Chapter 6:

Introduction to Western harmony

Emotion and story

Music is a form of communication. It’s used to express ideas and emotions, and to evoke the same in others.

Western harmony, which derives from the European classical music (“euroclassical”) tradition, provides tools to do this based on tension and release, consonance and dissonance, and shared cultural experience.

Tension and release

“Tension and release” is a basic biological mechanism, shared perhaps by all life.

It is the basis of locomotion—it physically moves us—and it moves us emotionally as well.

Consonance and dissonance

Tension and release are created with “consonant” and “dissonant” sounds.

Consonant sounds are mathematically simple and harmonically relaxed.

Dissonant sounds are mathematically complex and harmonically tense.

Expectation and surprise

Tension and release can also be created by playing with listeners’ expectations.

All the music we have previously heard creates expectations about music we will hear in the future.

If we have a similar enough background to the listener to understand their expectations, we can play with those expectations to provoke a response.

For example, we can produce something they expect, provoking familiarity; delay meeting their expectations, provoking tension; give them something they didn’t expect but are also familiar with, provoking surprise and hopefully delight; and so on.

The hero’s journey

“Harmonic functions” can be used to tell a kind of story, similar in some ways to the “hero’s journey” narrative pattern which is common in myths and stories across many cultures.

In both the hero’s journey and in harmonic progressions, the hero goes on an adventure, is victorious in a decisive crisis, and comes home transformed.

Intervals: the building blocks

Intervals are the basic building blocks of harmony.

They are responsible for the way music sounds, and how it makes us feel.

The difference between two pitches

An interval describes the difference in pitch between two notes. The greater the difference in pitch, the larger the interval. The difference in pitch creates the sound and feeling of the interval.

Pitch and frequency

Pitch is the frequency at which a sound wave vibrates. The higher the frequency, the faster the vibration, and the higher the pitch sounds to our ears.

An interval is defined as the difference in frequency between two pitches.

Interval names

Each interval is also given a name, like “major third” or “perfect fifth”.

Interval names are comprised of two parts: quality (like “major”) and number (like “third”).

Interval numbers

The number of an interval is the number of letter names from one note to the next in the musical alphabet.

For example, the interval from a C to a D is a second, from C to E is a third, and from C to G is a fifth. The interval from C to A is a sixth, because the musical alphabet goes to G and then wraps around to A again.

The interval from C to the next C above or below is called an “octave” (meaning 8), and from a note to itself is called a “unison” (meaning 1).

Perfect intervals

The quality of fourth, fifth, octave, and unison intervals is called “perfect”, because the sound waves of the pitches separated by those intervals vibrate together in an overlapping and mathematically elegant way. This makes them sound highly consonant.

Major and minor

The quality of the remaining, non-perfect interval numbers are called “major” or “minor”.

“Major” intervals are the intervals that comprise the “major scale” (described later).

“Minor” intervals are one fret below (or one “semitone” below) the major ones.

Semitones, tones, and frets

A “semitone” is the smallest interval in most Western music. On a guitar fretboard, it’s the distance across one fret on one string. Semitones are sometimes called “half steps”, or (on guitar) “frets”.

In Western music, there are twelve semitones in an octave.

The term “tone”, when used as an interval name, means two semitones, a distance of two frets. Sometimes the terms “whole tone” or “whole step” are used instead.

Be aware that the term “tone” has other meanings as well, such as “any note with a discernible pitch”. See “tone” in the glossary for details.

Interval table

This table shows the names of intervals based on how many semitones they contain, i.e. how many frets they span on one string.

Intervals on the fretboard

Notice that this table maps directly onto the fretboard. For example, place the first finger on the 2nd string, 1st fret, the note C, and consider that fret 0 in the interval table above. Every row in the table describes the matching fret down the 2nd string.

The table gives the corresponding note of a C chromatic scale for reference, and the short and long names commonly used for those scale degrees.

It also shows the dissonance of each interval, in terms of its frequency ratio to the tonic. Ratios with bigger numbers are more complex and therefore more dissonant.

Perfect intervals can be raised or lowered by a semitone / fret, changing their quality from perfect to “augmented” (raised) or “diminished” (lowered).

Chords

Western harmony is expressed through chords.

A chord is a group of two or more notes sounded together.

Stacking thirds

In most Western music, chords are constructed by “stacking thirds”.

Stacking thirds means adding successive intervals of a third until the chord has as many notes as we want it to.

Another way to think of stacking thirds is adding every other note in a scale. (And another way to say “every other note” is “every third note”. Get it?)

