Chords

A chord is two or more pitches sounding simultaneously. PhizMusic describes every chord in two complementary ways:

• Ratio-set — the frequency ratios between the notes, explaining why the chord sounds the way it does (physics)
• Step-combo — the chromatic step positions relative to the lowest note, specifying which notes to play (practical)

Ratio first, then step-combo. The physics explains the perception; the step-combo makes it playable.

🎯 Simple version: A chord is multiple notes at once. Some combinations sound smooth because they match the natural pattern of vibrating objects (the harmonic series). The chord {0, 4, 7} matches harmonics 4, 5, 6 — that’s why it sounds natural. PhizMusic describes chords by their frequency ratio (why it sounds good) AND their step positions (which keys to press).

Why Ratios First

The conventional approach teaches chord shapes — “stack a third then another third.” This tells you what to play but not why it works.

The physics-first approach starts from the harmonic series:

Harmonics:     1f    2f    3f    4f    5f    6f    7f    8f ...

Any subset of consecutive harmonics produces a naturally fused chord because:

1. The components are exact integer multiples of a shared fundamental
2. Their overlapping overtones reinforce rather than clash
3. The auditory system’s harmonic template matching (see ear-cochlea.md) recognizes the pattern and groups it as “one sound”

The most important example: harmonics 4, 5, and 6.

The 4:5:6 Chord

Take harmonics 4, 5, and 6 of any fundamental:

4f : 5f : 6f

Simplify by factoring out f:

Ratio-set: 4:5:6

What are the intervals?

• Between 4f and 5f: ratio 5/4 = 386 cents ≈ the 4-step-interval
• Between 5f and 6f: ratio 6/5 = 316 cents ≈ the 3-step-interval
• Between 4f and 6f: ratio 6/4 = 3/2 = 702 cents ≈ the 7-step-interval

So the step-combo is: {0, 4, 7}

This chord is maximally fused because every component is a harmonic of the implied fundamental (the frequency at 1f, two octaves below the lowest note). The auditory system hears this as a single, rich, stable sound.

Western theory calls this a “major triad.” PhizMusic calls it what it is: the 4:5:6 harmonic selection, played as step-combo {0, 4, 7}.

Common Chords

Triads (3 notes)

Why {0, 3, 7} sounds different from {0, 4, 7}: The ratio 10:12:15 involves larger numbers than 4:5:6. Larger ratio numbers = more complex relationship = less harmonic overlap = less perceptual fusion. The chord still sounds “good” (the outer 7-step-interval is the solid 3:2 ratio), but the internal structure is less aligned with the harmonic series, giving it a darker, more complex quality.

Why {0, 3, 6} sounds tense: Neither internal interval (step-3 and step-3) matches the strong 3:2 or 4:3 ratios. The outer interval (step-6, the tritone) sits at the point of maximum harmonic ambiguity. There is no implied fundamental that neatly generates this combination.

Tetrads (4 notes)

The {0, 4, 7, 10} tetrad deserves special attention. Its ratio-set 4:5:6:7 means it literally IS harmonics 4 through 7 of a fundamental. The 7th harmonic (ratio 7:4 = 969 cents) falls 31 cents below the 12-TET step-10 (1000 cents). This means the “natural” version of this chord doesn’t quite match the 12-TET approximation. Barbershop quartets and blues singers instinctively tune the top note lower than 12-TET to hit the 7:4 sweet spot. Western theory calls this chord “dominant seventh” and explains it as functional harmony. Physics simply says: “It’s harmonics 4-5-6-7.”

Building Chords from Step-Combos

Any combination of step-numbers is a valid chord. But some step-combos produce more perceptual fusion than others. A rough hierarchy:

1. Step-combos that map to small-integer ratio-sets (e.g., {0,4,7} → 4:5:6): maximum fusion
2. Step-combos with a strong 3:2 backbone (e.g., {0,3,7} — the outer 7 is solid): moderate fusion
3. Step-combos without any strong ratio anchors (e.g., {0,1,2}): minimal fusion, maximum roughness

This hierarchy is a description of physics, not a prescription of taste. Composers deliberately use all levels of fusion and roughness for expressive purposes. Tension and resolution are two sides of the same phenomenon (see chord-progressions.md).

Transposition

Because chords are step-combos, transposition is addition:

Original:      {0, 4, 7}
Transpose +5:  {5, 9, 0}  (mod 12)
Transpose +7:  {7, 11, 2} (mod 12)

The internal intervals — and therefore the ratios, the fusion quality, and the character — are invariant under transposition. In 12-TET, every transposition sounds identical. This is the payoff of equal temperament (see twelve-tet.md).

Voicing and Registration

The step-combo {0, 4, 7} specifies pitch classes — which of the 12 chromatic steps to include. It doesn’t specify:

• Which octave each note is in (registration)
• How the notes are spaced across octaves (voicing)
• How many times each note appears (doubling)

These choices affect the sound significantly — a closely spaced {0, 4, 7} in one octave sounds different from a widely spaced version across three octaves — but the harmonic identity (4:5:6 ratio relationship) is preserved. Voicing is an expressive dimension beyond the step-combo itself.

Translation Table

Connections

• Intervals — the step-intervals that make up a chord’s internal structure
• Harmonic Series — the physical source of the ratio-sets
• Frequency Ratios — why simple ratios produce fusion
• Chord Progressions — how chords connect in sequences over time
• Consonance & Dissonance — the perceptual mechanism behind chord “stability” and “tension”
• Scales — chords as simultaneous selections from a step-subset

Suggested References

• Sethares Dissonance Curves (aatishb.com) — 3D triad dissonance surface and heatmap showing why certain step-combos produce minimal roughness