Melody. It is the main musical line that we hear and hum. It is the essential ingredient of melodic music. (Groove is the essential ingredient of rap.)

There are a few different approaches to making melody.

We’ve discussed a lot about chords in this blog. A melody can follow chords. Once you’ve created a sequence of chord changes, you can place a melody on top which consists of notes found in those chords.

We’ve also discussed scales. Another approach to melody is building it from notes within a particular scale. This is commonly done with major and minor scales, but modal scales and altered scales are also fair game.

Melodies don’t even have to change note; they can be rhythmic patterns of a single note or a small collection of notes. This is found is some hip-hop vocal melodies.

Something important to note about melody is that notes of higher frequency tend to ring out more; our attention is drawn to them. Try to use these climatic highs at a focal point in the piece of music.

A melody must have a balance between conjunct movement and adjunct movement. Conjunct movement is when the notes of a melody do not leap farther than a whole or half step apart. Adjunct movement is when the notes of a melody leap farther than a whole step apart. Effective, memorable melodies often employ large adjunct movement, sometime leaping an octave or farther.

Melodies are often repetitive, but change slightly each time the repeated line is heard. Once you’ve created a small musical phrase, you can repeat it, but alter the notes or rhythm slightly. This makes the melody “inevitable yet surprising”, a phrase used by the great Leonard Bernstein.

Some melodies employ a technique called counterpoint. Instead of writing one single melodic line, write two melodies and interweave them. This was a musical form of which Bach is considered the master.

These are varying methods for going about melody. There are so many components that make up a song — harmony, rhythm, timbre, texture — but if you don’t have the melody, you don’t have a song…at least not one that listeners will remember.

All of the other components are a contextualization for the melody. A melody can convey a very different feeling depending on the context that surrounds it. As someone who aspires to write his own music, mastering melody is a large priority, and it’s something that all aspiring musicians should understand.

Have you ever wondered: what does music mean?

As I discuss in this blog’s very first post, we all consume it constantly. It’s clear that we as humans have a particular affinity for music. In fact, as I write this, a boy sitting a few seats down from me at the same table is jamming out on his invisible drum set very nonchalantly. He is oblivious to his surroundings, including me, who has been intently observing him for the past ten minutes.

But what does it mean? No doubt there are scientific reasons we find it alluring (harmonious frequency waves and the like), but that is not the subject of this discussion.

Music speaks to us. But what is it saying? What are the lyrics saying? What are the sounds saying?

This has bothered me more and more over the course of the past several months. Are pop songs consumed en mass glorifying indulgence in sex and drugs really spreading meaningful or instructive messages? Are soulful Motown love songs telling us anything insightful or new? Is the massive library of classical and jazz music anything more than a pleasing collection of sounds? On that note, what message does instrumental music communicate?

Music isn’t so explicit about what it’s saying. It seems to be more sensual and less literary or academic.

I suspect this is why I’ve been a long fan of musical theater (especially of the Sondheim variety, who tragically passed last year). By embedding music in a specific setting with a specific plot and characters, its meaning is made more explicit. In this setting, music is used as a hyper-stimulant which emotionally bolsters the storytelling on stage. When a musical has complex philosophical themes or social commentary, I am especially intrigued and invested in its music.

But musical theater is only a tiny fraction of the music which captures our (and my) attention. What about music that isn’t tied up in a larger narrative structure? It is true that pop songs sometimes tell stories (“Cat’s in the Cradle” by Harry Chapin) or have a social message (“Big Yellow Taxi” by Joni Mitchell). But many times this is not the case. Music can be quite abstract. Many young people have taken up the hobby of producing beats with Digital Audio Workstation technology, and don’t even put words to their music. They just like how it sounds–they like the “vibe.” It’s difficult to decipher what these homemade beats, or a solo cello concerto for that matter, are saying to us.

The truth is, music is an abstract art form. Sometimes the lyrics are important, sometimes they’re not. Sometimes there are no lyrics at all. Music seems to elicit emotional experiences which give listeners an intimation of profound meaning, beauty, and order. We just can’t quite articulate what this meaning is in words. That being said, music might possibly be the closest humans have come to uncovering the meaning of life of Earth (depending on your religious beliefs).

