Physics-first music theory
Physics-first music theory
Functional analysis asks: given a step-subset and a root, what role does each step-combo play? Every member of the step-subset can serve as the bottom note of a triad built from that subset. Some of those triads sound stable, others tense, others transitional. The physics of why comes down to ratio-set quality and step-interval relationships between roots.
🎯 Simple version: Pick a scale (step-subset) and a home note (root). You can build a chord on each scale note. Some chords feel like “home,” some feel like “tension,” some feel like “pull toward home.” Functional analysis labels those roles — Position 0 is home, Position 7 (root a 7-step-interval above home) pulls hardest toward home. Position numbers are chromatic step-numbers (0-11), so non-scale positions have no triad and are skipped.
A “key” in Western theory is two things combined:
In PhizMusic, this is explicit: “step-subset {0,2,4,5,7,9,11} rooted at step 0” means the major scale starting on Do. Change the root to step 7 and you get “step-subset {7,9,11,0,2,4,6}” — the same internal structure transposed, now anchored on So.
The root step-number defines the reference frequency. All other pitches in the subset are heard relative to this anchor. The auditory system builds expectations around returning to this reference — that pull is the foundation of tonal music.
Why a root creates gravity: the root’s harmonics overlap most strongly with the harmonic series of the step-subset’s most consonant intervals (the 7-step-interval, the 5-step-interval). The auditory system’s harmonic template matching reinforces the root as a perceptual anchor.
In Western theory, “Key of C major” is a compact label. In PhizMusic, it unpacks into two explicit pieces of information:
“Key of G major” is the same geometric pattern rooted at a different starting position: step 7 (So, ≈392.00 Hz). The resulting step-subset is {7, 9, 11, 0, 2, 4, 6} — identical interval structure, different absolute pitches.
Changing key = transposition = same geometry, different starting point on the 12-step circle. Every relationship between positions is preserved: the triad at the root is still 4+3, the triad 7 steps above the root is still 4+3 with a leading-tone pull, and so on. Only the absolute frequencies change.
The root frequency’s harmonics align most strongly with the step-subset’s most consonant intervals:
In “Key of step 0 major,” Do4 = 261.63 Hz is home. The most consonant interval is the 7-step-interval: 261.63 × 3/2 ≈ 392 Hz (So). This is why Position 7 has the strongest pull back to Position 0.
Modulation = changing the root = reorienting the listener’s harmonic template around a new reference frequency. If a piece moves from “step-subset rooted at step 0” to “step-subset rooted at step 7,” every position relationship is preserved — but the listener’s sense of “home” shifts from 261.63 Hz to 392.00 Hz. The brain’s harmonic template matcher recalibrates, and what was Position 7 becomes Position 0 in the new key.
The phrase “key of…” appears constantly in real musical situations. Each usage maps to specific PhizMusic operations — choosing a step-subset pattern, a root step-number, or both. Here are five common scenarios with their PhizMusic translations.
1. Band rehearsal: “Let’s play this in G”
A guitarist says “let’s play in G” before a jam. What they mean: use the major step-subset {0,2,4,5,7,9,11} with root at step 7 (So). The reference frequency is So4 = 392.00 Hz. Every instrument orients around this anchor — the bass player’s home note is 392.00 Hz (or 196.00 Hz an octave below), the keyboard player builds Position 0’s step-combo {7,11,2} as the home chord, and Position 7 relative to this root (step 2, Re, 293.66 Hz) provides the strongest pull back to the 392.00 Hz anchor.
PhizMusic: step-subset {0,2,4,5,7,9,11} rooted at step 7 (So, 392.00 Hz). Absolute step-subset members: {7,9,11,0,2,4,6}.
2. Singer asks to change key: “Can we do it in a lower key — maybe D?”
A singer finds that the melody’s highest note is uncomfortable. Moving from “key of G” to “key of D” means: keep the same step-subset pattern {0,2,4,5,7,9,11} but change the root from step 7 (So, 392.00 Hz) to step 2 (Re, 293.66 Hz). Every pitch in the song drops by 5 chromatic steps. The melody that peaked at 4.11 (Si4, 493.88 Hz) now peaks at 4.6 (Hu4, 369.99 Hz) — a difference of 123.89 Hz that may sit comfortably in the singer’s range. All internal relationships are identical: Position 7’s pull toward Position 0, leading-tone resolution, voice-leading distances — nothing changes except the absolute frequencies.
