Why do some chords feel "locked" (and others feel like mud) — there’s an equation for that!
You already know the feeling. You're tuning a guitar or playing in a string ensemble, and suddenly the perfect fifth just snaps into place — the sound clarifies, the beating stops, and it rings with a kind of effortless purity. Meanwhile, an interval like a minor second at the same volume sounds tense, thick, and ambiguous no matter how precisely you tune it.
Music theory gives us labels for this: consonance and dissonance. But it doesn't really explain why a perfect fifth snaps into place and a tritone doesn't. For that, we need to look at something developed in the 1990s and 2000s by music theorist and tuning researcher Paul Erlich: harmonic entropy.
This concept is one of the most powerful ideas to come out of the microtonal music community, and it's genuinely useful even if you never plan to write a note outside of 12-tone equal temperament. Here's what it is, how it works, and why it matters.
First: What Is Your Brain Actually Doing When It Hears an Interval?
When you hear any complex sound — a cello, a trumpet, a sung vowel — your auditory system doesn't just register a single pitch. It hears a whole stack of overtones: frequencies at 2×, 3×, 4×, 5× the fundamental, and so on. These are called the harmonic series.
Your brain's job is to take all those partial frequencies and figure out: what is the fundamental pitch here? Usually, this is easy. The harmonic series of a cello A-string is so orderly that your brain locks onto A440 instantly, even if the fundamental is quiet or absent.
Now play two notes simultaneously. Your brain is suddenly trying to fit two sets of overtones into a single coherent picture. It asks: "Are these two sounds part of the same harmonic series? If so, what's their shared fundamental?"
When the interval between two notes is a simple ratio — like 2:1 (an octave) or 3:2 (a perfect fifth) — the overtones line up neatly. Your brain finds the answer quickly and confidently. The interval feels stable.
When the interval is something messy — like a ratio of 17:12 or an irrational number with no simple integer relationship nearby — the overtones don't line up. Your brain can't find a clean answer. Multiple competing interpretations all seem roughly equally plausible. The result is a sense of ambiguity, instability, and what we call dissonance.
Harmonic entropy is a mathematical way of measuring exactly this: how confused is your brain when it tries to find a fundamental for a given interval?
The Key Idea: Confusion as a Number
In information theory, entropy is a measure of uncertainty or disorder. High entropy means many equally likely possibilities — your brain can't make up its mind. Low entropy means one possibility dominates — your brain has a clear answer.
Harmonic entropy, as developed by Paul Erlich, applies this concept directly to intervals. Here's the conceptual picture:
Imagine all the simple-integer ratios lined up on a number line measured in cents (the logarithmic unit of pitch, where an octave = 1200 cents). There's 1/1 at 0 cents, 16/15 at about 112 cents, 9/8 at 204 cents, 5/4 at 386 cents, 4/3 at 498 cents, 3/2 at 702 cents, and so on — a whole dense field of ratios, simpler ones spread further apart, more complex ones clustered together.
Now you play an interval. Because your auditory system isn't perfect — pitch perception has a natural fuzziness — it doesn't hear an exact ratio. It hears something like a bell curve of probabilities centered on what you played. Maybe you played exactly 700 cents, but your auditory system perceives a smeared-out region from roughly 683 to 717 cents.
Within that smeared region, every just intonation ratio picks up some probability of being "the right answer." The question is: how concentrated or spread out are those probabilities?
- Near 3/2 (702 cents), the bell curve sits on top of one simple ratio that dominates everything around it. One candidate wins clearly. Low entropy. High consonance.
- Near 600 cents (the tritone), you're far from any especially simple ratio. The bell curve spreads its probability mass across a large number of moderately complex competitors — 7/5, 10/7, 17/12, and others. No clear winner. High entropy. High dissonance.
The harmonic entropy function is simply this entropy value — calculated mathematically — for every possible interval from 0 to 1200 cents (or beyond). The result is a curve that dips down at consonant intervals and rises to peaks at dissonant ones.
The Shape of the Curve
When you actually plot harmonic entropy, you get a spiky, mountain-range-like curve with valleys and peaks. The valleys (low entropy = high concordance) fall at the simple intervals you'd expect:
- The octave (2/1, 1200¢): deepest valley of all
- The perfect fifth (3/2, 702¢): very deep
- The perfect fourth (4/3, 498¢): deep
- The major third (5/4, 386¢): clear valley
- The minor third (6/5, 316¢): valley, though shallower
- The harmonic seventh (7/4, 969¢): a notable valley that's often overlooked in standard theory
The peaks — high entropy, high dissonance — fall in the large gaps between these simple ratios, places where no simple fraction is nearby.