We feel it…but we can’t say what it is exactly–an incredibly frustrating reality.

Ah yes, ’tis a beautiful day. The sun is shining, the sky is clear, and the birds are chirping in impassioned tones: “give us more music theory!” I acquiesce to their melodious siren song, but do intend to also impart my musical musings upon you, the reader. After all, music theory is not just for the birds.

Today we’ll be talking about two rhythmic concepts: polyrhythms and swing percentages.

First up, polyrhythms.

A polyrhythm can be defined as the simultaneous playing of one rhythmic value – such as straight eighth notes – against another contrasting value – such as triplets – over a single meter. Polyrhythms have deep roots in non-Western music traditions from Indonesian and African cultures. They are commonly found in contemporary rock and jazz.

The interplay and juxtaposition of seemingly opposing rhythms are what give polyrhythms their character. A common way to think of these patterns in musical notation is to envision/hear them as ratios. Triplets against eighth notes would have a ratio of 3:2. Other simple polyrhythms are 3:4, 4:3, 5:4, 7:8 and so on.

3:2 in Western notation:

4:3 in a different visualization:

Polyrhythms can become quite complex, and multi-layered. Take, for example, 2:3:4:5:6 (as heard toward the end of this video.) They can add an interesting rhythmic flare to music, and are worth checking out.

Next, swing percentages.

Even if we are not conscious of it, we all have a feeling for “swing” versus “straight.” If you’ve ever played in a jazz band, you are keenly aware of this stylistic difference. Classical composers of centuries’ past tend to prefer straight timing, whereas swing was all the rage in the 1930’s era of big bands.

Swing refers to a type of rhythm where alternating subdivisions are given unequal durations, creating a long-short-long-short pattern. A common type of swing is a triplet swing in which the first note has twice the length of the second note.

Triplet swing is just one of many possible ways that subdivisions can be swung. One way to conceptualize these variations is with percentages. A swing percentage refers to the percentage of the beat that is taken up by the first note. For example, 50% corresponds to perfectly straight time where both notes have an equal duration. 57% is septuplet swing. 60% is quintuplet swing. 62.5% is golden ratio swing. 66.6% is the aforementioned triplet swing. Want to swing harder? Try 80% on for size.

All of these percentages are valid in the right musical context, and can be found in jazz, rock, and electronically-produced music.

Now, with the knowledge of these niche yet powerful concepts, you are free to spread your wings and make a really sick beat.

This week, we’ll be discussing altered chords and altered scales…you know, for all of you jazz buffs out there.

As I’ve stated before, a major concept in music is the oscillation between tension and release. Chords which often create tension in music are dominant chords (major chords with a flat seven). Dissonance is created between the major third and the flat seven, which sit a tritone apart. The chord wants to be resolved, most likely to a stable major or minor chord. While the standard dominant seventh chord produced adequate tension for many classical composers of centuries past, jazz musicians of the twentieth century simply were not satisfied. They wanted more. Hence, the altered dominant chord was born.

Altered dominant chords are extended chords, reaching into the territory of ninths, elevenths, and thirteenths. We’ve discussed these upper extensions before. To create an altered chord, we begin with the standard dominant chord–root, major third, fifth, flat seventh. Then, we add upper extensions. Then, we alter the fifth and/or the upper extensions by a half-step (and likely drop the fifth in our voicing). Some common C altered dominant chords are pictured below:

Or, if we’d like to get a little crazier:

Seems complicated right? These chords will become a lot easier to work with once we discuss the altered scale.

The altered scale is as follows:

This is the scale which many jazz musicians use to solo over altered chords. In sequence, it includes the flat ninth, sharp ninth, major third, sharp eleventh, flat thirteenth, and flat seventh.

This is perhaps the more “traditional” way of thinking about the upper reaches of tension. When attempting to create a lot of dissonance, I prefer to conceptualize it as clumping a bunch of notes together which clash, yet all can move smoothly into a resolving chord by motion of a half or whole step. This is laid out brilliantly by Jacob Collier, a young British musician and music theory wizard who you should check out if you enjoy this blog.

