Natural tone series

The natural tone row is a row of tones arranged in ascending pitch , which can be produced on wind instruments , but also on almost every pipe or hose, without shortening or lengthening the vibrating column of air, only by blowing them in different ways. In terms of their frequency relationships, the natural tone series, like the flageolet series , essentially coincides with the partial series , also known as the overtone series . However, natural tones such as the flageolet tones sound real, while partials are only spoken of as components of a musical tone (i.e. an acoustic sound ).

The five lowest natural vibrations ( standing waves ) of the air column in a conical tube. Dark: maximum, light: minimum of the pressure amplitude. Because of the conical shape, the distances between the pressure maxima in a wave are not the same and also no direct measure for the wavelength of the emitted sound

The sounds of wind instruments come from standing waves that form in the instrument. The wavelength of the first (lowest) natural note, the root or pedal note , is twice the length of the air column in most wind instruments and four times the length of the air column in certain reed instruments (clarinet). By so-called overblowing , various other natural frequencies of the air column can be excited above the fundamental tone , i.e. different natural tones can be generated. Natural tones play a particularly important role in valveless wind instruments such as the natural horn , natural trumpet or alphorn .

The phenomenon of stringed instruments that is physically related to natural tones is the flageolet tones.

Frequency relationships

The frequencies of the standing waves that can be generated in a given column of air are (approximately) integer multiples of the lowest possible frequency, the frequency of the fundamental. The following table shows an example of the first 16 tones of the natural tone series based on the key C. The colors used are based on the music-color synesthesia .

Simple model - comparison with keynote
Partial no:	1	2	3	4th	5	6th	7 *	8th	9	10	11 *	12	13 *	14 *	15th	16
Multiples of the basic frequency:	simple	double	triple	fourf.	five.	sixf.	sevenf.	eightf.	ninef.	tenf.	eleven.	twelve.	thirteenf.	fourteenf.	fifteenf.	sixteenf.
Example f in Hz:	66	132	198	264	330	396	462	528	594	660	726	792	858	924	990	1056
Grade:
Note name:	C.	c	G	c '	e '	G'	≈ b '	c ''	d ''	e ''	≈ f ''	G''	≈ as ''	≈ b ''	H''	c '' '
Relation to the tone below:	1: 1	2: 1	3: 2	4: 3	5: 4	6: 5	7: 6	8: 7	9: 8	10: 9	11:10	12:11	13:12	14:13	15:14	16:15
Interval to the tone below:	Prime	octave	perfect fifth	pure fourth	major third	minor third	-	-	large whole tone	small whole tone	-	-	-	-	-	diatonic semitone
Ratio of partial to root:	1: 1	2: 1	3: 1	4: 1	5: 1	6: 1	7: 1	8: 1	9: 1	10: 1	11: 1	12: 1	13: 1	14: 1	15: 1	16: 1
Interval above the fundamental:	Prime	octave	Duodecime	2 octaves	2 octaves + major third	2 octaves + perfect fifth	2 octaves + natural sept	3 octaves	3 octaves + major second	3 octaves + major third	3 octaves + alphorn company	3 octaves + perfect fifth	3 octaves + ≈ minor sixth	3 octaves + natural septime	3 octaves + major seventh	4 octaves

Partials / natural tones in the columns are each octave apart
1	2	4th	8th	16	...
3	6th	12	...
5	10	...
7 *	14 *	...

Distribution of partials / natural tones
First octave	1
Second octave	2	3
Third octave	4th	5	6th	7 *
Fourth octave	8th	9	10	11 *	12	13 *	14 *	15th

The notes marked with '≈' are outside the diatonic scale , while the rest of the notes correspond to the diatonic notes in pure tuning . The higher the octave reached, the closer the natural tones are and the more of them are outside the diatonic scale.