PhizMusic: same step-subset geometry, root changes from step 7 (392.00 Hz) to step 2 (293.66 Hz). This is transposition, not modulation — the listener never hears the shift as a reorientation because the song restarts in the new root.
3. Reading a lead sheet: “Cm7 — F7 — BbMaj7”
A jazz musician reads chord symbols on a lead sheet. The symbols encode both the root step-numbers and the step-combo structures. “Cm7” means: root at step 0 (Do, 261.63 Hz), step-combo {0,3,7,10} — a 4-note structure with intervals 3+4+3. “F7” means: root at step 5 (Fa, 349.23 Hz), step-combo {5,9,0,3} — intervals 4+3+3. “BbMaj7” means: root at step 10 (Ve, 466.16 Hz), step-combo {10,2,5,9} — intervals 4+3+4. The lead sheet implies the step-subset context: these three chords belong to the step-subset {0,2,3,5,7,8,10} rooted at step 10 (Ve, 466.16 Hz) — a major step-subset where the Cm7 functions as Position 2 (the “ii” in Western terms), F7 as Position 7 (the “dominant”), and BbMaj7 as Position 0 (home). The musician infers the full tonal context from three chord symbols.
PhizMusic: step-subset {0,2,4,5,7,9,11} rooted at step 10 (Ve, 466.16 Hz). The progression is Position 2 → Position 7 → Position 0 — the strongest cadential approach pattern.
4. Transposing for a different instrument: “The concert pitch is Eb, so you play in F”
A trumpet player (whose instrument sounds a step-interval of 2 lower than written) receives a part written “in F” while the rest of the ensemble plays in “Eb.” The physics: the ensemble’s reference frequency is step 3 (Xo, 311.13 Hz). The trumpet’s notation is shifted up by 2 chromatic steps to compensate for the instrument’s built-in transposition. When the trumpet player reads and fingers step 5 (Fa), the instrument produces step 3 (Xo, 311.13 Hz) — matching the ensemble. In PhizMusic terms, both players are operating in step-subset {0,2,4,5,7,9,11} rooted at step 3 (Xo, 311.13 Hz). The trumpet’s written part simply applies an offset of +2 to every step-number so that the sounding output aligns. The geometry is identical; only the notation layer differs.
PhizMusic: the sounding step-subset is rooted at step 3 (Xo, 311.13 Hz) for all instruments. Transposing instruments apply a fixed step-number offset in their notation — the acoustic result is the same root, same step-subset, same Positions.
5. Identifying a song’s key by ear: “This song is in A minor”
A musician listens to a recording and identifies the tonal center. What their auditory system is doing: harmonic template matching locks onto the frequency that the melody and harmony most consistently resolve toward. If that frequency is 440.00 Hz (La4, step 9) and the pitch collection sounds dark and warm (step-combo at the root has interval structure 3+4, ratio ≈ 10:12:15), the musician concludes: step-subset {0,2,3,5,7,8,10} rooted at step 9 (La, 440.00 Hz) — the natural minor pattern. The absolute step-subset members are {9,11,0,2,4,5,7}. Position 0 is La4 = 440.00 Hz. Position 7 is step 4 (Mi, 329.63 Hz), which in the natural minor produces a dark 3+4 step-combo rather than the bright 4+3 that would create a strong leading-tone pull — explaining why minor-key resolutions often feel less decisive than major-key ones.
PhizMusic: the ear identifies the root step-number (step 9, La, 440.00 Hz) by detecting which frequency functions as the perceptual anchor, then infers the step-subset pattern from the quality of the intervals surrounding it.
Given a 7-member step-subset, each member can generate a triad by stacking every-other member of the subset. This is the PhizMusic equivalent of “building a chord on each scale degree.”
The Position number equals the chromatic step-number of the triad’s root (0-11). Steps that are not in the step-subset have no diatonic triad and are marked “—” in the table.
