The harmonic series is a fundamental concept in physics and music. It refers to a sequence of frequencies that are integer multiples of a base frequency, called the fundamental frequency.
Let's look at the first few harmonics. The first harmonic is the fundamental frequency. The second harmonic has twice the frequency, completing two cycles in the same time. The third harmonic has three times the frequency, and so on.
The mathematical relationship is simple: the frequency of the nth harmonic equals n times the fundamental frequency. This relationship is crucial in music, as it determines the timbre or unique sound quality of instruments.
In music, the harmonic series plays a crucial role in determining the timbre or tone color of different instruments. Each instrument produces a unique blend of harmonics that gives it its characteristic sound.
Let's look at how different instruments emphasize different harmonics. A flute has a strong fundamental with fewer upper harmonics, giving it a pure, clear sound.
A violin has more prominent upper harmonics, creating a richer, more complex sound.
A trumpet has particularly strong middle harmonics, especially the third harmonic, giving it its bright, brassy sound.
These different harmonic profiles are what allow us to distinguish between instruments even when they're playing the same note at the same volume. The harmonic series is essentially the DNA of musical sound.
The harmonic series naturally creates the musical intervals that form the foundation of Western music. Let's explore how these harmonics relate to the notes on a piano keyboard.
The first harmonic is our fundamental note, which we'll call C. This is our reference pitch.
The second harmonic is exactly twice the frequency of the fundamental, creating an octave. This is the most basic and consonant musical interval, with a frequency ratio of 2 to 1.
The third harmonic creates a perfect fifth above the second harmonic, or a twelfth above the fundamental. This interval has a frequency ratio of 3 to 2 and is the next most consonant interval after the octave.
The fourth harmonic is two octaves above the fundamental, with a frequency ratio of 4 to 1 relative to the fundamental. Compared to the third harmonic, it creates a perfect fourth interval with a ratio of 4 to 3.
The fifth harmonic gives us a major third above the fourth harmonic, with a frequency ratio of 5 to 4. These natural intervals from the harmonic series form the basis of our musical scales and harmony.
The harmonic series is fundamental to sound synthesis. Different waveforms can be created by combining harmonics in specific ways. Let's start with a pure sine wave, which consists of only the fundamental frequency with no additional harmonics.
A square wave can be created by adding together odd-numbered harmonics. With just the fundamental, it looks like a sine wave. As we add more odd harmonics, it begins to approximate a square wave.
A sawtooth wave includes all harmonics, both odd and even. As we add more harmonics, we get closer to the characteristic sawtooth shape.
A triangle wave, like the square wave, uses only odd harmonics, but their amplitudes decrease much more rapidly. This creates the smoother triangular shape.
These different waveforms are the building blocks of sound synthesis. By controlling which harmonics are present and their relative strengths, synthesizers can create an infinite variety of timbres, from the mellow sounds of a flute to the bright, buzzy sounds of a synthesizer.
Now let's explore the mathematical harmonic series, which shares its name with the musical concept but represents something different. The mathematical harmonic series is the sum of reciprocals: 1 plus one-half plus one-third plus one-fourth, and so on.
Let's see what happens as we add more terms to this series. With just the first term, we have 1.
Adding the second term, one-half, gives us one and a half.
With the third term, one-third, our sum grows to about 1.83.
As we continue adding terms, the sum keeps growing, but at a slower and slower rate. With 10 terms, we reach about 2.93.
Unlike the musical harmonic series, which creates a finite set of frequencies, the mathematical harmonic series diverges—it grows without bound, albeit very slowly. This was proven by Nicole Oresme in the 14th century and is a famous result in mathematics.
Now let's explore the mathematical harmonic series, which shares its name with the musical concept but represents something different. The mathematical harmonic series is the sum of reciprocals: 1 plus one-half plus one-third plus one-fourth, and so on.
Let's see what happens as we add more terms to this series. With just the first term, we have 1.
Adding the second term, one-half, gives us one and a half.
With the third term, one-third, our sum grows to about 1.83.
As we continue adding terms, the sum keeps growing, but at a slower and slower rate. With 10 terms, we reach about 2.93.
Unlike the musical harmonic series, which creates a finite set of frequencies, the mathematical harmonic series diverges—it grows without bound, albeit very slowly. This was proven by Nicole Oresme in the 14th century and is a famous result in mathematics.
Now let's explore the mathematical harmonic series, which shares its name with the musical concept but represents something different. The mathematical harmonic series is the sum of reciprocals: 1 plus one-half plus one-third plus one-fourth, and so on.
Let's see what happens as we add more terms to this series. With just the first term, we have 1.
Adding the second term, one-half, gives us one and a half.
With the third term, one-third, our sum grows to about 1.83.
As we continue adding terms, the sum keeps growing, but at a slower and slower rate. With 10 terms, we reach about 2.93.
Unlike the musical harmonic series, which creates a finite set of frequencies, the mathematical harmonic series diverges—it grows without bound, albeit very slowly. This was proven by Nicole Oresme in the 14th century and is a famous result in mathematics.
Now let's explore the mathematical harmonic series, which shares its name with the musical concept but represents something different. The mathematical harmonic series is the sum of reciprocals: 1 plus one-half plus one-third plus one-fourth, and so on.
Let's see what happens as we add more terms to this series. With just the first term, we have 1.
Adding the second term, one-half, gives us one and a half.
With the third term, one-third, our sum grows to about 1.83.
As we continue adding terms, the sum keeps growing, but at a slower and slower rate. With 10 terms, we reach about 2.93.
Unlike the musical harmonic series, which creates a finite set of frequencies, the mathematical harmonic series diverges—it grows without bound, albeit very slowly. This was proven by Nicole Oresme in the 14th century and is a famous result in mathematics.
Now let's explore the mathematical harmonic series, which shares its name with the musical concept but represents something different. The mathematical harmonic series is the sum of reciprocals: 1 plus one-half plus one-third plus one-fourth, and so on.
Let's see what happens as we add more terms to this series. With just the first term, we have 1.
Adding the second term, one-half, gives us one and a half.
With the third term, one-third, our sum grows to about 1.83.
As we continue adding terms, the sum keeps growing, but at a slower and slower rate. With 10 terms, we reach about 2.93.
Unlike the musical harmonic series, which creates a finite set of frequencies, the mathematical harmonic series diverges—it grows without bound, albeit very slowly. This was proven by Nicole Oresme in the 14th century and is a famous result in mathematics.
Now let's explore the mathematical harmonic series, which shares its name with the musical concept but represents something different. The mathematical harmonic series is the sum of reciprocals: 1 plus one-half plus one-third plus one-fourth, and so on.
Let's see what happens as we add more terms to this series. With just the first term, we have 1.
Adding the second term, one-half, gives us one and a half.
With the third term, one-third, our sum grows to about 1.83.
As we continue adding terms, the sum keeps growing, but at a slower and slower rate. With 10 terms, we reach about 2.93.
Unlike the musical harmonic series, which creates a finite set of frequencies, the mathematical harmonic series diverges—it grows without bound, albeit very slowly. This was proven by Nicole Oresme in the 14th century and is a famous result in mathematics.