Calculator Inputs
Example Data Table
| Fundamental Hz | Harmonic | Frequency Hz | Ratio | Common Meaning |
|---|---|---|---|---|
| 110 | 1 | 110 | 1:1 | Fundamental |
| 110 | 2 | 220 | 2:1 | Octave |
| 110 | 3 | 330 | 3:1 | Octave plus fifth |
| 110 | 4 | 440 | 4:1 | Two octaves |
Formula Used
Direct fundamental: f0 equals the entered base frequency.
Open string or open pipe: f0 = v / 2L. Here v is wave speed, and L is resonator length.
Closed pipe: f0 = v / 4L. Closed pipes normally emphasize odd harmonics.
Harmonic frequency: fn = n × f0. The harmonic number n is a positive whole number.
Wavelength: wavelength = v / fn. This estimates the length of one wave cycle.
Cents: cents = 1200 × log2(fn / f0). This shows pitch distance from the fundamental.
Relative strength: strength = 100 / n^p. The p value is the entered rolloff power.
How to Use This Calculator
- Choose direct frequency when you already know the fundamental pitch.
- Choose a length method when the base frequency should come from wave speed and length.
- Enter wave speed in meters per second.
- Enter the number of partials you want to list.
- Select all, odd, or even harmonic patterns.
- Adjust A4 reference when comparing musical notes.
- Press the calculate button to show results above the form.
- Use CSV or PDF buttons to save the result table.
Understanding the Overtone Series
An overtone series shows how one base vibration creates higher partials. The first harmonic is the fundamental. The first overtone is the second harmonic. Each higher harmonic is a whole number multiple of the base frequency. This simple rule links music, acoustics, and wave mathematics.
Why Harmonics Matter
Harmonics explain tone color. A flute, guitar, and bell can share one main pitch, yet each sounds different. Their partial strengths are not the same. The calculator focuses on positions, not loudness. It lists the frequency, wavelength, ratio, cents, and nearest equal tempered note for each selected partial.
Useful Inputs
Start with a measured or chosen fundamental frequency. Then choose the number of partials to display. Add wave speed when wavelength matters. Air normally uses about 343 meters per second near room temperature. A string or structural member may use another speed. The boundary mode controls which partials appear. Open pipes and strings use every harmonic. Closed pipes use odd harmonics only.
Mathematical Value
The series is useful because it is predictable. If the fundamental is 110 Hz, the second harmonic is 220 Hz. The third is 330 Hz. The fourth is 440 Hz. These values produce clear frequency ratios. They also show why octaves, fifths, and major thirds appear early in many musical systems.
Reading Results
Frequency tells the vibration rate. Wavelength tells the physical wave length for the chosen speed. Ratio shows the harmonic relation to the base pitch. Cents show distance from the fundamental in logarithmic musical units. The nearest note field helps compare pure harmonics with equal temperament.
Practical Uses
Musicians can study tuning and resonance. Teachers can prepare harmonic examples. Builders can compare resonant behavior in pipes, strings, and simple wave paths. Students can check physics assignments with transparent formulas. Use results as estimates. Real systems also depend on material, tension, stiffness, damping, temperature, and measurement limits.
Better Decisions
Change one input at a time. Compare tables after each change. Export results when documenting a lesson, design note, or lab record. The table and downloads make the calculation repeatable. That helps you explain assumptions, review patterns, and share clear numerical evidence. It supports cleaner comparison between theoretical and measured harmonic values during reviews.
FAQs
What is an overtone series?
It is a list of higher partials created above a fundamental vibration. The first overtone is the second harmonic, not the fundamental itself.
What is the difference between harmonic and overtone?
The first harmonic is the fundamental. The first overtone is the second harmonic. So overtone number is usually one less than harmonic number.
How is each frequency calculated?
The calculator multiplies the fundamental frequency by the harmonic number. For example, harmonic 5 of 100 Hz equals 500 Hz.
Why does wavelength need wave speed?
Wavelength depends on speed and frequency. The calculator divides wave speed by partial frequency to estimate each wavelength.
What does the cents value mean?
Cents measure pitch distance on a logarithmic scale. There are 1200 cents in one octave.
Why are closed pipes different?
A simple closed pipe emphasizes odd harmonics. That is why the odd harmonic pattern is useful for closed pipe comparisons.
What does relative strength show?
It is a simple estimated rolloff value. It does not replace measured amplitude, but it helps compare partial levels quickly.
Can I export the calculated table?
Yes. After calculation, use the CSV button for spreadsheet use or the PDF button for printable records.