The procedure for building a triad at a given position:
The result is a 3-note step-combo built entirely from the step-subset. Different positions produce step-combos with different interval structures — and therefore different ratio-set approximations and different perceptual qualities.
For step-subset {0, 2, 4, 5, 7, 9, 11} (major) rooted at step 0:
| Position | Root in subset? | Step-combo | Intervals | Approx. ratio-set | Quality |
|---|---|---|---|---|---|
| 0 (Do) | ✓ | {0, 4, 7} | 4 + 3 | 4:5:6 | Bright, stable — home |
| 1 (Ka) | — | — | — | — | not in step-subset |
| 2 (Re) | ✓ | {2, 5, 9} | 3 + 4 | 10:12:15 | Dark, warm |
| 3 (Xo) | — | — | — | — | not in step-subset |
| 4 (Mi) | ✓ | {4, 7, 11} | 3 + 4 | 10:12:15 | Dark, warm |
| 5 (Fa) | ✓ | {5, 9, 0} | 4 + 3 | 4:5:6 | Bright, stable |
| 6 (Hu) | — | — | — | — | not in step-subset |
| 7 (So) | ✓ | {7, 11, 2} | 4 + 3 | 4:5:6 | Bright, tense — strongest pull to Pos 0 |
| 8 (Bi) | — | — | — | — | not in step-subset |
| 9 (La) | ✓ | {9, 0, 4} | 3 + 4 | 10:12:15 | Dark, gentle |
| 10 (Ve) | — | — | — | — | not in step-subset |
| 11 (Si) | ✓ | {11, 2, 5} | 3 + 3 | ≈25:30:36 | Tense, unstable — diminished |
| Position | Step-combo | Hz (root of triad at Do4 = 261.63 Hz) | Preview |
|---|---|---|---|
| 0 (Do) | {0, 4, 7} | 261.63, 329.63, 392.00 | |
| 1 (Ka) | — | — | |
| 2 (Re) | {2, 5, 9} | 293.66, 349.23, 440.00 | |
| 3 (Xo) | — | — | |
| 4 (Mi) | {4, 7, 11} | 329.63, 392.00, 493.88 | |
| 5 (Fa) | {5, 9, 0} | 349.23, 440.00, 523.25 | |
| 6 (Hu) | — | — | |
| 7 (So) | {7, 11, 2} | 392.00, 493.88, 587.33 | |
| 8 (Bi) | — | — | |
| 9 (La) | {9, 0, 4} | 440.00, 523.25, 659.26 | |
| 10 (Ve) | — | — | |
| 11 (Si) | {11, 2, 5} | 493.88, 587.33, 698.46 |
Pattern: Positions 0, 5, and 7 produce step-combos with interval structure 4+3 (ratio ≈ 4:5:6). Positions 2, 4, and 9 produce 3+4 (ratio ≈ 10:12:15). Position 11 produces 3+3 (diminished — no strong harmonic-series alignment). The quality of each position is determined entirely by the step-subset’s geometry. Five positions (1, 3, 6, 8, 10) are outside the major step-subset and have no diatonic triad.
🔊 Hz reference: All frequencies above assume Do4 = 261.63 Hz. For example, Position 7’s triad root (So4) is 261.63 × 2^(7/12) ≈ 392.00 Hz. The 3:2 ratio between 392.00 Hz and 261.63 Hz (= 1.4983… ≈ 1.5) is why this interval is the most consonant after the octave.
The most powerful directional pull in tonal music is Position 7 → Position 0. Why?
The root of Position 7 sits at the 7-step-interval from the root of Position 0. The 7-step-interval corresponds to the frequency ratio 3:2 — the strongest consonance after the octave. The step-7 cycle explains this relationship in depth: step-7 is the generator of the diatonic system itself.
But Position 7’s step-combo also contains step 11 — which sits at the 1-step-interval from step 0 (the root of Position 0). This 1-step-interval relationship creates maximum melodic tension: the ear expects step 11 to resolve upward to step 0. When it does, that tiny motion produces a disproportionately strong sense of arrival.
