# Equation relationship between frequency and wavelength of musical notes

### Wavelength - Wikipedia

It states the mathematical relationship between the speed (v) of a wave and its wavelength (λ) and frequency (f). Using the symbols v, λ, and f, the equation can . Table of Musical Frequencies. Note C C# D D# E F F# G G# A A# B Where did this formula come from? Here is a graph of that relationship: interval between A and the E above, say), we need to play a note which has 1. The wavelength and the speed of the wave determine the pitch, or frequency of the and speed are related by the equation speed = frequency * wavelength. Note that the violin's jagged waveform produces a sharper sound, while the.

The graphic below depicts the standing wave patterns for the lowest three harmonics or frequencies of a guitar string. Problem-Solving Scheme The wavelength of the standing wave for any given harmonic is related to the length of the string and vice versa.

## Fundamental Frequency and Harmonics

If the length of a guitar string is known, the wavelength associated with each of the harmonic frequencies can be found. And conversely, calculations can be performed to predict the natural frequencies produced by a known length of string.

Each of these calculations requires knowledge of the speed of a wave in a string. The graphic below depicts the relationships between the key variables in such calculations.

## The Wave Equation

These relationships will be used to assist in the solution to problems involving standing waves in musical instruments. To demonstrate the use of the above problem-solving scheme, consider the following problem and its detailed solution. Determine the fundamental frequency 1st harmonic of the string if its length is The solution to the problem begins by first identifying known information, listing the desired quantity, and constructing a diagram of the situation.

The problem statement asks us to determine the frequency f value. The speed is given, but wavelength is not known.

If the wavelength could be found, then the frequency could be easily calculated. In this problem and any problemknowledge of the length and the harmonic number allows one to determine the wavelength of the wave. For the first harmonic, the wavelength is twice the length. This relationship is derived from the diagram of the standing wave pattern and was explained in detail in Lesson 4. This would be the harmonic with the longest wavelength and the lowest frequency.

The lowest frequency produced by any particular instrument is known as the fundamental frequency. The fundamental frequency is also called the first harmonic of the instrument. The diagram at the right shows the first harmonic of a guitar string.

If you analyze the wave pattern in the guitar string for this harmonic, you will notice that there is not quite one complete wave within the pattern. A complete wave starts at the rest position, rises to a crest, returns to rest, drops to a trough, and finally returns to the rest position before starting its next cycle.

A standing wave pattern is not actually a wave, but rather a pattern of a wave. Thus, it does not consist of crests and troughs, but rather nodes and antinodes. The pattern is the result of the interference of two waves to produce these nodes and antinodes. In this pattern, there is only one-half of a wave within the length of the string.

### Fundamental Frequency and Harmonics

This is the case for the first harmonic or fundamental frequency of a guitar string. The diagram below depicts this length-wavelength relationship for the fundamental frequency of a guitar string. The second harmonic of a guitar string is produced by adding one more node between the ends of the guitar string.

And of course, if a node is added to the pattern, then an antinode must be added as well in order to maintain an alternating pattern of nodes and antinodes.

In order to create a regular and repeating pattern, that node must be located midway between the ends of the guitar string. This additional node gives the second harmonic a total of three nodes and two antinodes. The standing wave pattern for the second harmonic is shown at the right.

### What are the frequencies of music notes?

A careful investigation of the pattern reveals that there is exactly one full wave within the length of the guitar string. For this reason, the length of the string is equal to the length of the wave. The third harmonic of a guitar string is produced by adding two nodes between the ends of the guitar string.

**Sound Harmonics and Frequency**

And of course, if two nodes are added to the pattern, then two antinodes must be added as well in order to maintain an alternating pattern of nodes and antinodes. In order to create a regular and repeating pattern for this harmonic, the two additional nodes must be evenly spaced between the ends of the guitar string.