# What are overtones in music

## The overtone series

### definition

The overtone series is the chord of partials that vibrate simultaneously when a natural note is sounded.

The overtone series is the basis of all tone systems because it is the only natural scale. As soon as a note sounds, overtones resonate with it. They all sound at the same time. So the overtone series is actually a chord. The structure is always the same and corresponds to a mathematical harmonic series, hence the name series. You don't usually hear the overtones. Because they all vibrate as a chord at the same time, they appear to us like a single note.

The term overtone series means the harmonic partials (for the difference between overtones, partial tones and harmonics, see below). There are also sounds with inharmonic overtones. The more inharmonic overtones a sound contains, the more noisy it becomes.

Overtone series of A (110 Hz).

All sounds consist of overtone chords. Only sine tones have no overtones. One sound differs from the other mainly through the volume of the individual overtones (in addition, through noise components and temporal changes in the sound). The overtone series is not only the basis for music, it allows us to speak and sing, recognize people by their voices, locate sounds and distinguish a piano from a flute.

This scale does not come from humans, but arises directly from the laws of vibration. It follows a universal wave principle of the universe and makes it audible and tangible for us. The tones deviate from our usual equally tempered tone system. Nevertheless, the equally tempered system, like all other tone systems, is derived from the overtone series.

The reason is that we internally assemble tone sequences into small chords and then unconsciously compare them with the overtone series. Because we love matches with the overtone series, we invented tonal systems based on natural intervals. However, cultures have not always found the same intervals to be beautiful. Hence there are over 3000 different sound systems in the world.

Our western system with 12 semitones per octave, for example, is based on an idea from ancient Greece to take the interval between the second and third harmonic - the fifth - as a basis and then layer it twelve times on top of each other.

In the following video you can see the overtone series of *a* (220 Hz) see and hear. The video was recorded by Bodo Maass using our Overtone Analyzer software.

When singing and in instruments, all partials of the overtone series sound at the same time. Our brain combines these partial tone bundles into a single sound and assigns them to a sound source. The frequency spacing between the partials is perceived as the pitch, the volume distribution of the overtones as the timbre. Most singers are unaware that they are actually always whole** Partial chords **to sing. Our brain has an archaic knowledge of this chord. Apparently it can recognize partial chords as sounds from a sound source even before birth, e.g. B. the mother's voice.

*When singing the note c, the entire series of overtones sounds as a chord. (Overtone Analyzer screenshot).*

Such a chord of partials sounds like a single tone, but it has a timbre. While a tone with no overtones is colorless. A tone with overtones is called in physics **sound** designated. Musicians and physicists may mean something different with this word!

Different timbres are created by **different volume levels of the overtones** (in addition to noise components and transient behavior). The personal vocal sound of a person is thus created through a volume distribution of the overtones that is typical for each person. If two people sing the same note, then they only differ in the overtone volume. If you filter the overtones individually, you can no longer recognize the people. The volume distribution contains countless information: vowels, identification of the person, physical and psychological well-being, age, etc.

*Spectrum of a sung c with its overtones and the typical volume distribution for the vowel Ä. (Overtone Analyzer screenshot).*

The connection can be visualized with the help of sound spectra and spectrograms. Sound analysis programs break down the sound into its individual frequencies and display the volume levels in color. The Overtone Analyzer, developed by Bodo Maass and myself, specializes in making sound contexts easy to understand for musicians.

**The sound spectrum** (Frequency spectrum) is a way of representing the sound optically. The spectrum shows the volume distribution of the overtones in the sung vowel ä. Each peak corresponds to an overtone, the further to the right the peak, the louder the overtone. At the very bottom is the keynote. The frequency, i.e. the pitch, increases towards the top and the volume increases towards the right. (*Overtone Analyzer screenshot*)

**The spectrogram** (also called a sonagram) is another form of representation for sounds. The volume is shown here in colors, in the example the redder, the louder. Each cross line corresponds to an overtone. At the very bottom is the keynote. The frequency increases upwards, i.e. the pitch, from left to right the time runs. (*Overtone Analyzer screenshot*)

### Overtone series - intervals

**The interval sequence of the overtone series is always the same.** The intervals only depend on the position in the row. For example, the interval from the 2nd to the 3rd partial is always a fifth. Regardless of which note you start with, the same melody always results from the respective root note. The intervals become narrower and narrower towards the top (while the frequency spacing remains the same, see below). All neighboring intervals are unique and only appear once in the series. Each interval, after it has occurred once, is repeated in the octaves above with new intermediate tones.

