Harmonic Series Calculator
Calculate harmonic frequencies and explore the mathematical foundation of musical harmony
Calculate Harmonic Series
Root note of the harmonic series
Octave number (4 = middle octave)
How many harmonics to calculate
Fundamental Frequency
Harmonic Series Results
| Harmonic | Frequency (Hz) | Note | Interval | Ratio | Cents |
|---|---|---|---|---|---|
| 1 | 261.63 | D#4 | Unison | 1:1 | 0 |
| 2 | 523.25 | D#5 | Unison + 1 octave | 2:1 | 0 |
| 3 | 784.88 | A#5 | Perfect 5th + 1 octave | 3:1 | +2 |
| 4 | 1046.50 | D#6 | Unison + 2 octaves | 4:1 | 0 |
| 5 | 1308.13 | G6 | Major 3rd + 2 octaves | 5:1 | -14 |
| 6 | 1569.75 | A#6 | Perfect 5th + 2 octaves | 6:1 | +2 |
| 7 | 1831.38 | C#6 | Minor 7th + 2 octaves | 7:1 | -31 |
| 8 | 2093.00 | D#7 | Unison + 3 octaves | 8:1 | 0 |
| 9 | 2354.63 | F7 | Major 2nd + 3 octaves | 9:1 | +4 |
| 10 | 2616.26 | G7 | Major 3rd + 3 octaves | 10:1 | -14 |
| 11 | 2877.88 | A7 | Tritone + 3 octaves | 11:1 | -49 |
| 12 | 3139.51 | A#7 | Perfect 5th + 3 octaves | 12:1 | +2 |
| 13 | 3401.13 | B7 | Minor 6th + 3 octaves | 13:1 | +41 |
| 14 | 3662.76 | C#7 | Minor 7th + 3 octaves | 14:1 | -31 |
| 15 | 3924.38 | D7 | Major 7th + 3 octaves | 15:1 | -12 |
| 16 | 4186.01 | D#8 | Unison + 4 octaves | 16:1 | 0 |
Cents: Deviation from equal temperament tuning (0 = perfect match)
Color coding: Green ≤5¢, Yellow ≤10¢, Orange ≤20¢, Red >20¢
Common Harmonic Intervals
Perfect Intervals
Problematic Intervals
Common Fundamentals
Music Theory Tips
Lower harmonics sound more consonant than higher ones
The 7th and 11th harmonics create notable dissonance
Simple ratios (2:1, 3:2, 4:3) sound more pleasing
Harmonic series forms the basis of just intonation
Instrument timbre depends on harmonic content
Applications
Instrument Design
Understanding resonant frequencies
Audio Synthesis
Creating realistic instrumental sounds
Tuning Systems
Comparing equal temperament vs just intonation
Acoustics
Room acoustics and speaker design
Understanding the Harmonic Series
What is the Harmonic Series?
The harmonic series is a sequence of frequencies that are integer multiples of a fundamental frequency. When you hear a musical note, you're actually hearing the fundamental frequency plus many of these harmonic overtones, which give the instrument its unique timbre.
Mathematical Foundation
- •1st Harmonic: f (fundamental frequency)
- •2nd Harmonic: 2f (octave)
- •3rd Harmonic: 3f (perfect fifth above octave)
- •nth Harmonic: n × f
Just Intonation vs Equal Temperament
Just Intonation: Based on simple integer ratios from the harmonic series
Equal Temperament: Divides the octave into 12 equal semitones
Cents Deviation
Cents measure the difference between just intonation (harmonic series) and equal temperament. One semitone = 100 cents, so small deviations (±5 cents) are barely noticeable, while larger deviations (±30 cents) create noticeable dissonance.
Fun Fact: The harmonic series explains why certain intervals sound consonant (simple ratios like 2:1, 3:2) while others sound dissonant (complex ratios like 11:8).
Practical Applications
Instrument Timbre
Different instruments emphasize different harmonics, creating their unique sound character. A flute has mainly the fundamental, while a violin has rich harmonic content.
Chord Theory
Major and minor chords are built from the harmonic series. The major triad (4:5:6 ratio) appears naturally in the series, explaining its consonant sound.
Audio Engineering
Understanding harmonics is crucial for EQ, distortion effects, and synthesizer programming. Adding or removing specific harmonics shapes the sound.