For example, let’s look at a perfect cadence from G to C. With the G note in the bass, I can put any note I want above it…as long as it resolves nicely to C. Above the G, I can use an F#, Ab, Eb, C#, D, Bb–you name it. Together, these notes are extremely dissonant, but can move in tandem to notes within the C major 7 chord, creating a satisfying resolution. F# and Ab can move to G, Eb to E, Bb to B, C# to C. There’s an infinite number of dissonant combinations which can resolve to a stable major or minor chord with stepwise motion.

Just how “altered” can we get? The choice is up to you.

Disclaimer for new readers: Much like math courses, each blog post here builds upon the concepts which have been introduced in previous posts–it is cumulative. If you do not know much about music theory, I recommend checking out earlier posts.

Today, we continue our study of harmony. We have discussed modes in the past, but not exactly how to apply them. That’s where modal interchange comes in.

Before we go any farther, a few important terms.

Modulation is changing keys for an extended period of time. Tonicization is changing keys for a short period of time through the use of a secondary dominant (this was discussed in last week’s post). Modal interchange is using chords from a parallel key while retaining the same tonal center. Tonal center refers mainly to the root note of the 1 chord. Parallel key and mode are synonymous terms which refer to scales containing different notes but sharing the same root note.

The important difference between modulation and modal interchange is a modulation changes the tonal center, while modal interchange does not.

Note: Modulation results in a change in key signature, as pictured above.

Let’s review the modes of the major scale and their related seventh chords, pictured in this chart:

Scale	1 (Root)	2	3	4	5	6	7
Ionian	IMaj7	IIm7	IIIm7	IVMaj7	V7	VIm7	VIIm7♭5
Dorian	Im7	IIm7	♭IIIMaj7	IV7	Vm7	VIm7♭5	♭VIIMaj7
Phrygian	Im7	♭IIMaj7	♭III7	IVm7	Vm7♭5	♭VIMaj7	♭VIIm7
Lydian	IMaj7	II7	IIIm7	#IVm7♭5	VMaj7	VIm7	VIIm7
Mixolydian	I7	IIm7	IIIm7♭5	IVMaj7	Vm7	VIm7	♭VIIMaj7
Aeolian	Im7	IIm7♭5	♭IIIMaj7	IVm7	Vm7	♭VIMaj7	♭VII7
Locrian	Im7♭5	♭IIMaj7	♭IIIm7	IVm7	♭VMaj7	♭VI7	♭VIIm7
Melodic Minor	ImMaj7	IIm7	♭IIIMaj7#5	IV7	V7	VIm7♭5	VIIm7♭5
Harmonic Minor	ImMaj7	IIm7♭5	♭IIIMaj7#5	IVm7	V7♭9	♭VIMaj7	VIIo7

Modal interchange is sometimes referred to as “chord borrowing,” as essentially it is borrowing chords from a different mode. The most common use of modal interchange is between the major scale and a parallel minor mode (usually aeolian or harmonic minor).

For example, the minor four (iv) chord, one of the most popular tricks in the songwriter’s toolkit, is an example of modal interchange. You can hear it featured prominently in the music of the Beatles, Stevie Wonder, Radiohead, and way more. The minor seventh dominant (bVII7) chord–sometimes referred to as the “backdoor”–is another common example derived from interchange with a minor mode.

Modal interchange is a technique that can make a chord progression more interesting, exciting, and flavorful. By briefly departing into the non-diatonic territory of another mode, the listener is surprised but not completely disoriented, as the progression quickly finds its way back to the tonic. It is a concept that unlocks many new doors for composers.

Welcome back, readers. The music theory nerd’s paradise continues to expand. Today’s topic: secondary dominants.

We’ve discussed dominant chords in the past. They create tension/instability that can be resolved by moving back to the tonic chord in a given key. The perfect cadence is a standard example of this idea, moving from V to I.

However, in the Western tradition of music, staying confined to the dominant chords found within a single diatonic key (traditionally, the V and vii0) is simply not enough to quench our undying thirst for interesting harmonic material. To remedy this problem, musicians utilize secondary dominants.