Building chords from stacked thirds is called “tertian harmony”, which is the most common type of harmony in Western music. There are others, like “quartal harmony”, which is based on stacking fourths.

Triad chords

Triads are the most common chords in Western music.

A triad chord is a three-note chord, built by stacking thirds.

The three notes in a triad chord are called the “root”, “third”, and “fifth” factors of the chord, which are generic intervals of (some quality of) a third and a fifth from the root. The specific intervals determine the chord type.

Major triads

The major triad is the most common chord in Western music.

It is considered the most “stable” chord, constructed from a major third and a perfect fifth, the first two distinguishable harmonics in the overtone series.

As Bert Ligon said in Comprehensive Technique for Jazz Musicians, the major triad consists of “the first three pitches in the overtone series and the natural laws of physics insist that the planet vibrates with these tones when the winds blow, which may explain the universal occurrence of the triad in melodies.” (Ligon, 1999)

Types of triad chord

There are four types of triad chord. The chord type is determined by the intervals between its factors and the root.

Triad thirds

If the third factor is a major third (4 semitones), it is a major chord. If the third is a minor third (3 semitones), it’s a minor chord.

• Stacking a major third (4 semitones) on a minor third (3 semitones) gives a perfect fifth (7 semitones).
• Stacking a minor third (3 semitones) on a major third (4 semitones) gives a perfect fifth (7 semitones).
• Stacking two minor thirds (3 semitones) gives a tritone / diminished fifth (6 semitones).
• Stacking two major thirds (4 semitones) gives an augmented fifth (8 semitones).

Harmonizing a scale

Harmonizing a scale refers to assembling chords from the notes of a given scale, by starting on each degree of the scale and stacking thirds to form a triad chord. The result is one chord for each degree in the scale.

The chords will be of different types depending on how the scale degree intervals fall within the chord.

Chords are numbered with Roman numerals matching the number of their scale degree. Uppercase Roman numerals are used for major chords, and lowercase Roman numerals are used for minor chords.

Harmonizing the major scale

For example, the major scale has degrees
1-2-3-4-5-6-7 (C-D-E-F-G-A-B).

Harmonizing the major scale gives the following chords:

The type of each chord is determined by the intervals of its chord factors.

See more about intervals for information about determining the intervals between chord factors.

Harmonizing the natural minor scale

As another example, the natural minor scale has degrees
1-2-b3-4-5-b6-b7 (C-D-Eb-F-G-Ab-Bb).

Harmonizing the natural minor scale gives the following chords:

Note that minor key harmony is slightly more complicated than just harmonizing the natural minor scale. See Chapter 9. Minor key harmony for details.

Keys

Major, minor, and modal

Most songs in Western music revolve around a keycenter or tonic—a note which serves as the musical “home” of the composition. Such compositions are said to be “tonical” (or, confusingly but commonly, “tonal”).

Songs in a major or minor key are said to be “in the key of” the tonic note. Sometimes a song changes key one or more times, to add interest.

A song in one key can be played in another key, and it will sound the same to all known listeners, but at an overall higher or lower pitch. Changing the key of a song is called “transposing” it to another key.

Most Western music is in a major or minor key. See Chapter 8. Major key harmony and Chapter 9. Minor key harmony for details.

Some songs are based around a mode, rather than being based on a major or minor key. See Chapter 18. Diatonic modes for details.

The musical alphabet

The name of a key, and of each note within it, comes from one of the seven letters of the “musical alphabet”. A-B-C-D-E-F-G, in English.

Each letter represents a standard fundamental pitch. For example, A (above middle C) is 440 Hz. Each letter also represents the “pitch class” of all other pitches a whole number of octaves apart. For example, the pitch class A consists of all the As in all octaves.

A whole tone interval separates each note of the musical alphabet, except for semitones between B-C and E-F. When starting from C, the musical alphabet describes a major scale.

See Chapter 16. Major scale for details.

Sharps and flats

To accommodate the twelve tones of Western music, the seven letters of the musical alphabet are supplemented with “accidentals”: sharp (#) and flat (b) symbols that indicate a semitone above (#) or below (b) a given note from the musical alphabet.

For example, F# describes the pitch between F and G, a semitone above F and a semitone below G.

The note name Gb also describes the pitch between F and G. F# and Gb describe the same pitch, so they are called “enharmonic” symbols for that pitch.

It’s useful for the same pitch to be described by two different letters in this way, because it allows every key to have seven notes with exactly one of each letter from the musical alphabet, A-B-C-D-E-F-G, with some different combination of sharps or flats.

