Physics-first music theory
Physics-first music theory
An interval is the distance between two pitches, measured as a step-interval — the number of chromatic steps separating them. Step-interval is the PhizMusic primary measure: unambiguous, arithmetic-friendly, and independent of any scale or key. Each step-interval corresponds to a frequency ratio that explains its acoustic character.
🎯 Simple version: The distance between two notes is counted in steps (0 to 12). Each step-interval has a natural frequency ratio. The 7-step-interval (what musicians call a “fifth”) is special because it’s the 3:2 ratio — the simplest ratio after the octave. Small-number ratios sound smooth; big-number ratios sound rough.
Every interval has two complementary descriptions:
Neither description alone is complete. The step-interval tells you which keys to press; the ratio tells you why the result sounds consonant, tense, or neutral.
All intervals within one octave (0 through 12 steps), with both the just-intonation ratio (from the harmonic series) and the 12-TET approximation:
| Step-interval | Just ratio | Just cents | 12-TET cents | Error | Example (Hz) | Character | Preview |
|---|---|---|---|---|---|---|---|
| 0 | 1:1 | 0.0 | 0 | 0.0 | 261.63 Hz | Identity — same pitch | |
| 1 | 16:15 | 111.7 | 100 | -11.7 | 261.63 → 277.18 Hz | Maximum tension, semitone "rub" | |
| 2 | 9:8 | 203.9 | 200 | -3.9 | 261.63 → 293.66 Hz | Mild tension, melodic step | |
| 3 | 6:5 | 315.6 | 300 | -15.6 | 261.63 → 311.13 Hz | Dark, warm fusion | |
| 4 | 5:4 | 386.3 | 400 | +13.7 | 261.63 → 329.63 Hz | Bright, sweet fusion | |
| 5 | 4:3 | 498.0 | 500 | +2.0 | 261.63 → 349.23 Hz | Open, stable | |
| 6 | √2:1 | 600.0 | 600 | 0.0 | 261.63 → 369.99 Hz | Maximum ambiguity, symmetry point | |
| 7 | 3:2 | 702.0 | 700 | -2.0 | 261.63 → 392.00 Hz | Maximum fusion after octave | |
| 8 | 8:5 | 813.7 | 800 | -13.7 | 261.63 → 415.30 Hz | Inversion of step-4 | |
| 9 | 5:3 | 884.4 | 900 | +15.6 | 261.63 → 440.00 Hz | Bright, open | |
| 10 | 9:5 | 1017.6 | 1000 | -17.6 | 261.63 → 466.16 Hz | Tense, wants to resolve | |
| 11 | 15:8 | 1088.3 | 1100 | +11.7 | 261.63 → 493.88 Hz | Extreme tension, leading tone | |
| 12 | 2:1 | 1200.0 | 1200 | 0.0 | 261.63 → 523.25 Hz | Octave — perceptual "reset" |
1200 × log₂(ratio))freq = 261.63 × 2^(step/12)The 6-step-interval (the tritone) is unique: it exactly bisects the octave. Its ratio in 12-TET is √2:1 — the only irrational ratio that is exactly representable. In just intonation, there’s no single “correct” ratio for this interval; it sits at the boundary between two harmonic territories (the 7:5 from below, the 10:7 from above).
Step-intervals 5 and 7 are the best-approximated intervals in 12-TET (within 2 cents of just). This is not coincidence — 12-TET was designed to optimize these critical intervals.
Step-intervals 4 and 3 have the largest errors (~14-16 cents). These are the intervals most affected by the equal-temperament compromise, and the ones where the difference between 12-TET and just intonation is most audible in sustained chords.
Every interval has a complement (inversion) that together with it completes an octave:
step-interval + inversion = 12
| Step-interval | Inversion | Ratio | Inversion ratio |
|---|---|---|---|
| 0 | 12 | 1:1 | 2:1 |
| 1 | 11 | 16:15 | 15:8 |
| 2 | 10 | 9:8 | 9:5 |
| 3 | 9 | 6:5 | 5:3 |
| 4 | 8 | 5:4 | 8:5 |
| 5 | 7 | 4:3 | 3:2 |
| 6 | 6 | √2:1 | √2:1 |
Step-6 is its own inversion — it divides the octave exactly in half.
When two tones are in a simple frequency ratio, their harmonic overtones align:
Example: the 7-step-interval (ratio 3:2, tones at 200 Hz and 300 Hz)
Lower tone harmonics: 200 400 600 800 1000 1200 1400 1600 ...
Upper tone harmonics: 300 600 900 1200 1500 1800 2100 2400 ...
^^^ ^^^^
Shared frequencies!
Every 3rd harmonic of the lower tone matches every 2nd harmonic of the upper tone. These shared frequencies reinforce rather than interfere. The harmonics that don’t match are well-separated — outside each other’s critical bandwidth — so they don’t create roughness.
Example: the 1-step-interval (ratio ~16:15, tones at 200 Hz and 212 Hz)
Lower tone harmonics: 200 400 600 800 1000 ...
Upper tone harmonics: 212 424 636 848 1060 ...
^^^ ^^^ ^^^ ^^^
12Hz 24Hz 36Hz 48Hz apart — within critical bandwidth!
Nearly every pair of harmonics falls within critical bandwidth, creating roughness (rapid beating). This is the physics of dissonance (see consonance-dissonance.md).
Intervals larger than 12 steps span more than one octave. They are heard as the basic interval (mod 12) at a wider spacing:
Compound intervals are common in chords and orchestration but maintain the acoustic character of their mod-12 equivalent, colored by the additional spacing.
| PhizMusic | Western | Notes |
|---|---|---|
| The 0-step-interval | Unison | — |
| The 1-step-interval | Minor 2nd / semitone | “Half step” |
| The 2-step-interval | Major 2nd / whole tone | “Whole step” |
| The 3-step-interval | Minor 3rd | — |
| The 4-step-interval | Major 3rd | — |
| The 5-step-interval | Perfect 4th | — |
| The 6-step-interval | Tritone / augmented 4th / diminished 5th | Multiple Western names for one interval |
| The 7-step-interval | Perfect 5th | — |
| The 8-step-interval | Minor 6th | — |
| The 9-step-interval | Major 6th | — |
| The 10-step-interval | Minor 7th | — |
| The 11-step-interval | Major 7th | — |
| The 12-step-interval | Octave | — |
Note: Western names count from 1 (unison = “1st”), count only diatonic steps, and use quality labels (minor, major, perfect, augmented, diminished). PhizMusic counts from 0, counts all chromatic steps, and uses no quality labels — the number IS the description.