The Demystification of the Harmonic Series

by Francis Koerber (of

Demonstration A : Understanding the Mathematics of the Harmonic Series



3 (+2)


5 (-14)

6 (+2)

7 (-31)


9 (+4)

10 (-14)

11 (-49)

12 (+2)

13 (+41)

14 (-31)

15 (-12)


17 (+5)

18 (+4)

19 (-2)

20 (-14)

In order to fully comprehend the intervalic relationships between notes in the harmonic series, a number of demonstrations have been created using audio files of each harmonic and of combinations thereof.

It is helpful to understand the mathematics of the harmonic series before using these demonstrations. In particular, the variation of cents of a particular partial in relation to the tempered scale.

In the table to the left, the numbers listed represent first the number of the harmonic, and then in parenthesis, the number of cents that it deviates from the tempered scale.

The first demonstration allows the user to play any series of harmonics together in a chord. It is especially interesting to play combinations of harmonics that bring definition to many of the scales in use today.

To play chords, select the play button next to each harmonic to hear the blend of harmonics.


The entire harmonic series demonstrated in glissando



Demonstration B : Naturally Occurring Pentatonic Scale within the Harmonic Series

Harmonics (8, 9, 10, 12, 13)

The primary series of notes for one of the simplest scales utilized throughout history is the pentatonic scale. It is argueably the foundational scale that is utilized in both Eastern and Western civilizations, and could be considered the basis for all other scales that are derivations of the pentatonic system.

This scale occurs because the 13th harmonic (representing the major 6th from the tonic) is approximately half way between that of the tempered scale minor 6th and its respective major. In a tempered tuning, the intervals are all stretched a little in order to create equal intervals of a minor 2nd throughout the octave. Therefore, the 8-13 harmonic combination actually represents the pure interval of a major 6th with respect to the harmonic series.


Demonstration C : Naturally Occurring Chords within the Harmonic Series

Harmonics (4, 5, 6, )

Major triad

Harmonics (4, 5, 6, 7)

Dominant 7th

Harmonics (5, 6, 7)

Diminished Triad

Harmonics (10, 12, 15)

Minor triad

Harmonics (7, 9, 11)

Augmented Triad

Harmonics (10, 12, 15,18)

Minor 7th


Demonstration D : Harmonic Additive Progression

Harmonics (1-20)

Harmonics in Glissando
Composition by Francis Koerber demonstrating use of randomly selected Harmonics


Copyright © 2011 Francis Koerber All rights reserved.