In Hz terms: Position 7’s root is So4 = 392.00 Hz. Position 0’s root is Do4 = 261.63 Hz. The ratio 392.00/261.63 ≈ 1.498 ≈ 3/2. Meanwhile, step 11 (Si4 = 493.88 Hz) wants to resolve upward to step 0 in the next octave (Do5 = 523.25 Hz) — a distance of only 29.37 Hz, which the ear perceives as a strong semitone pull.
The combination of:
makes Position 7 → Position 0 the most powerful resolution in the 7-member step-subset system.
Position 5 → Position 0 is the second-strongest pull. The root of Position 5 sits at the 5-step-interval from Position 0’s root — the ratio 4:3 (349.23/261.63 ≈ 1.335 ≈ 4/3), also highly consonant. But Position 5 lacks the leading-tone (step 11 is not in its step-combo {5, 9, 0}), so the resolution is softer.
Progressions expressed as Position sequences (for the major step-subset rooted at step 0):
| Pattern | Step-combos | Character |
|---|---|---|
| 0 – 5 – 7 – 0 | {0,4,7} → {5,9,0} → {7,11,2} → {0,4,7} | The foundational closed loop. Depart, build tension, resolve. (Western: I–IV–V–I) |
| 0 – 7 – 9 – 5 | {0,4,7} → {7,11,2} → {9,0,4} → {5,9,0} | Tension first, then gradual descent back toward stability. (Western: I–V–vi–IV) |
| 0 – 9 – 5 – 7 | {0,4,7} → {9,0,4} → {5,9,0} → {7,11,2} | The “pop progression” — cycles through all three quality types. (Western: I–vi–IV–V) |
| 2 – 7 – 0 | {2,5,9} → {7,11,2} → {0,4,7} | Strong cadential approach — Position 2’s dark quality sets up Position 7’s tension. (Western: ii–V–I) |
These patterns are not rules. They are commonly observed sequences whose perceptual effects (tension, resolution, surprise) are explained by the voice-leading distances and ratio-set qualities described in chord-progressions.md.
The position table above uses the major step-subset. Different step-subsets generate different quality distributions:
Natural minor {0, 2, 3, 5, 7, 8, 10} rooted at step 0:
| Position | Root in subset? | Step-combo | Intervals | Quality |
|---|---|---|---|---|
| 0 (Do) | ✓ | {0, 3, 7} | 3 + 4 | Dark, warm — home |
| 1 (Ka) | — | — | — | not in step-subset |
| 2 (Re) | ✓ | {2, 5, 8} | 3 + 3 | Tense, unstable |
| 3 (Xo) | ✓ | {3, 7, 10} | 4 + 3 | Bright, stable |
| 4 (Mi) | — | — | — | not in step-subset |
| 5 (Fa) | ✓ | {5, 8, 0} | 3 + 4 | Dark, warm |
| 6 (Hu) | — | — | — | not in step-subset |
| 7 (So) | ✓ | {7, 10, 2} | 3 + 4 | Dark — no leading-tone pull |
| 8 (Bi) | ✓ | {8, 0, 3} | 4 + 3 | Bright, stable |
| 9 (La) | — | — | — | not in step-subset |
| 10 (Ve) | ✓ | {10, 2, 5} | 4 + 3 | Bright, stable |
| 11 (Si) | — | — | — | not in step-subset |
Notice: Position 7 in the natural minor produces a 10:12:15 ratio-set instead of 4:5:6. The absence of the leading-tone (step 11, one chromatic step below the root) weakens the resolution pull. This is why Western composers developed the harmonic minor — step-subset {0, 2, 3, 5, 7, 8, 11} — which raises step 10 to step 11, restoring the 4:5:6 ratio-set at Position 7 and the leading-tone resolution.
