Physics-first music theory
Physics-first music theory
Start on any step-number. Add 7. Take mod 12. Repeat. You will visit all 12 chromatic steps before returning to where you started. This is the step-7 cycle — the organizing structure that emerges from repeatedly stacking the interval with the simplest frequency ratio after the octave (≈ 3:2, or the 7-step-interval in 12-TET).
Western theory calls this the “circle of fifths.” PhizMusic calls it what it is: a cycle generated by the 7-step-interval under modular arithmetic.
🎯 Simple version: Pick any note. Jump 7 keys up on a piano (counting every key, black and white). Keep jumping by 7. You’ll land on every single note exactly once before coming back to where you started. That’s the step-7 cycle — it’s the only “tour” of all 12 notes where each jump is the most harmonically powerful interval available.
The step-7 cycle is a direct consequence of arithmetic in the integers modulo 12 (written ℤ₁₂). Here is the procedure:
| Iteration (n) | Calculation | (n × 7) mod 12 | Dodeka syllable |
|---|---|---|---|
| 0 | 0 × 7 = 0 | 0 | Do |
| 1 | 1 × 7 = 7 | 7 | So |
| 2 | 2 × 7 = 14 → 14 − 12 | 2 | Re |
| 3 | 3 × 7 = 21 → 21 − 12 | 9 | La |
| 4 | 4 × 7 = 28 → 28 − 24 | 4 | Mi |
| 5 | 5 × 7 = 35 → 35 − 24 | 11 | Si |
| 6 | 6 × 7 = 42 → 42 − 36 | 6 | Hu |
| 7 | 7 × 7 = 49 → 49 − 48 | 1 | Ka |
| 8 | 8 × 7 = 56 → 56 − 48 | 8 | Bi |
| 9 | 9 × 7 = 63 → 63 − 60 | 3 | Xo |
| 10 | 10 × 7 = 70 → 70 − 60 | 10 | Ve |
| 11 | 11 × 7 = 77 → 77 − 72 | 5 | Fa |
| 12 | 12 × 7 = 84 → 84 − 84 | 0 | Do ← back to start |
The cycle visits every step-number exactly once: 0 → 7 → 2 → 9 → 4 → 11 → 6 → 1 → 8 → 3 → 10 → 5 → (0).
The key property: gcd(7, 12) = 1 — that is, 7 and 12 share no common factors. They are coprime.
In the group ℤ₁₂ (integers mod 12 under addition), an element g generates the entire group if and only if gcd(g, 12) = 1. The proof is straightforward: if gcd(g, 12) = d > 1, then all multiples of g mod 12 are also multiples of d, so you can only reach 12/d distinct values — not all 12.
The generators of ℤ₁₂ are exactly the elements coprime to 12:
Generators: 1, 5, 7, 11 (each visits all 12 steps)
Step-intervals 1 and 11 are trivial (chromatic steps up/down). Step-intervals 5 and 7 are the musically interesting generators — and they are related: 12 − 7 = 5. Moving by the 7-step-interval clockwise is the same as moving by the 5-step-interval counterclockwise.
What happens when gcd(g, 12) > 1? The cycle closes early:
| Step-interval (g) | gcd(g, 12) | Steps generated | Cycle |
|---|---|---|---|
| 2 | 2 | 6 of 12 | {0, 2, 4, 6, 8, 10} — whole-tone subset |
| 3 | 3 | 4 of 12 | {0, 3, 6, 9} — diminished subset |
| 4 | 4 | 3 of 12 | {0, 4, 8} — augmented subset |
| 6 | 6 | 2 of 12 | {0, 6} — tritone pair |
Each generates only 12 / gcd(g, 12) distinct steps. These partial cycles are themselves musically significant — they correspond to the equal divisions of the octave — but they cannot serve as generators of the full chromatic set.
🎯 Simple version: 7 and 12 don’t share any divisors (besides 1), so jumping by 7 never falls into a repeating loop until you’ve hit every note. Jump by 2 instead, and you only hit the even-numbered notes — you’re stuck in a 6-note loop forever.
For any step-interval g and number of iterations n:
step-number = (n × g) mod 12
If gcd(g, 12) = 1, this visits all 12 step-numbers for n = 0, 1, …, 11.
Among the four generators of ℤ₁₂ (1, 5, 7, 11), the 7-step-interval holds a privileged acoustic position. Its frequency ratio in 12-TET is:
2^(7/12) ≈ 1.4983
This is the closest 12-TET interval to the 3:2 ratio (= 1.5000) — the simplest frequency ratio after the octave (2:1). The 3:2 ratio is acoustically special because:
Harmonic fusion. A tone’s 3rd harmonic (3f) and another tone’s 2nd harmonic (2 × 1.5f = 3f) coincide exactly. This spectral overlap creates the sensation of consonance — the two tones fuse perceptually. See Consonance & Dissonance.
Simplicity of ratio. After 1:1 (unison) and 2:1 (octave), the ratio 3:2 uses the smallest possible integers. Simpler ratios → more harmonic overlap → stronger perceptual fusion.
Universality. The 3:2 ratio appears in virtually every tuning system across cultures — Pythagorean tuning, Chinese sānfēn sǔnyì (三分損益), Indian ṣaḍja-pañcama relationship — because it is a mathematical fact about vibrating objects, not a cultural preference.
The 5-step-interval (the generator in the opposite direction) approximates the 4:3 ratio — the octave complement of 3:2 (since 2/1 ÷ 3/2 = 4/3). It is equally valid as a generator, and produces the same cycle in reverse order.
The step-7 cycle is not just a mathematical curiosity — it is a map that reveals deep structure in how pitch-subsets relate to each other.
On the cycle, each position is exactly 7 chromatic steps from its neighbors. This means proximity on the cycle indicates harmonic closeness — pitches that share strong spectral overlap.
Select any 7 consecutive positions on the cycle and collect their step-numbers. You get a set of 7 steps from the 12 available — and this set is always one of the diatonic step-subsets (what Western theory calls major/minor scales, modes):
7 consecutive from position 0: {0, 7, 2, 9, 4, 11, 6} → sorted: {0, 2, 4, 6, 7, 9, 11}
7 consecutive from position 1: {7, 2, 9, 4, 11, 6, 1} → sorted: {1, 2, 4, 6, 7, 9, 11}
... and so on for all 12 starting positions
Any 5 consecutive positions yield a pentatonic step-subset. Any 6 consecutive yield a hexatonic. The cycle organizes step-subsets by how many positions they share — the more overlap on the cycle, the more closely related the subsets sound.
Both traversals visit all 12 steps. They are mirror images of the same mathematical structure.
| PhizMusic | Western | Notes |
|---|---|---|
| Step-7 cycle | Circle of fifths | Same structure; “step-7” names the generator directly |
| The 7-step-interval | Perfect fifth | 7 chromatic steps ≈ 3:2 ratio |
| The 5-step-interval | Perfect fourth | 5 chromatic steps ≈ 4:3 ratio; generates the cycle in reverse |
| Step-5 cycle | Circle of fourths | Same cycle traversed counterclockwise |
| Generator of ℤ₁₂ | — | PhizMusic makes the group-theory basis explicit |
| Contiguous arc of 7 | Diatonic key | 7 adjacent positions on the cycle = a diatonic step-subset |