Table footnotes

↑ The frequency of 66 Hz for the fundamental tone C corresponds to the choice of 440 Hz for the pitch A ': a small third (frequency ratio ⁶ / ₅ ) via a' = 440 Hz the sound c '' with 440 × ⁶ / ₅ = 528 Hz. The C, which is three octaves lower, has the frequency 528: (2 ³ ) = 66 Hz.
↑ 7th partial tone = 462 Hz ( natural seventh ). Deviation from b '= 475.2 Hz of the pure tuning ≈ 49 cents. Note: The unit of cents is used primarily for the representation of the subtle differences in size of the intervals , with a (equal) semitone = 100 cents and an octave = 1200 cents. The calculation is made using the logarithm of two lb of the frequency ratio. Here 1200lb (475.2 / 462) ≈ 49 cents.
↑ 11th partial tone = 726 Hz ( Alphorn-Fa ). Deviation from f '' = 704 Hz or f sharp '' = 742.5 Hz of the pure tuning ≈ 53 cents or 39 cents.
↑ 13th partial tone = 858 Hz. Deviation from a flat '' = 844.8 Hz of the pure tuning ≈ 27 cents
↑ 14th partial tone = 924 Hz ( natural seventh ). Deviation from b '' = 950.4 Hz of the pure tuning ≈ 49 cents
↑ The musical interval of an octave corresponds to a doubling of the frequency.

Music making practice

The playable series of notes of a natural trumpet with a length of approx. 240 cm. Listen (MIDI)

Brass instruments

On brass instruments, the player creates the scale of natural tones by changing the lip tension and the blowing pressure. The pitch can be varied ( intoned ) by about + 50 / -50 cents .

The first natural tone, the fundamental or pedal tone, can be used by trained wind players to intonate the brass instruments with a wide length and is required in the literature especially for the bass trombone and tuba . The first natural tone is not or only rarely used on instruments with a narrow scale such as the trumpet and the French horn (see instrumentation ). The scale is open to the top. The natural horn in F, for example, blows up to the 24th natural note. Simple hunting music gets along with diatonic natural tones. With the alphorn , the natural seventh and even the alphorn fa , which sounds unusual for ears used to classical music, are sometimes played.

Diatonic and chromatic scales as well as a clean intonation in the high second range (from 7th natural tone) can only be blown by extending the length of the pipe. This is best illustrated with the slide trombone : From each natural tone, the seven slide positions each result in a further half step down. Valve instruments extend the pipe using rotary or pump valves .

	Pipe length through	Keynote	small second	great second	minor third	major third	Fourth	Tritone
Slide trombone	Train position	1	2	3	4th	5	6th	7th
Valve instrument	Valve combination	0	2	1	1/2 (3)	2/3	1/3	1/2/3

Diatonic and chromatic scales are very difficult to play on the natural horn and natural trumpet and baroque trumpet . To simplify matters, in the area of early music z. B. developed the keyed horn and the keyed trumpet . In contrast to modern brass instruments, the change in the air column length is achieved here, similar to woodwind instruments, by shortening the oscillating air column from the longest pipe by releasing finger holes or opening flaps and thereby shortening the air column.

With the help of various blowing techniques (over- or under-blowing), a correction of the "impure" natural tones is possible on all brass instruments. In the case of horns, it is also possible to correct them by “plugging” the hand into the lintel. Due to the physics of sound generation, the shape of the bell of brass instruments has an effect on the timbre and also on the purity of the intervals between the natural tones.

Woodwind instruments

The natural tones are important here when overblowing . Open flutes and reed instruments with a conical tube can blow over all natural tones. In practice, overblown is usually a maximum of the 4th natural tone. An exception are overtone flutes (open flutes without finger holes or keys), on which only the series of natural notes can be played. These instruments are overblown up to the 8th natural note or even higher.

On flat flutes and reed instruments with a cylindrical tube, you can only blow over the odd-numbered natural notes. In practice only the 3rd and 5th natural tone are used; the 7th natural tone is very difficult to achieve, and its intonation differs significantly from the corresponding diatonic or equally tuned tone.

organ

In the organ, the natural tones play a role in overblowing pipes that deliver an overtone instead of their fundamental. A distinction must be made between the aliquot registers , which are used as an additive mixture of overtones to change the timbre.

Inaccuracy of real natural and overblown tones

It is generally assumed that the natural tones form pure intervals with one another and that their frequencies are integral multiples of the fundamental tone frequency. However, this only applies approximately and with certain restrictions, which are explained in more detail under Overtone, Section: Limits of the Simple Model .

The corresponding real overblown tones can deviate from the theoretical integer even more than the natural tones themselves. For example, the overblowing is gedackter pipes resulting Blasquinte almost ⅛ sound smaller than the pure or tempered fifth.