Harmonic minor {0, 2, 3, 5, 7, 8, 11} rooted at step 0:
| Position | Root in subset? | Step-combo | Intervals | Quality |
|---|---|---|---|---|
| 0 (Do) | ✓ | {0, 3, 7} | 3 + 4 | Dark, warm — home |
| 1 (Ka) | — | — | — | not in step-subset |
| 2 (Re) | ✓ | {2, 5, 8} | 3 + 3 | Tense, unstable |
| 3 (Xo) | ✓ | {3, 7, 11} | 4 + 4 | Bright, augmented |
| 4 (Mi) | — | — | — | not in step-subset |
| 5 (Fa) | ✓ | {5, 8, 0} | 3 + 4 | Dark, warm |
| 6 (Hu) | — | — | — | not in step-subset |
| 7 (So) | ✓ | {7, 11, 2} | 4 + 3 | Bright, tense — leading-tone restored |
| 8 (Bi) | ✓ | {8, 0, 3} | 4 + 3 | Bright, stable |
| 9 (La) | — | — | — | not in step-subset |
| 10 (Ve) | — | — | — | not in step-subset |
| 11 (Si) | ✓ | {11, 2, 5} | 3 + 3 | Tense, unstable — diminished |
The key difference: Position 7 now contains step 11 (the leading-tone), restoring the Position 7 → Position 0 resolution pull that defines tonal music.
A mode in PhizMusic is nothing more than the same step-subset with a different root step-number. The set of absolute pitches — the actual Hz values played — remains identical. What changes is which pitch the listener’s brain treats as the perceptual anchor, the reference frequency for the harmonic template matcher.
This is where Western terminology becomes most confusing. “C major” and “A minor” are said to be “related keys,” yet conventional theory obscures the deep physical truth: they use exactly the same seven frequencies. The only difference is which frequency your ear treats as “home.”
Consider step-subset {0,2,4,5,7,9,11} rooted at step 0 (Do4 = 261.63 Hz). This is “C major” in Western terminology:
| Step | Syllable | Hz (octave 4) |
|---|---|---|
| 0 | Do | 261.63 |
| 2 | Re | 293.66 |
| 4 | Mi | 329.63 |
| 5 | Fa | 349.23 |
| 7 | So | 392.00 |
| 9 | La | 440.00 |
| 11 | Si | 493.88 |
Root triad at Position 0: {0, 4, 7} = Do-Mi-So = 261.63 Hz, 329.63 Hz, 392.00 Hz
Now take these exact same seven frequencies and re-root at step 9 (La4 = 440.00 Hz). This is “A minor” in Western terminology. From La’s perspective as step 0, the step-subset becomes {0,2,3,5,7,8,10}:
| Step (from La) | Syllable | Hz (octave 4) | Original step (from Do) |
|---|---|---|---|
| 0 | La | 440.00 | 9 |
| 2 | Si | 493.88 | 11 |
| 3 | Do | 523.25 (octave 5) | 0 |
| 5 | Re | 587.33 (octave 5) | 2 |
| 7 | Mi | 659.26 (octave 5) | 4 |
| 8 | Fa | 698.46 (octave 5) | 5 |
| 10 | So | 783.99 (octave 5) | 7 |
Root triad at Position 0: {9, 0, 4} = La-Do-Mi = 440.00 Hz, 523.25 Hz, 659.26 Hz
🎯 Simple version: “C major” and “A minor” use the same 7 piano keys. The only difference: in C major, you treat C as home; in A minor, you treat A as home. Same physical frequencies, different perceptual anchor.
The frequencies don’t change. What changes:
Crucially, the Position 7 triad also changes:
This is why Western music treats them as distinct keys. The root triad’s quality changes, and critically, the tension-resolution pattern at Position 7 → Position 0 changes. In major, Position 7 is bright and contains the leading-tone (step 11, a half-step below step 0). In natural minor, Position 7 is dark and lacks that leading-tone pull.
If the frequencies are identical, why do modes produce different emotional qualities?
The answer lies in harmonic template matching. When you hear a sequence of pitches, your brain attempts to fit them onto a harmonic series template rooted at some fundamental frequency. The pitch you treat as “root” determines which template your brain tries to use.
When the root changes:
Perceptually, the listener’s brain locks onto different reference frequencies, producing genuinely different emotional landscapes despite identical pitch material. The physics is the same; the neuroscience is different.
See Scales — Modes for the gap-pattern perspective on this same phenomenon.
