# Harmonic Series Calculator

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD and Rijk de Wet
Last updated: May 24, 2023

Our harmonic series calculator will help you find the harmony in your music: keep learning to discover the underlying elegance of music and much more!

The harmonic series may well be the most important concept in music: the timbre of a musical instrument (which is the sound we associate with the music itself) is intimately connected to the harmonic series.

What will you learn here?

• What is the harmonic series in music?
• Why is it so important?
• How do we create a complex sound using the harmonic series frequencies?
• What are the differences between the just intonation and the equal temperament tunings?
• How to calculate the harmonic series?

And much more: get in tune with music at Omni Calculator! And if your curiosity is not satisfied, check out the other tools the music nerds created, like the music scale calculator or the music interval calculator.

## What is the harmonic series?

In music, a harmonic series is a set of frequencies corresponding to integer multiples of a fundamental frequency or note.

A tone is a note on a musical instrument. If you listen closely, you can identify multiple pitches — the psychoacoustic concept corresponding to the frequency. In pitched instruments (like pianos and guitars), one of the frequencies will be more prominent: that's our fundamental frequency.

🙋 In this text, we will use the word "frequency" to identify the corresponding sine wave with the equation $\sin{(2\pi\cdot t \cdot f)}$, where $t$ is time.

Each frequency that appears in a tone is called partial. We can identify two types of partials:

• Harmonic partials: frequencies that match the mathematical harmonicity;
• Inharmonic partials: frequencies that deviate from the ideal harmonic, creating a certain dissonance.

The distance from an inharmonic partial to the nearest ideal harmonic is measured in cents.

🔎 A cent is a division of the interval between two notes that are a semitone apart (like $\text{E}4$ and $\text{F}4$). Each semitone interval contains exactly $100$ cents, for a total amount of $1200$ cents per octave. Remember that the value of a cent in hertz varies.

Now that you know what the harmonic series is, let's keep on learning more about it and its implications!

Are you interested in calculating the sum of the first n terms in a harmonic series? Use our harmonic series sum calculator.

You may hear the word "overtone" in the same context as "partial" from time to time. When talking of harmonic series, overtones are the partial frequencies above the fundamental frequency.

If we consider the fundamental frequency to be the first partial frequency, then we have a discrepancy between the numbering of the partials and the overtones (the first overtone corresponding to the second partial, and so on).

The usage of the term "overtone" may be helpful in certain situations: a "pure" frequency (a single sine wave) is entirely devoid of overtones while it still maintains a partial.

🙋 Don't mistake the overtones of musical theory with overtone singing! The singing style of barbershop choruses or Tuvan throat singers uses a psychoacoustic phenomenon called combination tones, which is only loosely related to the harmonic series.

## A few words on tunings: just intonation vs. equal temperament

The definition of the frequency and pitch of each note in a musical system requires the study of the intervals and relationships between the elements of a scale.

Western music usually uses the equal temperament tuning system. In this system, the octaves are divided into equal intervals. This implies that the ratio of frequencies of two adjacent notes (with a semitone between them) is always equal to the twelfth root of two: $\sqrt{2}$.

This tuning sounds most pleasing to our ears. However, it is not the only one possible: let's talk about just intervals.

If we decide to define the interval between each note and some reference note (usually middle C) as integer numbers ratios, what we obtain is the just intonation tuning (in contrast to the equal temperament system, where we fix the ratio between any two adjacent notes). The harmonic series is based on multiple integers of a fundamental frequency: it naturally produces just interval scales.

A just intonation scale would sound slightly off to us. That's only a matter of habits, though — it is as good as (if not better than) Western music's "traditional" tuning. You can listen to some examples of comparison between just intonation, equal temperament, and other tuning systems online. We found a version of , and the (in a spacesuit!). You be the judge!

## Where do we calculate the harmonic series frequencies?

Calculating the harmonic series is straightforward: take your fundamental frequency $f$, and multiply it by subsequent integer numbers. By the way, if you don't know the frequency of your fundamental, don't worry: we made a calculator exactly for that reason: the note frequency calculator!

Back to harmonies, now. Here is the series for $f$:

$f,\ 2f,\ 3f,\ 4f,\ 5f,...$

It goes on this way indefinitely; however, due to our perception of sound, the higher the multiple, the more similar two harmonics will sound.

🔎 The harmonic series in music is intimately connected to the harmonic series in math. In the latter, the elements are the inverse of subsequent integer numbers, eventually multiplied by a constant factor.

Let's consider an example. Take as a fundamental the note $\text{C}4$:

$\text{C}4 = 261.6256\ \text{Hz}$

And calculate the first $16$ partials:

Partial

Freq. (Hz)

Note

Cents

1

261.6256

C4

0

2

523.251

C5

0

3

784.877

G5

2

4

1046.5

C6

0

5

1308.13

E6

-14

6

1569.75

G6

2

7

1831.38

A#6

-31

8

2092.3

C7

0

9

2354.63

D7

4

10

2616.26

E7

-14

11

2877.88

F#7

-49

12

3139.44

G7

2

13

3401.06

A♭7

41

14

3662.68

B♭7

-31

15

3924.30

B7

-12

16

4185.92

C8

0

As you can see, the distance between two pairs of adjacent notes is equal to the frequency of the fundamental.