"There are also deviations with wind instruments: the overtones and - even more - the overblown tones do not exactly correspond to the multiple of the fundamental, but still precisely enough to be perceived by us as belonging together." ( Jobst Fricke , 1962 )

literature

Michael Dickreiter: Handbook of the recording studio technology. 6th edition, K. G. Saur Verlag KG, Munich 1997, ISBN 3-598-11320-X .
Archimandrite Johannes Pfeiffer: The way to natural cult song. The musical system of German Orthodox church chant, its spiritual and historical requirements, its symbolism and the harmonic structure of the overtones , Verlag Kloster Buchhagen , Bodenwerder- Buchhagen 2012, ISBN 978-3-926236-09-8 .
Michael Magleitner: On the variety of tonal design options. PDF , University of Vienna, 2009.

Web links

Voice frequency graphics and audio examples of the sung natural tone series

Individual evidence

↑ J. Wolfe: "Pipes and Harmonics"
↑ Matthias Bertsch. (2002) Studies on sound generation on the trumpet (intonation on trumpets.) Vienna: IWK.MDW.AC.AT, 2002.
^ Hector Berlioz , Richard Strauss : Instrumentationslehre . New edition 1955 edition. C. F. Peters, Frankfurt 1955, p. 264 ff .
↑ Willibald Gurlitt, Hans Heinrich Eggebrecht (Ed.): Riemann Musik Lexikon , Sachteil, Mainz: Schott 1967, p. 111 f.
↑ The interior mood of the natural tone series and the sounds , Jobst Fricke in: Festschrift KG Fellerer for the 60th birthday. Hrsg. Hüschen, Regensburg 1962, page 162, and intonation and musical hearing, Habil.-Schr. Cologne 1968

Remarks

↑ Overblowing is not to be understood in the sense of overblowing natural notes, but means a slightly sharper blowing, which leads to a slightly higher intonation. Correspondingly, the (softer) blowing under blows a slightly deeper intonation.

[Grundfrequenz-2] The frequency of 66 Hz for the fundamental tone C corresponds to the choice of 440 Hz for the pitch A ': a small third (frequency ratio ⁶ / ₅ ) via a' = 440 Hz the sound c '' with 440 × ⁶ / ₅ = 528 Hz. The C, which is three octaves lower, has the frequency 528: (2 ³ ) = 66 Hz.

[Naturseptime-3] 7th partial tone = 462 Hz ( natural seventh ). Deviation from b '= 475.2 Hz of the pure tuning ≈ 49 cents. Note: The unit of cents is used primarily for the representation of the subtle differences in size of the intervals , with a (equal) semitone = 100 cents and an octave = 1200 cents. The calculation is made using the logarithm of two lb of the frequency ratio. Here 1200lb (475.2 / 462) ≈ 49 cents.

[Alphorn-Fa-4] 11th partial tone = 726 Hz ( Alphorn-Fa ). Deviation from f '' = 704 Hz or f sharp '' = 742.5 Hz of the pure tuning ≈ 53 cents or 39 cents.

[as-5] 13th partial tone = 858 Hz. Deviation from a flat '' = 844.8 Hz of the pure tuning ≈ 27 cents

[Naturseptime2-6] 14th partial tone = 924 Hz ( natural seventh ). Deviation from b '' = 950.4 Hz of the pure tuning ≈ 49 cents

[Oktve-7] The musical interval of an octave corresponds to a doubling of the frequency.

[Wolfe-1] J. Wolfe: "Pipes and Harmonics"

[8] Matthias Bertsch. (2002) Studies on sound generation on the trumpet (intonation on trumpets.) Vienna: IWK.MDW.AC.AT, 2002.

[9] Hector Berlioz , Richard Strauss : Instrumentationslehre . New edition 1955 edition. C. F. Peters, Frankfurt 1955, p. 264 ff .

[11] Willibald Gurlitt, Hans Heinrich Eggebrecht (Ed.): Riemann Musik Lexikon , Sachteil, Mainz: Schott 1967, p. 111 f.

[12] The interior mood of the natural tone series and the sounds , Jobst Fricke in: Festschrift KG Fellerer for the 60th birthday. Hrsg. Hüschen, Regensburg 1962, page 162, and intonation and musical hearing, Habil.-Schr. Cologne 1968

[unterblasen-10] Overblowing is not to be understood in the sense of overblowing natural notes, but means a slightly sharper blowing, which leads to a slightly higher intonation. Correspondingly, the (softer) blowing under blows a slightly deeper intonation.