To adequately explain this concept, we must look at the Circle of Fourths. What is the circle of Fourths? Here it is:

Those of you who are classically trained may be shouting in vigorous protest at your computer screen: “No, no! That’s the Circle of Fifths!” Whether or not this is the case, I invite you to take a closer look.

This image visually represents the twenty-four keys of Western music and how they are related. The major keys are located in the outer rung of the circle, and the minor keys are located in the inner rung of the circle. Each major key’s relative minor key (sharing the same notes) is located within the same “pizza wedge.” The sequence of keys are arranged in a manner such that (1) as one travels clockwise around the circle, one travels a fifth between each key, and (2) as one travels counter-clockwise around the circle, one travels a fourth between each key.

For the purpose of secondary dominants, conceptualizing the circle as the Circle of Fourths is more helpful. To explain why, let’s traverse the circle counterclockwise.

We begin with F, which is the fifth of Bb, the next note in the traversal. This means we can create the perfect V-I cadence F7-Bb. Let’s continue. Next after Bb is Eb. Bb is the fifth of Eb. We notice a pattern–why not make Bb a dominant chord resolving to Eb? Now our progression is F7-Bb7-Eb. We can keep moving counterclockwise around the circle in this manner until we again reach F: F7-Bb7-Eb7-Ab7-…-C7-F

Obviously, such a progression would be excessive, but this harmonic idea is used all the time in music. Take a look at the chords which sit under the titular lyrics of “Make You Feel My Love” by Bob Dylan (D7-G7-C). Or, take a look at the chords of the intro to “Bohemian Rhapsody” by Queen (Bb-C7–F7–Bb-Gm7-Bb7–Eb-Cm7-F7).

Notice how in Bohemian Rhapsody a few ii-V-I’s are thrown into the mix. This is called interpolation, and will be explained in a later post.

Up until now, we have operated within the clean and unambiguous framework of functional harmony. Modes are a whole new way of conceptualizing harmony.

So what are modes in music? Vaguely stated, modes are variations of scales derived by changing the beginning note of the scale. The major and natural minor scales, which we have discussed previously, are two of the seven Greek modes. This chart should efficiently convey how each mode is obtained and how it is structured:

	Mode name	Interval sequence	Example
	Ionian	W–W–H–W–W–W–H	C–D–E–F–G–A–B–C
	Dorian	W–H–W–W–W–H–W	D–E–F–G–A–B–C–D
	Phrygian	H–W–W–W–H–W–W	E–F–G–A–B–C–D–E
	Lydian	W–W–W–H–W–W–H	F–G–A–B–C–D–E–F
	Mixolydian	W–W–H–W–W–H–W	G–A–B–C–D–E–F–G
	Aeolian	W–H–W–W–H–W–W	A–B–C–D–E–F–G–A
	Locrian	H–W–W–H–W–W–W	B–C–D–E–F–G–A–B

Note: W = whole step. H = half step.

The ionian, lydian, and mixolydian are considered major. The ionian is the major scale. The lydian is the major scale with an augmented fourth. The mixolydian is the major scale with a minor seventh.

The dorian, phrygian, and aeolian are considered minor. The aeolian is the natural minor scale. The phrygian is the natural minor scale with a minor second. The dorian is the natural minor scale with a major sixth.

The locrian is considered diminished. It is more theoretical than practical, as its tonic is a diminished chord, which is unstable.

Each mode has a distinctive sound that composers use to convey a certain level of brightness.

Note that these are the modes which are derived from the major/natural minor scale. These modes are the most used and most commonly studied. There also exists a set of modes from the harmonic and melodic minor scales, as well as the harmonic major scale. These are less commonly used, but still have their place in niche corners of the jazz realm.