What's striking is how quickly the valleys drop off as you detune an interval. A perfect fifth that's 10 cents flat suddenly has significantly higher entropy — your brain loses its confident lock, and the interval sounds less pure, more "searching." This matches the lived experience of any ensemble player: you can hear when a fifth snaps into tune because the ringing clarity arrives suddenly.
Concordance: A Subtler Concept Than Consonance
Before going further, it helps to distinguish harmonic entropy from the broader concept of consonance and dissonance.
"Consonance" in Western music theory is partly a cultural and contextual judgment. A tritone resolving upward to a major third sounds "dissonant" in a Mozart sonata, but that's partly because we've been trained to hear it that way. Consonance and dissonance in this sense is about musical function and expectation.
What harmonic entropy measures is something more fundamental, which the Xenharmonic Wiki calls psychoacoustic concordance — the raw perceptual tendency of an interval to fuse into a clear, stable sound. Concordance is what barbershop quartets exploit when they lock into pure tuning. It's what happens when a flexible-pitch string ensemble subtly adjusts away from equal temperament to make a chord ring. It's a psychoacoustic phenomenon, not a cultural one.
This distinction matters because:
- Many intervals that are "dissonant" by common practice rules (the major seventh, for instance) actually have moderate harmonic entropy. They're sonically stable but functionally unstable in tonal harmony.
- An interval can be culturally "consonant" (the major third in equal temperament) while having somewhat higher entropy than its just intonation version — which is why equal-tempered major thirds don't quite ring the way they do in just intonation.
Why Simple Ratios Win: The Role of the Farey Series
To understand the math more concretely, it helps to know about the Farey series — a concept from number theory that turns out to be surprisingly musical.
The Farey series of order N lists all fractions between 0 and 1 whose numerator and denominator don't exceed N. A remarkable property: when you list these fractions in order, the simpler fractions (those with smaller numerators and denominators) are more spread out from their neighbors. The fraction 1/2 has a lot of "room" around it. The fraction 13/23 is packed tightly among many other nearby complex fractions.
This turns out to be directly relevant to harmony. When the bell curve of your pitch perception lands near a simple ratio like 3/2, that ratio has a wide "domain" — it captures a large portion of the bell curve's probability. When you land near a complex ratio, many competing fractions share the probability almost equally.
In Erlich's model, the harmonic entropy of an interval is computed by:
- Placing a Gaussian (bell curve) centered on the interval
- Assigning each just-intonation ratio a probability proportional to the area of the bell curve that falls in that ratio's "domain" (defined by the mediants between adjacent fractions)
- Computing the Shannon entropy of the resulting probability distribution
The sole free parameter is s, the width of the bell curve — roughly, the fuzziness of pitch perception. A value of about 1% frequency deviation (around 17 cents) is typically used as a generic value representing average listening conditions, though the right value depends on timbre, register, and the listener.
Concordance vs. Roughness: Two Different Things
Harmonic entropy is one of two major psychoacoustic components of what we perceive as dissonance. The other is roughness (sometimes called sensory dissonance), which is what William Sethares explored extensively in his work on tuning and timbre.
Roughness is caused by close pairs of overtones beating rapidly against each other — the harsh, grating quality of, say, a minor second played loudly on an organ. It's a more local, pairwise phenomenon: each pair of harmonics that falls within the critical band contributes roughness.
These two phenomena behave quite differently, and this matters a lot for chords:
A major triad (4:5:6) and an inverted utonal triad (1/6:1/5:1/4, or roughly 10:12:15 rearranged) have exactly the same set of intervals between pairs of notes. So their roughness is identical. But their harmonic entropy is very different: the major triad has a clear, simple shared fundamental (the root), while the utonal version doesn't. Listeners consistently hear the major triad as more fused, more stable, with a stronger sense of root. Harmonic entropy accounts for this difference. Roughness alone cannot.
This is part of why Erlich developed harmonic entropy as a complement to roughness-based models.
What This Means for Chords
The harmonic entropy framework extends naturally from intervals (dyads) to triads and larger chords. Instead of asking "how well does this interval match a simple ratio?", you ask "how well does this chord match a simple harmonic series?"