Notes in each key

See Chapter 14. Keys and their notes for details, and some tricks for determining which notes are in each key.

Harmonic function

“Harmonic function” describes the theoretical tendency of certain chords to “want” to progress to other chords, or remain at rest.

This approach originated in euroclassical music in the 18th and 19th centuries, and still heavily influences jazz music. It is less evident in modern and non-Western music.

Euroclassical scale degree names

Harmonic function terminology is based on the names given to the scale degrees in the euroclassical tradition.

The most important are the tonic (1), the dominant (5), and the leading tone (7). The rest are defined in relation to those.

The leading tone

The leading tone is the note of a scale that is one semitone below the tonic.

The minor second interval between the leading tone and the tonic is very dissonant. Most listeners will find it tense and wish for it to resolve somewhere more consonant.

Western listeners tend to expect the 7 to be followed by the 1, based on past experience. If in most music we have heard in the past, C-D-E-F-G-A-B… has almost always been followed by another C, we will expect this because it’s what always happens. (If that has not been our experience, expectations might be different.)

This combination of tension and expectation results in “wanting” the dissonant leading tone to resolve to the consonant tonic.

This is the basic assumption of Western harmony: that the leading tone functions in this way for most listeners, pulling toward the tonic. The other harmonic functions are derived from this function of the leading tone.

Tonic, subdominant, and dominant

Chords can be grouped into three basic harmonic functions.

• Tonic: Home, a place of stability
• Subdominant: Away from home, but still stable.
• Dominant: A place of tension, pulling toward home.

A harmonic sequence will often start at home (a tonic), move away from it (to a subdominant), then to a place of tension (a dominant), before providing emotional release or resolution by returning home (to the tonic).

This progression mirrors a classic storytelling arc. Compare it to the “hero’s journey” narrative pattern:

1. The hero goes on an adventure (subdominant),
2. is victorious in a decisive crisis (dominant), and
3. comes home transformed (tonic).

In a major key, the diatonic chords are generally assigned harmonic functions as follows:

• Tonic: I, vi, iii
• Subdominant: IV, ii
• Dominant: V, vii

One way to understand this is by comparing the intervals of the chords to the tonic.

The harmonic functions are named after the scale degree of the I, IV, and V chords. These are the “primary chords”, and serve as archetypes of the chord functions.

• Tonic I (1-3-5) is home.
• Subdominant IV (4-6-1) is away from home, but it does include the 1̂. It is stable, being a major triad and having no leading tone. The 4̂ is a dissonant semitone away from the 3̂ in the tonic.
• Dominant V (5-7-2) is away from home. It is more dissonant than the subdominant, because it has the leading tone (7̂), and it contains the dissonant tritone between the 4̂ and 7̂. The 7̂ and 4̂ are each a dissonant semitone away from tonic chord tones, so we “want” them to resolve to the 1̂ and 3̂ in the I chord.

The remaining chords sort into functions by similarity:

• Tonic vi (6-1-3) and sometimes iii (3-5-7) are close to home, sharing two notes with the I and having no 4̂. (The iii does have the leading tone (7̂) however, so in some cases it can act more as a dominant than a tonic.)
• Subdominant ii (2-4-6) is away from home, having a 4̂, sharing no notes with I and one note with vi and iii; and it is close to the IV (sharing two notes with no dissonant intervals).
• Dominant vii° (7-2-4) is tense, containing a highly dissonant tritone interval between its root and diminished fifth (the 7̂ and 4̂ scale degrees). It has the leading tone as its root, pulling to the tonic.

V-I cadence

A defining characteristic of most euroclassical and jazz music is the use of “cadences”, standard chord movements that end a musical phrase and establish a key center.

Cadences use the primary chord functions to establish their tension and resolution. The basic cadence is called an “authentic cadence”, which moves from the V chord (the dominant) to the I chord (the tonic).

This chord movement in a descending fifth (or ascending fourth) is a standard aspect of jazz and euroclassical idioms.

Listeners of these types of music have heard V-I cadences so often, they have formed an expectation that the V will often lead to the I. This common expectation reinforces and enables the tricks of harmonic function. On the other hand, listeners who have little experience of hearing music with V-I cadences will not have the same expectations, and will not have the same response to harmonic functions.

Circle of fifths progressions

Jazz takes the cadential movement of descending fifths even further, by putting a ii chord in front of the V-I cadence.

The ii is a perfect fifth above the V, so the ii-V-I cadence gives the effect of repeatedly falling by perfect fifths until we reach the tonic.

In this context, the ii is said to be “preparing” the V, and it is referred to as having “pre-dominant” function.