Modes built off of the melodic minor scale:

• Melodic minor scale
• Dorian b2
• Lydian #5
• Lydian b7
• Mixolydian b6
• Aeolian b5
• Locrian b4

Modes built off of the harmonic minor scale:

• Harmonic minor scale
• Locrian natural 6
• Ionian #5
• Dorian #4
• Phrygian b6
• Lydian #2
• Locrian b4 bb7

The modes built off of the harmonic major scale:

• Ionian b6 (Harmonic major)
• Dorian b5
• Phrygian b4
• Lydian b3
• Mixolydian b2
• Lydian #2 #5
• Locrian  bb7

Okay, so these are the modes–but why should you care? A brief history lesson is in order. In the late 1950s, musicians like John Coltrane and Miles Davis led a shift from tonal music to modal music in the jazz world and beyond. No longer were musicians confined to the chains of the subdominant, dominant, and tonic within a singular major or minor key–they could explore new and exciting melodies and progressions. This revolution makes it necessary that we thoroughly understand the modes and their application.

Pictured: Miles Davis (trumpet) and John Coltrane (saxophone).

Before we continue on our musical journey, I would be remiss to neglect an important harmonic concept: inversions.

Essentially, inversions are different ways of playing the same chord. Here they are:

Root position is when the root of the chord is in the bass.

First inversion is when the third of the chord is in the bass.

Second inversion is when the fifth of the chord is in the bass.

Here are the inversions of a C major triad:

Third inversion is when the seventh of the chord is in the bass (if applicable).

Here are the inversions of a G dominant seventh:

Generally, inversions do not go any farther than this. There a few different ways of notating inversions. We will discuss figured bass and popular music notation.

Figured bass was most commonly used in the Baroque period of Western classical music–the lead sheets of the 17th century. The chart below should efficiently illustrate the concept:

Triads
Inversion	Intervals above bass	Symbol	Example
Root position	5 3	None
1st inversion	6 3	6
2nd inversion	6 4	6 4
Seventh chords
Inversion	Intervals above bass	Symbol	Example
Root position	7 5 3	7
1st inversion	6 5 3	6 5
2nd inversion	6 4 3	4 3
3rd inversion	6 4 2	4 2 or 2

Two important notes:

Intervals in this chart are relative to the bass note, not the root (as we’ve seen in the past). For example, in first inversion of a C major triad, the figured bass notation “6” refers to the note (C) which sits a sixth above the bass note (E).

As we’ve seen in previous notation schemes, certain notes in the chord are often implied. However, it is important to note that figured bass relies on the key signature of the piece to specify the notes to be played. For example, let’s look at the G dominant seventh chord. Just as “G7” implies the major third (B) and perfect fifth (D) from the root note (G), the figured bass notation “2” implies the fourth (B) and sixth (D) from the bass note (F).

In popular music, inversions are simply notated by writing the chord name followed by a forward slash and the bass note. For example, Ab major in first inversion would be notated as Ab/C. This notation is also sometimes used when the bass note is not present in the triad. For example, Bb/C is a way of voicing Bbadd9.

Inversions are often used to create stepwise motion in the baseline of a progression (moving a step/half step from one chord to the next). From Elton to James Taylor, inversions have long been an essential tool in the songwriter’s kit.

Next week, we will explore entirely new territory–modal music!

If you’ve been following up until this point, you have amassed a significant amount of knowledge about harmony–more specifically, chords, their function, and their notation. Now, we can begin to explore how these chords are actually used in music. Let’s take a look at some common chord progressions.

I-V-vi-VI

Perhaps the most famous chord progression, the I-V-vi-VI has been used across the decades in many different genres. Some examples include “Let It Be” by the Beatles, “Don’t Stop Believin” by Journey, “Take On Me” by Aha, and “I’m Yours” by Jason Mraz.

While this particular order tends to be popular, the I, V, vi, and VI chords are often rearranged. In the ’50s and early ’60s, the I-vi-IV-V progression was especially popular. An example is “Those Magic Changes” by Sha Na Na.

Another popular variation is iv-IV-I-V, as heard in “Otherside” by the Red Hot Chili Peppers and “Not Afraid” by Eminem.

Some songs don’t even bother with the vi chord. I-IV-V is another very popular progression. Some examples include “La Bamba” by Ritchie Valens, “Louie Louie” by the Kingsmen, and “Should I Stay or Should I Go” by the Clash.

