal "upside down" version of the movement from tonic to dominant.



Here we see the following:
1. The tonic (C) is found in all of the chords.
2. The dominant (G) moves to the submediant (A) and back by step in opposite directions.
3. The mediant (E) moves to the subdominant (F) and back by step in opposite directions.

In this case, because the tonic is found in all of the chords, the overall effect is that of a double suspension, a movement away from the tonic that is largely illusory because the tonic is still there. It is, nonetheless, still immensely satisfying as the strong dissonance between the tonic- subdominant (1:11, F♮) resolves downward to the closest factor of two (1:10, E). ⁸

Viewed from this perspective, the subdominant works because it is an inversion of the dominant. However, it "works" because it inverts, and thus replicates, the same powerful voice-leading principles as the dominant, albeit not as powerfully because the tonic is still present in the chord, unlike the previous example.

None of the other chords in the key have similar relationships with the tonic. Indeed, other chords share pitches with the tonic, but none retain important common pitches (tonic in the case of I-IV-I and dominant in the case of I-V-I). In fact, the other chords function as weaker substitutes of I, IV, and V:



Movements between all of the other chords function with similar voice- leading principles as we have seen with the primary three chords, which means that the relationship between tonic and dominant governs them. Dissonances are created in the foreground (melody, chords), and they function within a broader background hierarchy in which the key and its tonal center are perceived by the listener, even when the tonic is not being sounded in the foreground level. (That being said, given that the secondary chords function as weaker versions of the primary chords, the ethereal ghost of the tonic asserts itself quietly and regularly until there is a change of key, when another tonic takes the helm.)

V. In Conclusion

The entire harmonic enterprise is thus founded upon the relationship between the tonic and dominant, which is itself powered by the harmonic series. All of the other harmonic relationships are iterations of the tonic-dominant voice-leading tendencies and inherent hierarchy of consonance and dissonance contained therein. Our aural faculty recognizes those consonances (simple waveforms whose nodes⁹ are shared or align symmetrically—1:2, 1:3, 1:4. etc.) and dissonances (complex waveforms that do not have nodes that align symmetrically with the nodes of the simple waveforms—1:7. 1:11, 1:13, etc.) and automatically calculates the simple solution to the complex waveforms by conjuring up the very pitch that most readily "resolves" the complexity. Our brains provide the imagined expectation of the nearest simple waveform that is a member of the very same binary sequence (y=2^x) found at the heart of our computers. It is a truly miraculous process made possible by a similarly miraculous human brain that processes it as it does. Without that, music would not be possible—we can only imagine what an impoverished existence it would be without this glorious art form to accompany us, entertain us, console us, enthrall us, and ultimately, elevate and inspire us to realize our full potential as creative and expressive beings.

VI. Epilogue

Heinrich Schenker was an Austrian music theorist whose theory I wrote about in a recent article on jazz and pop music. His theory, in a nutshell, is that classical music, even the most significant works—the best ones at least—all reduce to I-V-I. Thus, it was impossible to delve into this topic and not see the Schenkerian connections. From the perspective in this article, Schenker was right, but perhaps too limited in scope, for it is not just singular works of music that reduce to a structural I-V-I—instead, it appears that at the ultimate macro level, I-V-I is indeed the engine of all tonal music, fueled by the binary sequence in the harmonic series and processed by the organic computer that is our aural faculty.

Endnotes

¹ For some background and context, I suggest reading the article on Music and Evolution: Hearing Math, Seeing Sound, and Other Unanswered Questions that prompted the question which this article attempts to answer. ↩

² The chords on scale degrees 3 and 6 are similarly named in reference to their relationship to the tonic/dominant. The mediant (iii) is the midway point between the tonic and dominant, and the submediant is the midway point between the tonic and the subdominant. The remaining two chords, on scale degrees 2 and 7 are the supertonic and the leading tone, respectively. The supertonic means that it is one step "over the tonic" and the leading tone means that the pitch on scale degree 7 "leads" (resolves) to the tonic. Thus, all of the names of the chords rest on their relationship to the tonic and the tonic/dominant.↩↩

³ In pop music, some songs end without resolving to the tonic, ending on the subdominant, which does not happen in classical music or jazz. I hear this phenomenon as a "harmonic fadeout" that gives the impression that the song continues on in perpetuity. ↩

⁴ After the first 8 partials, the notes in the series start to show marked tuning issues. The subdominant is there as the 21st partial, but it is seven cents sharp.↩

⁵ i) Some have made a strong case for the fact that the major scale with the raised fourth music should used as the "natural" scale. Most notably in the jazz world, it was George Russell's influential text from 1953 entitled "The Lydian Chromatic Concept of Tonal Organization" that brought the idea of using this scale (and by extension, using modes) to jazz musicians in the mid- 1950s. Similarly, in classical music, the late 19th and early 20th Century composers like Debussy, Ravel, Stravinsky, Bartok, Holst, and others leaned heavily on the Lydian mode (and others) as they moved away from the major-minor scale system in search of new means of expression.
ii) Additionally, others have noted that the distance between the tonic and the subdominant going up is a perfect fourth, an interval which does appear between the third (G) and fourth (C) partial. It is not, however, clear to me how that provides an argument for the perfect fourth above the tonic (F) as somehow being "in the series."↩

⁶ This should not be read as a criticism of the modes or music based on the modes—the modes provide expressive capabilities that the major-minor scale system does not, and vice versa.↩

⁷ The third (E) of the tonic chord moves down to the supertonic (D) and back up to the third of the tonic. Regardless of that motion, the supertonic is still a dissonance against the background tonic.↩

⁸ Depending on the position of the pitch of resolution, the ratio may be 1:5, or 1:10, but that is not an important distinction. I am mostly ignoring this type of octave displacement in order to make the resolution easily apparent to the reader. ↩

⁹ A "node" is a point on the wave that is not vibrating.↩

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