undertone

Undertone is a term from music or acoustics that also applies to other physical vibrations . In music it is also called subharmonic .

definition

According to Hermann von Helmholtz , the " harmonic undertones" of a " sound " are those to whose "harmonic overtones " it belongs. The frequencies of the undertones are therefore integral divisors of the fundamental frequency or the fundamental frequency is an integral multiple of the frequencies of the undertones:

nth undertone	n + 1th subharmonics	Frequencies	Example for the concert pitch A (440 Hz ) as the fundamental	the keynote is ... that undertone
0. undertone	1st subharmonic	${\ displaystyle f _ {\ text {UT0}} = {\ tfrac {1} {1}} \ cdot f _ {\ text {keynote}} = f _ {\ text {GT}}}$	${\ displaystyle f _ {\ text {UT0}} = f _ {\ text {GT}} = 440 \; {\ text {Hz}}}$	0th overtone or 1st harmonic
1. undertone	2nd subharmonics	${\ displaystyle f _ {\ text {UT1}} = {\ tfrac {1} {2}} \ cdot f _ {\ text {GT}} \ Leftrightarrow f _ {\ text {GT}} = 2 \ cdot f _ {\ text {UT1}}}$	${\ displaystyle f _ {\ text {UT1}} = 220 \; {\ text {Hz}}}$	1st overtone or 2nd harmonic
2. undertone	3rd subharmonics	${\ displaystyle f _ {\ text {UT2}} = {\ tfrac {1} {3}} \ cdot f _ {\ text {GT}} \ Leftrightarrow f _ {\ text {GT}} = 3 \ cdot f _ {\ text {UT2}}}$	${\ displaystyle f _ {\ text {UT2}} = 147 \; {\ text {Hz}}}$	2nd overtone or 3rd harmonic
3. undertone	4. Subharmonics	${\ displaystyle f _ {\ text {UT3}} = {\ tfrac {1} {4}} \ cdot f _ {\ text {GT}} \ Leftrightarrow f _ {\ text {GT}} = 4 \ cdot f _ {\ text {UT3}}}$	${\ displaystyle f _ {\ text {UT3}} = 110 \; {\ text {Hz}}}$	3rd overtone or 4th harmonic
4. undertone	5th subharmonics	${\ displaystyle f _ {\ text {UT4}} = {\ tfrac {1} {5}} \ cdot f _ {\ text {GT}} \ Leftrightarrow f _ {\ text {GT}} = 5 \ cdot f _ {\ text {UT4}}}$	${\ displaystyle f _ {\ text {UT4}} = 88 \; {\ text {Hz}}}$	4th overtone or 5th harmonic
...	...	...	...

The periods of the undertones are integer multiples of the period of the reference tone (i.e. they are 2, 3, 4 etc. times larger). This can be visualized by not dividing the length of a string as with the overtones, but doubling, tripling, etc.

The undertone series can be thought of from the overtone series or the subharmonics can be thought of from the harmonics by mirroring the fundamental or the fundamental downward.

Occur

While the overtones sound more or less strongly with every sound generator or musical instrument , the undertones do not appear in the frequency spectrum of most instruments. In the case of string instruments , something similar can be achieved by applying too much bow pressure. However, these are not undertones, but anomalous low frequencies , which result from slipping of the bow hairs with the stick-slip effect .

Undertones formed the basis of the Mixtur trautonium constructed and played by Oskar Sala .

Objects lying loosely on vibrating surfaces sometimes create subharmonics. Subharmonics are better known in technical processes such as ultrasonic applications ( ultrasonic cleaning devices , ultrasonic welding , sonotrodes ). They express themselves in chirping and squeaking when using a fundamental vibration that is beyond the audible range .

Connection with chords

While the tone for the stock from the first harmonics of a fundamental tone Dur - chord results, the first overtones give the minor chord. The doctrine widespread in (speculative) music theory that the minor triad is thus a "sub-sound" that is mirror-symmetrical to the major triad is called harmonic dualism :

Example for root C:

Overtones are

Keynote	1	2	3	4th	5	6th	7th
C.	C.	G	C.	E.	G	B.	C.

This results in the first overtones of the C major chord CEG .

Undertones are

Keynote	1	2	3	4th	5	6th	7th
C.	C.	F.	C.	As	F.	D.	C.

This results in the F minor chord F-Ab-C from the first undertones .

In modern music there are always experiments to include the tones with 7 times or a seventh of the basic frequency in the harmonies. In the above example, the tones B and D .

literature

Hermann von Helmholtz : The theory of tone sensations as a physiological basis for the theory of music . Vieweg, Braunschweig 1863 ( online ).

Individual evidence

↑ Helmholtz 1863, p. 76.

↑ Wave analysis of a string bowed to anomalous low frequencies. (PDF) (No longer available online.) Formerly in the original ; Retrieved March 26, 2012 . ( Page no longer available , search in web archives ) Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.@1@ 2Template: Dead Link / www.nmh.no
↑ The search for the 7th note. Retrieved March 26, 2012 .

[1] Helmholtz 1863, p. 76.