In the world of music theory, harmonic intervals are the building blocks of every chord we hear. Understanding how these intervals work, and how they resolve, is the key to creating emotional tension and release in the pieces we study, analyse, or even compose.
What are Harmonic Intervals?
A harmonic interval, as we have already studied in Chapter 1 of the intervals, occurs when you play two notes at the same time. We divide these intervals into two main categories: Consonant and Dissonant.
1. Consonant Intervals (Σύμφωνα Διαστήματα)
Consonant intervals sound stable, “pleasant”, and at rest. We further divide them into two groups:
- Perfect Consonances (Τέλειες Συμφωνίες) : Perfect Unison (P1), Perfect Octave (P8), Perfect 4th (P4), and Perfect 5th (P5).
- Imperfect Consonances (Ατελής Συμφωνίες) : Major and Minor 3rds, Major and Minor 6ths.
2. Dissonant Intervals (Διάφωνα Διαστήματα)
Dissonant intervals sound “unstable” or create a sense of friction. These include 2nds, 7ths, and all Augmented or Diminished intervals.
Note: Intervals are the DNA of chords. If a chord contains a dissonant interval, the entire chord becomes unstable. To achieve a satisfying acoustic result, these dissonant intervals must “resolve” into consonant ones.
The Resolution
A resolution happens when a dissonant interval moves into a consonant one. Follow these specific rules to guide the “tension” back to “rest”. These are some of the most common resolutions of dissonant intervals.
- Minor 2nd (m2): Resolves to a Minor 3rd. Keep the top note steady and move the base down a tone.
- Major 2nd (M2):
- To a Minor 3rd: Move the base down a semitone.
- To a Major 3rd: Move the base down a tone.
- Augmented 2nd (A2): Resolves to a Perfect 4th. The top moves up a semitone, and the base moves down a semitone.
- Diminished 3rd (d3): Resolves to a Perfect Unison. Both notes move inward by a half step, so the top moves one semitone up, and the base one semitone down.
- Augmented 4th (A4):
- To a Minor 6th: Top up a semitone, base down a semitone.
- To a Major 6th: Top up a semitone, base down a tone.
- Diminished 4th (d4): Resolves to a Minor 3rd. Keep the top note steady; move the base up a semitone.
- Diminished 5th (d5):
- To a Minor 3rd: Top down a tone, base up a semitone.
- To a Major 3rd: Top down a semitone, base up a semitone.
- Augmented 5th (A5): Resolves to a Major 6th. Top moves up a semitone; base stays the same.
- Augmented 6th (A6): Resolves to a Perfect Octave. Top up a half step; base down a semitone.
- Minor 7th (m7):
- To a Minor 6th: Top down a tone; base stays the same.
- To a Major 6th: Top down a semitone; base stays the same.
- Major 7th (M7): Resolves to a Major 6th. Top down a tone; base stays the same.
- Diminished 7th (d7): Resolves to a Perfect 5th. Top down a semitone; base up a semitone.
Caution: Rabbit Hole
The Physics of Sound: Why Do We Hear Tension?
Why does a Perfect 5th feel “right” while a Minor 2nd feels “wrong”? It comes down to frequency ratios.
- Consonant Intervals have simple mathematical ratios (like 2:1 for an octave). The brain processes these easily as “pleasant” information.
- Dissonant intervals have complex ratios that create “beats” or acoustic fiction.
Composers use this intentionally. When a composer wants to create tension or anxiety, they use dissonant intervals to force the listener’s brain to “wait” for a resolution.
A Historical Revolution: From the Renaissance to Bach
Our modern musical aesthetic didn’t happen overnight. What we perceive as pleasant is largely shaped by Western musical aesthetics developed from the Renaissance and especially the Baroque period.
- The Renaissance: This era laid the foundation. Before this, even 3rds and 6ths were considered dissonant!