A chord like 4:5:6:7 — a major triad with an added harmonic seventh — is an extremely simple-ratio chord. Its four notes are literally the 4th, 5th, 6th, and 7th harmonics of a common fundamental. Its harmonic entropy is very low. You can hear this: the chord has a distinctive, buzzing richness and a clear, powerful root, quite unlike anything you can build in standard 12-tone equal temperament.
The 7th partial — the harmonic seventh — is about 969 cents above the root, or roughly 31 cents flatter than the equal-tempered minor seventh (b♭). This makes it a "microtonal" interval by standard definitions, but from a harmonic entropy perspective it's one of the most concordant intervals that exists. Its exclusion from Western music theory is a historical accident of how tuning systems developed, not a reflection of its sonic qualities.
This is one of the reasons harmonic entropy has been so important to the xenharmonic and just intonation communities: it gives a principled, psychoacoustics-based reason to take intervals like 7/4, 11/8, and 13/8 seriously as musical materials, regardless of whether they appear in standard Western scales.
Harmonic Entropy and Tuning Systems
One of the most illuminating applications of harmonic entropy is evaluating tuning systems — asking which scales and temperaments give you the most access to low-entropy intervals.
Work summarized on the Low Harmonic Entropy Linear Temperaments page of the Xenharmonic Wiki found that when you survey different scales and rank them by average harmonic entropy across their intervals, the winners are temperaments you might already know:
- Meantone temperament (the basis of standard Western tuning since the Renaissance) comes out on top for scales up to 7 notes. The pentatonic and diatonic scales have the lowest average harmonic entropy in their size class — which the article calls "a theoretical explanation for the fact that these are the most popular scales in the world."
- Pajara (the temperament underlying scales like the 10-note "pajara decatonic") ties with meantone chromatic for scales larger than 7 notes.
- Various other temperaments — porcupine, miracle, orwell — appear in finer-resolution analyses.
This is a remarkable result. Harmonic entropy gives us an independent, physics-based reason to understand why the major and minor scales have been so universally successful: they happen to maximize access to concordant intervals given the constraints of a manageable number of notes.
Concordance and the History of Western Tuning
It's worth stepping back to appreciate how harmonic entropy reframes the entire history of Western tuning.
From Pythagorean tuning, to meantone, to the various well-temperaments of the Baroque, to modern 12-tone equal temperament — every step was a negotiation between competing concordance demands. Pure fifths vs. pure thirds. Good keys vs. playable modulation. The whole story can be understood as an attempt to give musicians access to as many low-entropy intervals as possible, within the constraint of a fixed, finite set of pitches per octave.
Harmonic entropy provides a rigorous quantitative framework for this story, one that confirms many intuitions while also surfacing surprises. Equal temperament, for instance, does very well on fifths and fourths, moderately well on thirds, and notably less well on the 7th harmonic relationships that are prominent in jazz harmony and blues.
A Note on Limits
It's worth being honest about what harmonic entropy doesn't do.
A 2025 study by a researcher on the Xenharmonic Wiki (User:Sintel/Validation of common consonance measures) compared harmonic entropy against other consonance models using a dataset of listener ratings for 38 just intervals. The result was a bit humbling: the Tenney complexity measure (a much simpler formula: just log₂(n×d) for a ratio n/d) outperformed harmonic entropy in predicting perceived consonance, with an R² of 0.888 vs. harmonic entropy's 0.807.
The study's conclusion was that for relatively simple intervals, raw complexity correlates with perceived dissonance about as well as the more elaborate harmonic entropy model. However, the author also noted that harmonic entropy (and roughness models) fail to account for proximity — the fact that a complex interval near a simple one sounds smoother than you'd expect from its nominal ratio alone. A proposed "continuous Tenney" measure that accounts for this proximity performed even better (R² = 0.944).
None of this invalidates harmonic entropy as a tool. It remains uniquely valuable for understanding chords (not just intervals), for explaining why otonal chords sound stronger than utonal ones, and for the theoretical project of grounding tuning theory in psychoacoustics. But it's useful to know it's not the final word on consonance perception.
Exploring It Yourself
The easiest way to actually see harmonic entropy is through Scale Workshop (version 3), a free browser-based tool for microtonal scale design. Under the Analysis tab, it includes an interactive harmonic entropy graph where you can see exactly where your scale's intervals fall relative to the low-entropy valleys.
You can explore intervals like 7/4, 11/8, or 13/7 and see them sitting in their own little valleys — not the deep troughs of a fifth or octave, but real local minima, with a sonic character all their own.
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