A Fundamental Frequency Calculator is a physics-based tool that determines the lowest natural frequency at which a vibrating object, such as a guitar string or a column of air in a pipe, will oscillate. This lowest frequency, known as the fundamental or the first harmonic, is what our ears perceive as the primary pitch of the musical note. The calculator uses specific physical properties of the object—for a string, this includes its length, tension, and mass—to compute this frequency. This tool is invaluable for musical instrument designers, physicists studying acoustics, and musicians who want to understand the scientific principles that govern the sound their instruments produce.
formula of Fundamental Frequency Calculator
The formula to calculate the fundamental frequency of a vibrating string is based on the principles of wave mechanics.
1. Fundamental Frequency of a String
This formula is used for stringed instruments like guitars, pianos, and violins. It is governed by the string's length, tension, and linear mass density.
Formula:
Fundamental Frequency (f₀) = (1 / (2 * L)) * sqrt(T / μ)
- f₀: The fundamental frequency, in Hertz (Hz).
- L: The vibrating length of the string (the distance from the nut to the bridge), in meters (m).
- T: The tension on the string, in Newtons (N).
- μ (mu): The linear mass density of the string (its mass per unit length).
- sqrt: The square root function.
1.1. Sub-Formula: Linear Mass Density (μ)
This must be calculated before finding the frequency.
Linear Mass Density (μ) = Mass of String / Length of String
The units for this must be kilograms per meter ( kg/m ) for the main formula to work correctly.
2. Fundamental Frequency of an Open Pipe
This formula is used for wind instruments that are open at both ends, like a flute.
Formula:
Fundamental Frequency (f₀) = v / (2 * L)
- v: The speed of sound in air (approximately 343 m/s at room temperature).
- L: The length of the air column in the pipe, in meters (m).
3. Fundamental Frequency of a Closed Pipe
This formula is used for wind instruments that are closed at one end, like a clarinet.
Formula:
Fundamental Frequency (f₀) = v / (4 * L)
Fundamental Frequencies of Musical Notes (A4 = 440 Hz)
This table shows the standard fundamental frequencies for the notes in the octave starting from Middle C.
| Note Name | Frequency (Hz) |
| C4 (Middle C) | 261.63 Hz |
| D4 | 293.66 Hz |
| E4 | 329.63 Hz |
| F4 | 349.23 Hz |
| G4 | 392.00 Hz |
| A4 | 440.00 Hz |
| B4 | 493.88 Hz |
| C5 | 523.25 Hz |
Example of Fundamental Frequency Calculator
Let's calculate the fundamental frequency of the high E string on a standard electric guitar.
First, we gather the typical physical properties of the string.
- Vibrating Length (L): A typical guitar scale length is 25.5 inches, which is 0.648 meters.
- Tension (T): A standard light-gauge high E string is under about 16 pounds of tension, which is approximately 71 Newtons.
- Linear Mass Density (μ): We need to calculate this. A typical high E string (0.010 inch gauge) has a mass of about 0.23 grams (0.00023 kg) over its vibrating length.
μ = Mass / Length = 0.00023 kg / 0.648 m ≈ 0.000355 kg/m
Now, we apply the main formula.
f₀ = (1 / (2 * L)) * sqrt(T / μ)
f₀ = (1 / (2 * 0.648)) * sqrt(71 / 0.000355)
f₀ = (1 / 1.296) * sqrt(200,000)
f₀ = 0.7716 * 447.2
f₀ ≈ 345 Hz
Therefore, the calculated fundamental frequency is approximately 345 Hz. This is very close to the standard tuning for the high E string (E4), which is 329.63 Hz. The small difference can be attributed to slight variations in the string's mass and the exact tension.
Most Common FAQs
When a string vibrates, it doesn't just vibrate as a whole. It also vibrates in smaller segments at the same time. These higher frequency vibrations are called harmonics or overtones. They are whole-number multiples of the fundamental frequency (2f₀, 3f₀, 4f₀, etc.). These overtones are what give a musical instrument its unique sound quality, or timbre. The fundamental frequency determines the pitch, while the overtones determine the character of the sound.
Yes, temperature can have a noticeable effect, especially on stringed instruments. As the temperature rises, the metal strings will expand slightly, which can cause their tension (T) to decrease. According to the formula, a decrease in tension will lead to a decrease in the fundamental frequency, causing the instrument to go slightly flat.