The overtone slider (picture) shows the intervals from partial to partial and from partial to fundamental. It represents, so to speak, the piano keys for overtone singing. On a sington you can only sing intervals that appear in the slider; usually only a part of it (cf. ambitus of overtone singing). The interval to the root determines the harmonic relationship. For example, the 5th partial is perceived as a major third. If you want to sing a fourth jump in an overtone melody, you can find it between the 3rd and 4th partial and choose the fundamental accordingly.

*Intervals of the overtone series - the overtone slider.*

### Overtone series - frequency relationships

**The frequencies of the partials are integer multiples of the fundamental frequency**. This mathematical relationship is called a "harmonic series".

That means, **the frequency spacing between two partials is always identical to the fundamental frequency**z.

The *Frequencies* can therefore be calculated very easily and displayed clearly. The calculation of the *Intervals* from the frequencies. While the frequency of any partial tone is quickly calculated in your head, it is better to learn the intervals by heart (unless you can calculate logarithms in your head ...).

**Frequencies of the partials of A. (110 Hz) calculate:**

1st partial tone - 1-fold frequency = 110 Hz

2nd partial tone - 2 times the frequency = 220 Hz,

3rd partial tone - 3 times the frequency = 330 Hz,

…

11th partial tone - 11 times the frequency = 1210 Hz etc.

All partials of *A.* so have the same frequency spacing of 110 Hz.

The distance between the partials of the fundamental c (130.8 Hz) would then - you guessed it - 130.8 Hz. So always identical to the fundamental frequency.

**Frequencies of the partials of c (130.8 Hz) calculate:**

1st partial tone - 1-fold frequency = 130.8 Hz

2nd partial tone - 2 times the frequency = 261.6 Hz,

3rd partial tone - 3 times the frequency = 392.4 Hz,

…

11th partial tone - 11 times the frequency = 1,439 Hz etc.

All partials of *c* so have the same frequency spacing of 130.8 Hz.

You can display the frequencies of the overtone series linearly or logarithmically. Musicians prefer the logarithmic representation because the intervals look like we hear them. Physicists often represent the frequencies linearly. The following graphs illustrate the difference.

**1. Logarithmic frequency representation for musicians**: In the musical representation they are *Intervals* important. They are therefore presented as we hear them. Musicians rarely need frequency information, mainly for tuning.**Advantages of this representation**: Musicians understand them intuitively, the intervals correspond to our hearing. (*Overtone Analyzer screenshot*)

**2. Linear frequency representation for physicists**: In the physical representation, the *Frequencies *so shown that frequency*distances* are immediately recognizable.**Advantage of linear representation**: It's easier to count on. You can see at a glance that all partials have the same frequency spacing, namely that of the fundamental frequency. (*Overtone Analyzer screenshot*)

### Convert frequencies into tones

Our ear hears intervals as frequency ratios. A multiplication with the same number is heard as the same interval.

**example**: 100 and 200 Hz have a frequency*distance* of 100 Hz. 200 Hz is twice the frequency of 100 Hz, the frequency*relationship* is 2: 1. A doubling (or halving) of a frequency is heard as an octave. Between 200 and 300 Hz there is again a distance of 100 Hz. But 300 Hz is 3/2 times 200 Hz. These 100 Hz intervals are heard as a fifth. So we don't hear the same distances, but rather the same relationships as the same interval. 2: 1 and 1: 2 are octaves. 2: 3 and 3: 2 correspond to the fifth. 3: 4, 4: 3 are major thirds, etc. You can find the corresponding intervals for the ratios in the section Overtone Series - Intervals.