The table above contains a lot of information: let's dig through it!

The harmonic series contains all of the higher octaves of the fundamental note. You can see that both $\text{C}5$,$\text{C}6$, $\text{C}7$, and $\text{C}8$ appears. The first interval of this harmonic series (between $\text{C}5$ and $\text{C}4$) is $2:1$. The interval is an octave. Any two notes separated by the same $2:1$ ratio are separated by an octave — in our series, we can identify some of them:

• $\text{C}6$ and $\text{C}5$, with $4:2$;
• $\text{C}7$ and $\text{C}6$, with $8:4$; and
• $\text{E}7$ and $\text{E}6$, with $10:5$.

An octave is a "boring" interval 😛. That's because it corresponds to the simplest ratio, $2:1$; this is also the reason it sounds so good to our ears (the simpler the ratio, the better the harmony). Math and music really share some connections! Another interesting observation is the interval between the third and second note of our harmonic scale, $\text{G}5$ and $\text{C}5$. Their frequencies have a ratio of $3:2$, and define what musicians call a perfect fifth.

Being defined by a relatively simple ratio, the perfect fifth is a nice sounding harmony — so good that Pythagoras decided to create an entire tuning system uniquely based on $3/2$. Spoiler: it's not that perfect.

The next step is to consider the fourth and third notes on the harmonic series: they have a ratio of $4/3$, and define a perfect fourth. And here we fall into the first conflict between the tuning systems we will explore.

#### Pitch wars!

In the just intonation system, a fourth corresponds to an interval (measured in cents) of:

$1200\cdot \text{log}_2\left(\frac{4}{3}\right) = 498.05\ \text{cents}$

Meh. Admittedly, we can hardly say it sounds off. However, this interval doesn't fit in the equal temperament scale. Since it spans five semitones (hey, that's why it's called a fourth... no wait), the interval should be $500$ cents, which corresponds to a ratio of:

\begin{align*} 500\ \text{cents} &: 498.05\ \text{cents} \\ %3\cdot 2^{500/1200} &:4\\ 1.0039&:1 \\ \end{align*}

The difference is barely noticeable, but it's there.

Progressing with the intervals, we meet a major third. Since it encompasses the fourth and fifth harmonics, its ratio is $5:4$. Although it's counterintuitive, the major third covers fourth semitones, an interval that should correspond to $400$ cents — we are sure you've guessed it by now! However, when you consider the harmonic scale, the interval becomes $1200\cdot \text{log}_2\left(5/4\right)=386.31\ \text{cents}$. This finally makes a difference you can appreciate.

The harmonic series proceeds with other intervals. The table above shows the differences between the pitches identified by the harmonic series (corresponding to the just intonation scale) and the equal temperament scale. You can clearly see that some intervals are particularly problematic: the 7th, the 11th, and the 14th are where the dissonance is higher.

## How to use our harmonic series calculator

Simply select the pitch and the octave of the fundamental note, then choose the number of notes you want to generate. Our harmonic series calculator will print a table of the values of the frequencies, alongside the name of the relative note, its octave, and the cents of difference from the same note in the equal temperament tuning system.

What can you do with it? Experiment with this different tuning system, or tune your instrument (if you can: unless you have a fretless guitar, it is impossible to use a 12TET fretboard with any different tuning). You can also use the values of the frequencies to synthesize a complex sound. There are programs you can use for this — we can give you the math, you do the music!

🙋 Harmonies are at the basis of chords: find out more at our dedicated tools, the chord calculator, and the chord progression calculator!

## FAQ

### What are the first harmonics of A440?

The first harmonics of the note A4, with the frequency of 440 Hz, are 880 Hz, 1320 Hz, and 1760 Hz. These numbers correspond to the first four integer multiples of the fundamental frequency. Among them, you can identify the notes:

• A4, A5, and A6 (respectively, the first, second, and fourth multiples); and
• E5, which together with A5 creates a perfect fifth.

### How to calculate the harmonic series from a fundamental note?

To calculate the partials of a harmonic series, multiply the frequency f of a fundamental note by consecutive integers:

• The first partial is 1 × f;
• The second partial is 2 × f;
• The third partial is 3 × f;

And so on. Doing so, you will obtain a scale of notes related by simple ratios of integers numbers: the first two harmonics are in ratio 1:2, the second and third 3:2, etc.

### What are the differences between just intonation and equal temperament?

Just intonation and equal temperament are two tuning systems that differ in the way they define the 12 notes composing each octave:

• Just intonation defines each note as an integer multiple of a fundamental, thus relating each pair of notes to a simple ratio of integers;
• Equal temperament: each note's frequency is equal to the frequency of the previous note, multiplied by a constant ratio (the twelfth root of two).

### What are the harmonics in a complex sound?

Each complex sound is made of a different composition of harmonics. Taking the frequency of a fundamental note and its multiples, the different proportions of each multiple (the partials) confer a unique timbre to the sound:

• A sound poor in higher partials will be "purer" like the one of a recorder or flute.
• A sound rich in higher partials (but skipping the more dissonant ones) will feel more layered and refined, like the violin.
Davide Borchia
Note
A
Octave
4 (1-line)
Number of partials
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