- The Baroque Era: This is when the “laws” of harmony we study today were perfected. Most modern music theory focuses on the techniques developed from the Baroque period onwards. The tuning from these era, slightly adjusted these frequency ratios (tempered – Well Tempered Clavier from J.S. Bach) So on a modern piano, due to the Tuning System, an Augmented 4th and a Diminished 5th actually use the same keys (they “merge”- enharmonic), even though they resolve differently in theory!
Culture and the “Ear of the Listener”
What we find “pleasant” is largely a matter of musical cultivation and tradition.
- Context Matters: If Bach played one of his Oratorios in the Middle Ages, people would have found the sound absolutely repulsive.
- Global Traditions: Different cultures use different systems. Greek traditional music or Indian Maqams, may sound “strange” or “out of tune” to a Western ear, but they perfectly harmonious within the traditions.
Famous Examples in Classical Music
- The Tritone (A4): Heard in the opening of The Simpsons or Leonard Bernstein’s “Maria” (resolving to a P5).
- Diminished 7th (d7): Used for high drama in Bach’s Toccata and Fugue in D Minor.
- Augmented 6th (A6): A favorite of Beethoven (Symphony No. 5) to create a powerful drive toward the dominant.
- Minor 2nd (m2): The legendary tension in John Williams’ Jaws Theme.
The Math Behind the Ear
To explain Frequency Ratios, we look at the mathematical relationship between the vibrations (Hertz) of two notes. The simpler the ratio, the more “consonant” (stable) the interval sounds to our brain.
Here are the primary ratios for the intervals you mentioned, based on the Pure (Just) Intonation system:
1. Consonant Ratios (Simple)
- Perfect Unison (P1): 1:1 (The frequencies are identical).
- Perfect Octave (P8): 2:1 (The top note vibrates exactly twice as fast as the base).
- Perfect Fifth (P5): 3:2 (Very stable; the waves align every few cycles).
- Perfect Fourth (P4): 4:3.
- Major Third (M3): 5:4.
- Minor Third (m3): 6:5.
- Major Sixth (M6): 5:3.
- Minor Sixth (m6): 8:5.
2. Dissonant Ratios (Complex)
These ratios are “messy,” which is why our brain perceives them as tension:
- Major Second (M2): 9:8 (The waves clash frequently).
- Minor Second (m2): 16:15 (Very complex; high tension).
- Major Seventh (M7): 15:8.
- Minor Seventh (m7): 16:9 or 9:5.
- The Tritone (A4/d5): 45:32 (In pure tuning). This is the most complex ratio, which is why it was called the “Devil in Music.”
The “Equal Temperament” Twist
As you correctly noted regarding Bach and the Well-Tempered Clavier, modern pianos don’t use these “pure” ratios. Instead, we use Equal Temperament. This means that, on a piano, a Perfect Fifth is actually 1.498:1 (slightly “out of tune” compared to the pure 1.5:1). This “compromise” is what allows us to play in all 24 keys, but it slightly changes the “purity” of the frequencies we hear.
The distinction between perfect/imperfect consonances and the various dissonances lands cleanly here, especially the point about simple ratios feeling at rest while complex ones create those acoustic beats that pull the ear toward resolution. It makes the old rules feel less arbitrary and more rooted in how sound actually behaves, particularly when building chords or working through progressions at the piano. The equal temperament note is useful too; it explains why some intervals on a modern instrument sit slightly off pure intonation yet still drive the music forward. Examples like the tritone in familiar themes help show how these intervals have kept their dramatic bite across centuries. Overall it reinforces why paying attention to this layer matters: it shapes the emotional arc of a piece more than any single melody note ever could. Solid foundation work for anyone trying to hear harmony rather than just play the right keys.
Thank you for your comment! Exactly! These are some notes I wrote to make intervals, more understandable for my students! Mainly for harmony students who are starting to study and compose polyphonic music, and also for students who are working on more advanced piano literature (from more complex jazz chords and their resolution to the harmonic patterns of Schoenberg!)
Best,
Helena