The ii-V-I is the most common chord progression in jazz music.

Jazz arrangers typically don’t stop there; they also pepare the pre-dominant chord with another chord a fifth above it, and onward until the chord progressions are often based on a long series of descending fifths. Often they will alter the intermediate chords in the progression by making them dominant sevenths, to increase tension and propel the progression along to its final V-I cadence.

More about intervals

Finding intervals on the fretboard with the major scale

The easiest way to find any interval on the fretboard is to know the major scale on the fretboard.

Every degree of the major scale is a major or perfect interval from the tonic. Degree 2̂ of the major scale is a major second from the tonic, degree 5̂ is a perfect fifth, degree 7̂ is a major seventh, and so on. To find a major third interval on the fretboard, imagine the lower note is the tonic of a major scale, and find the 3̂. The minor intervals are a fret below the major ones.

See Chapter 16. Major scale for more information.

Inverting intervals

“Ascending intervals” describe the distance from the lower to the higher pitch, and are often described as going “up”, such as “up a (perfect) fourth”.

“Descending intervals” describe the distance from the higher to the lower pitch, and are often described as going “down”, such as “down a (perfect) fifth”.

Because we hear two notes an octave apart as the same note, it is somewhat counterintuitive that the interval from the lower to the higher pitch is different than the interval from the higher to the lower pitch. The descending interval is an “inversion” of the ascending interval.

For example, the interval from C to E is a major third. But the interval from E to C is a minor sixth. This is because the interval from C to E spans C-D-E, but the interval from E to C spans E-F-G-A-B-C.

Similarly, the interval from C to F is a perfect fourth, and the interval from F to C is a perfect fifth. A perfect fourth inverts to a perfect fifth. Therefore, moving “up a fourth” or “down a fifth” arrives at the same note (in a different octave).

A simple formula for inverting an interval is to (1) invert the quality of the original interval (major to/from minor, augmented to/from diminished, perfect stays perfect); and (2) subtract the number of the original interval from 9.

Therefore, up a major third is the same as down a minor sixth, up a perfect fifth is the same as down a perfect fourth, up a minor seventh is the same as down a major second, and so on.

Interval arithmetic

The intervals of the chord factors can be determined by counting the number of semitones between the factors and the root.

To find the chord factor intervals using the interval table, look up the number of semitones in the factor’s scale degree, and subtract the number of semitones in the root scale degree. If the number is negative, add 12 semitones (because the chord spans the end of the scale into the next octave). The resulting difference is the number of semitones in the interval, which can then be looked up from the table.

For example, the V chord has scale degrees 5̂-7̂-2̂. (Scale degree numbers are sometimes written with “hats” (5̂) when it adds clarity, to distinguish them from other numbers.) The chord’s root is degree 5̂, which is 7 semitones from the tonic. The chord’s third factor is degree 7̂, which is 11 semitones from the tonic. The interval between the root and third factors is 11-7=4 semitones, which is a major third. So the chord’s third factor is a major third.

The V chord’s fifth factor is degree 2̂, which is 2 semitones from the tonic. The interval between the root and fifth factors is 2-7=-5 semitones, which is a negative number because the chord spans the end of the scale into the next octave. Adding 12 semitones to account for the octave, -5+12=7 semitones, which is a perfect fifth.

So the V chord’s factor intervals are R 3 5 (root, major third, perfect fifth), which makes it a major chord, represented by an uppercase Roman numeral (V and not v).

Consonance, dissonance, and frequency ratios

When two pitches sound together, their sound waves overlap. The more closely they overlap, the more consonant and harmonically relaxed they sound. The less they overlap, the more dissonant and harmonically tense they sound.

This overlap between sound waves can be measured by the ratio between the frequencies of the two pitches. The more closely they overlap, the simpler the ratio. The frequency ratio of each interval is listed in the interval table.

In a unison interval, since the two pitches are the same, their sound waves oscillate at the same frequency. The ratio between the frequencies is one-to-one (1:1).

In an octave, one pitch vibrates twice as fast as the other. The pitch frequency ratio of an octave is therefore two-to-one (2:1).

The ratio of a perfect fifth is 3:2, and a perfect fourth is 4:3.

Because these small-integer ratios are mathematically elegant, and because they sound particularly consonant to our ears, they are called perfect intervals.

Other intervals get mathematically more complex, and therefore more dissonant.

The “tritone” interval, considered the most dissonant, has a pitch frequency ratio of the square root of 2.

These much more complex sound waves require more attention and processing from our brains, which may account for their perceived dissonance.