A different type of progression is the 12-bar blues. It is structured as follows:

I7-IV7-I7-I7

IV7-IV7-I7-I7

V7-IV7-I7-V7

An example of the Bb blues:

Note that there are many variations on the 12-bar blues. A key feature of this progression is the dominant seventh that is appended on every chord. Some examples include “Call It a Stormy Monday” by Stevie Ray Vaughan, “Roll Over Beethoven” by Chuck Berry, and “Hound Dog” by Elvis Presley.

A staple progression in jazz music is the ii-V-I, also referred to as the ii-V-I turnaround. In this progression, the ii and V lead to the tonic, I. The chords of this progression are commonly played with sevenths (ii7-V7-Imaj7), and can be built upon further with upper extensions.

An example of the ii-V-I in C:

This progression can be incorporated into the blues.

F7	B♭7	F7	Cm7  F7
B♭7	B♭7	F7	Am7  D7
Gm7	C7	F7	Gm7  C7

Note: Cm7 and F7 are the ii and V of Bb, Am7 and D7 are the ii and V of G, and Gm7 and C7 are the ii and V of F.

The ii chord can be altered to create the progression II-V-I. The ii can also be treated as its own temporary tonic, extending the progression to iii-VI-ii-V-I.

The V7 chord can be substituted with the dominant chord built off of its tritone, the bII7, creating the progression ii7-bII7-Imaj7.

In a minor key, the ii7-V7-Imaj7 becomes ii7b5-V7-i7.

A variation on this progression is the “backdoor” ii-V, which approaches the tonic as follows:

iv7-bVII7-Imaj7

This is getting exciting. Our musical toolkit is growing. Next week, we will continue to explore the theory of musical harmony.

Before we can begin to discuss chord progressions, we must understand another important form of chord notation: roman numerals. Within a major or minor key, there is a chord built off of each step of the eight-note scale. Using each note as the root of the chord, the quality of the third, fifth, seventh (and beyond) intervals are determined by the other notes in the scale. A helpful way to label these chords is by the root’s scale degree number. Major chords are denoted as capital Roman numerals and minor and diminished chords are denoted as lower-case Roman numerals. All other notation should be familiar from last week.

Let’s look at the chords of C major and D major:

Note: The Roman numeral notation is identical for all major keys and for all minor keys.

Now let’s look at the chords of A natural, harmonic, and melodic minor (remember these?):

Now let’s discuss theoretical harmonic function of the chords of any given key.

First up, the tonic–the tonal center. In tonal music, the tonic is the chord that a piece returns to for a sound of stability, rest, and completion. It is “home.” It is the place to which we are always trying to return. The first, third, and sixth can function as the tonic. The third and sixth are considered to have weaker function than the first.

Next, the dominant. Dominant chords convey a sound of instability, tension, and incompletion. They want to move back toward the tonic. The fifth and seventh can function as the dominant. The seventh degree is considered to have weaker function than the fifth.

Finally, the subdominant. Subdominant chords occupy the middle ground between tonic and dominant. Like the dominant, they convey a feeling of preparation, but with less intensity. They may migrate to the dominant to increase tension, or the tonic to relieve tension. The second and fourth can function as the subdominant. The second is considered to have weaker function than the fourth.

We’ll begin our discussion of chord progressions with cadences, of which there are four. Cadences are two-chord progressions that occur at the end of a phrase.

The authentic cadence is a move from the dominant to the first. The perfect authentic cadence uses the fifth. The imperfect authentic cadence uses the seventh. In a major key, V/vii0 to I. In a minor key, V/vii0 (from harmonic/melodic scale) to i.

The plagal cadence is a move from the fourth to the first. In a major key, IV to I. In a minor key, iv (from natural/harmonic scale) to i.

Both of the authentic and plagal cadences create a satisfying resolution. These next two take us into new, uncertain harmonic territory.

The half cadence is a move from any chord to the fifth.

The deceptive cadence is a move from the fifth to any chord but the first (often the sixth).

Music, it is often said, can be boiled down to an interchange of tension and release. These harmonic concepts reflect that fundamental idea. Next week, we will look at common chord progressions found in different genres of music.