Consonance, Dissonance, and Frequency Ratios

What is Consonance? A Multifaceted View

At first glance, consonance might seem purely subjective or cultural. While culture undoubtedly plays a role, there are robust physical and psychoacoustic reasons why certain pitch combinations are perceived as more stable than others across many contexts. We can approach this from several angles.

First, let's explore an intuitive connection between rhythm and harmony.

This rhythmic perspective provides a compelling mental model. However, the dominant physical explanation, pioneered by Hermann von Helmholtz, is rooted in the acoustic properties of sound waves themselves. Modern psychoacoustics identifies three primary contributing theories to our perception of consonance:

1. Periodicity/Harmonicity: Consonant intervals combine to form a composite waveform that is highly periodic, with a spectrum whose partials align neatly with a single, lower fundamental frequency.
2. Interference between Partials: Dissonance arises from the unpleasant "roughness" caused by interference (beating) between frequency components that are close but not identical.
3. Cultural Familiarity: We learn to prefer the sounds we are most exposed to in our musical culture.

While all three contribute, the interference model is particularly powerful from a predictive, physical standpoint. Let's dissect it.

The Physics of Interference and Dissonance

To understand how interference creates dissonance, we'll build from the simplest case—two pure sine waves—to the complex reality of musical notes.

Dissonance of Pure Tones: Beating and Critical Bands

When two sine waves of very similar frequencies are played together, their wave crests and troughs periodically align (constructive interference) and misalign (destructive interference). You perceive this as a fluctuation in amplitude, a phenomenon known as beating. The beat frequency is simply the difference between the two source frequencies, $f_{\text{beat}}=|f_2-f_1|$.

• When the beat frequency is very low (< 5 Hz), we hear distinct pulses.
• As it increases (approaching ~20-30 Hz), the sensation changes from a smooth "wah-wah" effect to a jarring, unpleasant "roughness."
• As the frequency difference increases further, the two tones become distinct enough that the roughness subsides.

This unpleasantness is tied to the resolving power of our inner ear. The basilar membrane in the cochlea responds to different frequencies at different locations. However, this response is not infinitely precise. A single frequency stimulates a range of nerve endings, conceptualized as an auditory filter or critical band. If two frequencies are close enough to fall within the same critical band, the auditory system struggles to distinguish them, leading to the sensation of roughness or sensory dissonance.

This explains why two pure sine waves a semitone apart sound dissonant. But musical notes are not pure sine waves.

Dissonance of Complex Tones: The Sum of All Parts

As you know from our first module, real musical notes consist of a fundamental frequency plus a series of overtones (partials), which for many instruments form a harmonic series ($f,2f,3f,4f,\dots$). When you play an interval, you're not just hearing the interaction between the two fundamentals; you're hearing the sum of the interactions between every possible pair of partials from both notes.

The total dissonance of an interval is the sum of the roughness generated by all pairs of partials from the two notes.

This is where frequency ratios become paramount.

• Simple Ratios (e.g., Perfect Fifth, 3:2): If we play two notes with fundamentals $f_1=200$ Hz and $f_2=300$ Hz, their harmonic series are:

  • Note 1: 200, 400, 600, 800, 1000, 1200, ...
  • Note 2: 300, 600, 900, 1200, 1500, ...
    Notice the significant overlap. The 3rd harmonic of the first note coincides exactly with the 2nd harmonic of the second note. The 6th harmonic of the first note coincides with the 4th of the second, and so on. Many partials either align perfectly (zero dissonance) or are far apart, minimizing the number of pairs falling into the "roughness" zone. The result is low total dissonance, or high consonance.

• Complex Ratios (e.g., Minor Second, 16:15): The partials of two notes in a complex ratio will rarely align. Instead, they will create a dense forest of frequencies with many pairs falling into each other's critical bands, generating significant roughness and a high total dissonance.

This theoretical model, developed by Plomp and Levelt and expanded by others like William Sethares, allows us to plot a "dissonance curve" for any interval.

The resulting dissonance curve for a complex tone with a harmonic series shows distinct valleys, or local minima of dissonance, precisely at the simple integer ratios that define musical intervals.

Ranking Intervals by Consonance

Based on this physical model, we can now rank the intervals you learned to classify in the previous lesson. The ranking generally follows the complexity of their just intonation frequency ratios.

Spectrum of Consonance (Most Consonant to Most Dissonant):

1. Perfect Unison (1:1) & Perfect Octave (2:1): The most consonant intervals. The octave's harmonics are a perfect subset of the lower note's harmonics.
2. Perfect Fifth (3:2): The strongest and most stable harmony after the octave, due to its very simple ratio and significant harmonic overlap.
3. Perfect Fourth (4:3): Also highly consonant. It's the inversion of the perfect fifth.
4. Major Third (5:4) & Major Sixth (5:3): These are 'imperfect' consonances. Their ratios are slightly more complex, giving them more color and character than the perfect intervals.
5. Minor Third (6:5) & Minor Sixth (8:5): Still considered consonant, but more "colored" and less stable than their major counterparts due to more complex ratios.
6. Major Second (9:8) & Minor Seventh (16:9): Clearly on the dissonant side. The partials interfere significantly.
7. Major Seventh (15:8): Highly dissonant and "striving," often used to create tension that demands resolution.
8. Minor Second (16:15) & Tritone (~45:32): The most dissonant intervals within the octave. They produce maximum roughness and are functionally used to create extreme tension.

The Crucial Role of Timbre

A key insight from this theory is that consonance is not absolute; it is dependent on the timbre (i.e., the harmonic spectrum) of the sounds. The dissonance curves we've seen are for sounds with a full harmonic series, like a string or a sawtooth wave. If you change the spectrum, you change the consonance profile.

This has profound implications for you as an electronic musician. The sounds you design will have unique harmonic content, and therefore their own unique landscape of consonance and dissonance.

This principle is why instruments with non-harmonic overtones, like bells or many FM synthesis patches, can sound "out of tune" when playing standard Western scales, but sound perfectly consonant within their own unique tuning systems, as seen in Indonesian Gamelan music.

Conclusion

In this lesson, we have moved from the abstract naming of intervals to a physical understanding of their sonic effect. You now have a robust, physics-based model for why intervals create varying degrees of tension and stability.

Key Takeaways:

• Consonance is a spectrum: It ranges from the perfect stability of the octave to the harsh roughness of the minor second.
• Dissonance arises from interference: The primary physical cause of dissonance is the "roughness" produced by acoustic beating between partials that lie close together within the ear's critical bands.
• Simple frequency ratios are consonant: Intervals with simple integer ratios (2:1, 3:2, 4:3) are consonant because their harmonic partials either align perfectly or are spaced far apart, minimizing interference.
• Timbre dictates consonance: The specific consonance and dissonance profile of an interval depends directly on the harmonic content of the sounds involved. This is a powerful concept for sound design and experimental music.

You now possess the theoretical tools to not only classify intervals but also to predict their emotional impact. This knowledge of the tension inherent in each interval is a foundational skill. In our next lesson, we will focus on training your ear to connect these theoretical concepts to sound, as you learn to aurally recognize the key intervals that form the backbone of melody and harmony.