The major step-subset {0,2,4,5,7,9,11} generates seven modes, one rooted at each member. Here they are with their physical characteristics:
| Mode name (Western) | Root step | Root Hz (octave 4) | Root triad steps | Root triad intervals | Root triad quality | Position 7 has leading-tone? |
|---|---|---|---|---|---|---|
| Ionian (major) | 0 (Do) | 261.63 | {0, 4, 7} | 4 + 3 | Bright | ✓ (step 11 → 0) |
| Dorian | 2 (Re) | 293.66 | {2, 5, 9} | 3 + 4 | Dark | ✗ (step 0 not adjacent) |
| Phrygian | 4 (Mi) | 329.63 | {4, 7, 11} | 3 + 4 | Dark | ✗ (step 2 not adjacent) |
| Lydian | 5 (Fa) | 349.23 | {5, 9, 0} | 4 + 3 | Bright | ✗ (step 4 not adjacent) |
| Mixolydian | 7 (So) | 392.00 | {7, 11, 2} | 4 + 3 | Bright | ✗ (step 5 not adjacent) |
| Aeolian (natural minor) | 9 (La) | 440.00 | {9, 0, 4} | 3 + 4 | Dark | ✗ (step 7 not adjacent) |
| Locrian | 11 (Si) | 493.88 | {11, 2, 5} | 3 + 3 | Diminished | ✗ (step 9 not adjacent) |
Notice:
The “leading-tone” criterion checks whether Position 7 (the chromatic step 7 steps above the root) contains a step-subset member that is 1 chromatic step below the root. In Ionian (major) rooted at step 0, Position 7 contains step 7, and the step-subset includes step 11, which is 1 step below step 0 (mod 12). This creates strong directional pull: step 11 “wants” to resolve up by 1 step to step 0.
In the other modes, either Position 7 doesn’t contain a bright triad, or the step-subset doesn’t include a member 1 step below the root, so the leading-tone pull is absent.
When you compose or analyze music:
(step-subset, root) tuple.The deep insight: modes are a perceptual phenomenon, not a physical one. The acoustics are identical. The neuroscience — which harmonic template your brain locks onto — is what changes. This is why modes can evoke such different emotions while using the same collection of physical frequencies.
| Western term | PhizMusic equivalent | Notes |
|---|---|---|
| Key of C major | Step-subset {0,2,4,5,7,9,11} rooted at step 0 (Do, 261.63 Hz) | “Key” = step-subset pattern + root step-number |
| Key of G major | Step-subset {0,2,4,5,7,9,11} rooted at step 7 (So, 392.00 Hz) | Same pattern, different root |
| Key of A minor | Step-subset {0,2,3,5,7,8,10} rooted at step 9 (La, 440.00 Hz) | Minor pattern, root = La |
| Scale degree | Chromatic step-number of the position root (0-11) | Position number = chromatic step; non-subset steps have no triad |
| I (tonic) | Position 0 | Home step-combo; root = reference frequency |
| ii (supertonic) | Position 2 | First dark-quality position (root 2 steps above home) |
| iii (mediant) | Position 4 | Second dark-quality position (root 4 steps above home) |
| IV (subdominant) | Position 5 | Bright, second-strongest pull (root 5 steps above home) |
| V (dominant) | Position 7 | Bright, strongest pull — root at the 7-step-interval (≈ 3:2 ratio) |
| vi (submediant) | Position 9 | Dark, gentle (root 9 steps above home) |
| vii° (leading tone) | Position 11 | Diminished, maximum instability (root 11 steps above home) |
| Tonic function | Position 0 step-combo (highest stability) | Stable arrival point |
| Dominant function | Position 7 step-combo (maximum pull toward Position 0) | Tension seeking resolution; contains leading-tone |
| Subdominant function | Position 5 step-combo (departure from stability) | Transitional; ratio 4:3 root relationship |
| Roman numeral analysis | Position-number analysis | Same concept, using chromatic step-numbers (0-11) |
| Cadence (V → I) | Position 7 → Position 0 | The 7-step-interval root relationship + leading-tone |
| Plagal cadence (IV → I) | Position 5 → Position 0 | The 5-step-interval root relationship |
| Modulation | Changing the root step-number | Same step-subset pattern, new reference frequency |
| Relative minor | Same step-subset members, root shifts to step 9 | e.g., C major → A minor share the same 7 pitches |