The logarithm is used to convert distance into ratios. Here is the formula for converting frequency spacing into intervals.

a = first frequency

b = second frequency

The result of this formula are cent values, that is 1/100 semitones. 100 cents is exactly one (equally tempered) semitone. 1200 cents are 12 semitones, or one octave. 700 cents are 7 semitones, i.e. a fifth.

If values deviate from smooth 100s, e.g. B. 702 c, then this means that a note deviates by this cent amount from the tuning of the same level as usual on the piano. 702 c, the natural (pure) fifth from the overtone series, is 2 cents, 2 hundredths of a semitone larger than on the piano. Except for the octaves, the tones of the overtone series all deviate from the equal pitch.

Overtones are numbered in two different ways. This is because some count the root note, others don't. Actually, overtones are just the tones above the fundamental. Since the numbering from the root has advantages, it is better to speak of partials, since the root is also viewed as the partial of the sound.

I prefer that** Partial numbering** (or the harmonics, see below).

- Partial tone numbers have the advantage that the frequency ratio results directly from the number. The 14th partial has 14 times the frequency of the fundamental, 23 partial, etc.
- The frequency ratios of the intervals also result directly from the partial tone numbers. Example: The 3rd partial is the fifth in the 2nd octave. It has 3 times the frequency of the fundamental. The 2nd partial is the octave to the fundamental. The two adjacent partials 2 and 3 have a frequency ratio of 2: 3 (viewed upwards) or 3: 2 (viewed downwards) and sound as a fifth. Their doubling is again fifths. So the partial tone pairs 4/6, 8/12 etc. also form fifths with the same frequency ratio (since the fractions can be shortened to 2/3).

In the figure you can see that the octaves are always even numbers in the partial tone numbering, while they are odd in the overtone numbering. This can be important for musicians: The clarinet can e.g. B. by overblowing only the *odd partials* generate, so no octaves. This is sometimes wrong even in textbooks. It is of particular importance for overtone singers because their music is notated with the partial numbers.

### Natural tone series vs. overtone series

The natural tone series has the **same tonal structure as the overtone series**but it is not the same. While the partials of the overtone series are pure sine tones, the tones of the natural tone series each consist of partials and have their own overtone series.

Natural tone series | Overtone series |
---|---|

Tone sequence that can be generated on wind instruments (pipes) by overblowing or changing the lip frequency. | Partials of an acoustic sound. |

Natural tones are real tones and each have their own overtones. | Partials (overtones) are sine tones and have no overtones themselves. |

### Examples: overtone rows on F and E.

With the Overtone Analyzer software you can immediately display overtone series of any tone, including the tone name, frequency and cent deviation from the tempered system, and you can also listen to them right away. Download the trial version here for free.

You can download the overtone series (1-24) below as a MuseScore file (→ MuseScore free music notation program) and MusicXML file and transpose and listen to them as required. The cent deviations always remain the same for every position of the harmonic in the series, regardless of the fundamental. The notes are already tuned accordingly.

Sometimes numbers are useful.

Partial no. | Overtone no. | Interval to the keynote | Cent to the keynote | Interval to the partial below | Cent to partial below |
---|---|---|---|---|---|

18 | 17 | 4 octaves + large second + 4ct | 5004 | kl. Second -1ct | 99 |

17 | 16 | 4 octaves + kl. Second + 5ct | 4905 | kl. Second + 5ct | 105 |

16 | 15 | 4 octaves | 4800 | kl. Second + 12ct | 112 |

15 | 14 | 3 octaves + major seventh -12ct | 4688 | kl. Second + 19ct | 119 |

14 | 13 | 3 octaves + kl. 7th -31ct | 4569 | kl. Second + 28ct | 128 |

13 | 12 | 3 octaves + kl. 6th + 41ct | 4441 | kl. Second + 39ct | 139 |

12 | 11 | 3 octaves + fifth + 2ct | 4302 | 3/4 tone | 151 |

11 | 10 | 3 octaves + above m. Fourth -49ct | 4151 | Size Second + 35ct | 165 |

10 | 9 | 3 octaves + major third -14ct | 3986 | Size Second (Kl. Whole tone) + 18ct | 182 |

9 | 8 | 3 octaves + large second + 4ct | 3804 | Size Second (large whole tone) + 4ct | 204 |

