Musical Interval Frequency Ratios
Frequency ratios of musical intervals in just intonation and equal temperament.
Perfect fifth = 3/2, octave = 2/1.
Essential music theory reference.
The Formula
A musical interval is the relationship between two pitches, expressed as a frequency ratio. The simplest and most consonant interval is the octave (ratio 2:1) — doubling the frequency produces a note that sounds the same but higher. All other intervals fall between these octave doublings.
There are two main systems for interval ratios: just intonation (simple integer ratios, most consonant) and equal temperament (12th root of 2 system, allows modulation to any key).
| Interval | Semitones | Just ratio | ET ratio |
|---|---|---|---|
| Unison | 0 | 1/1 = 1.000 | 1.000 |
| Minor 2nd | 1 | 16/15 = 1.067 | 1.0595 |
| Major 2nd | 2 | 9/8 = 1.125 | 1.1225 |
| Minor 3rd | 3 | 6/5 = 1.200 | 1.1892 |
| Major 3rd | 4 | 5/4 = 1.250 | 1.2599 |
| Perfect 4th | 5 | 4/3 = 1.333 | 1.3348 |
| Tritone | 6 | 45/32 = 1.406 | 1.4142 |
| Perfect 5th | 7 | 3/2 = 1.500 | 1.4983 |
| Major 6th | 9 | 5/3 = 1.667 | 1.6818 |
| Major 7th | 11 | 15/8 = 1.875 | 1.8878 |
| Octave | 12 | 2/1 = 2.000 | 2.000 |
Example 1
A4 is 440 Hz. What is E5 (a perfect 5th above A4) in just intonation?
f₂ = 440 × (3/2) = 440 × 1.500
f₂ = 660 Hz (just intonation E5)
Example 2
E5 in equal temperament (7 semitones above A4)?
f₂ = 440 × 2^(7/12) = 440 × 1.4983
f₂ = 659.26 Hz (ET E5 — slightly flat vs. just 660 Hz, but unnoticeable to most ears)
When to Use It
- Understanding why certain note combinations sound harmonious
- Designing tuning systems for acoustic instruments
- Audio synthesis and additive synthesis applications
- Music theory and ear training coursework
Key Notes
- Frequency ratios of pure intervals: Perfect octave = 2:1; perfect fifth = 3:2; perfect fourth = 4:3; major third = 5:4; minor third = 6:5. These simple integer ratios produce the most consonant sounds because the overtones align, minimizing beating.
- Equal temperament — the modern compromise: Modern Western music divides the octave into 12 equal semitones, each with a ratio of 2^(1/12) ≈ 1.05946. This means every interval (except the octave) is slightly out of tune from the pure ratio — but every key works equally well. Older systems (just intonation, meantone) were more in-tune in some keys but unusable in others.
- Semitone counting: Unison = 0; minor 2nd = 1; major 2nd = 2; minor 3rd = 3; major 3rd = 4; perfect 4th = 5; tritone = 6; perfect 5th = 7; minor 6th = 8; major 6th = 9; minor 7th = 10; major 7th = 11; octave = 12 semitones.
- Cents for fine measurement: One cent = 1/100 of a semitone = 2^(1/1200) ≈ 1.000578. Used in tuning measurement and temperament comparison. Equal temperament 5th: 700 cents; pure 5th: 701.96 cents — a difference of ~2 cents, just barely perceptible.
- Applications: Musical interval theory underpins chord construction, transposition, harmony analysis, digital pitch shifting (Auto-Tune), sample-based synthesis (pitch mapping), microtonal music systems, and psychoacoustics research on consonance and dissonance.