8 | 7 | 3 octaves | 3600 | Size Second + 31ct | 231 |

7 | 6 | 2 octaves + kl. 7th -31ct | 3369 | 5/4 tone | 267 |

6 | 5 | 2 octaves + fifth + 2ct | 3102 | Kl. Third + 16ct | 316 |

5 | 4 | 2 octaves + major third -14ct | 2786 | Size Third -14ct | 386 |

4 | 3 | 2 octaves | 2400 | Quart -2ct | 498 |

3 | 2 | Octave + fifth + 2ct | 1902 | Fifth + 2ct | 702 |

2 | 1 | octave | 1200 | octave | 1200 |

1 | Keynote | Prime | 0 | Prime | 0 |

*Column: Numbering of the partials including the root. This numbering is more useful.**Column: Numbering of the overtones, the basic tone is not counted.**Column: interval to the root with**Cent deviation to the nearest equally tempered tone.*.*Column: Interval to the root in cents (100th semitone).**Column: Interval between the partials (always to the one below) with**Cent deviation from the equally tempered interval.**Column: Interval between the partials in cents (100ths of a semitone).*

With every oscillation, faster oscillations occur above the basic frequency, which overlap. This is a universal behavior of nature, whether it is sound or some other vibration.

Strings vibrate harmoniously. This means that in addition to the fundamental vibration, the string also vibrates in whole-numbered sections, i.e. over half the length, 1/3, 1/4, 1/5 etc. of the string length. These vibrations all occur at the same time and are superimposed to form the overall vibration. The partial vibrations look isolated as in the following picture.

### Videos: Creation of a wave

Propagation of a wave in a string.

Standing waves.

The wave in a string is created by the migration of impulses.

### glossary

Overtones, partials, harmonics ...

*Images: Spectra (top) and spectrograms from left: 1. Sinus tone, 2. Sound (synthetic: sawtooth tone), 3. Sound (voice), 4. Inharmonic sound (singing bowl), 5. Noise (white noise). (Overtone Analyzer screenshot)*

**Sine tones** (Fig. 1. from the left) have no overtones, so they are vibrations with only one frequency. The perceived pitch does not always correspond to that of a natural tone played on the same fundamental frequency. High sine tones often seem too deep to us. Sine tones are a mathematical construct. There are no real, completely overtone-free tones, there is always a sound or noise component.

**sound** (Fig. 2 and 3 from the left) is a tone with overtones in acoustics. In music, the term is usually used differently and can e.g. B. mean the timbre. Every real natural tone is a sound. There are only approximate sine tones in nature.

**Sounds** (Fig. 5. from left) are sound events that have such dense overtones or overtones that constantly change in frequency that we can no longer perceive a pitch. But there are flowing transitions to sounds. Depending on the characteristics, one speaks of noises with a tone character to sounds with a noise component.

**Partials** (synonymous: partial tones) are all (sine) tones that make up a sound, including the fundamental. They are counted from the root (the one with the lowest frequency).There can be harmonic and inharmonic partials, both mixed. Harmonic partials vibrate with integer multiples of the fundamental frequency, inharmonic partials with non-integer ones.

**Harmonics.** Short for "harmonic partials". Partials with integer multiples of the fundamental frequency are also called *Harmonics* designated. In the case of harmonics, the intervals always correspond to the natural overtone series. Harmonics are always partials as well. Most melody instruments and the human voice have harmonic overtones.

**Overtones** are all partials *above* of the keynote. The numbering begins above the root, i.e. at the 2nd partial. Therefore the numbering of the overtones is always 1 lower than that of the partials. There can be harmonic and inharmonic overtones, also mixed both.

**Inharmonic partials / overtones** (Fig. 4. from left): Drums, bells, gongs, singing bowls or xylophones are examples of instruments with inharmonic partials / overtones. This means that the frequencies of the overtones are not integral multiples of the lowest frequency. There are also sounds that contain both harmonic and inharmonic overtones. In the case of inharmonic sounds, one cannot often speak of the fundamental tone because the pitch heard is sometimes not that of the lowest partial tone, e.g. B. the sound of a bell.

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