Cent (music)

Physical unit
Unit name	cent
Unit symbol	¢, ${\displaystyle \mathrm {C} }$
Physical quantity (s)	Pitch interval
Formula symbol	${\displaystyle i\!\,,n\,,c}$
dimension	${\displaystyle {\mathsf {{\frac {T^{-1}}{T^{-1}}}=1}}}$

The cent (from the Latin centum "hundred") is an additive unit of measurement (more precisely: auxiliary unit of measurement ) with which a very precise comparison of the sizes of musical intervals is possible.

Diatonic intervals
Prime second third fourth fifth sixth seventh octave none decime undezime duodecime tredezime semitone / whole tone
Special intervals
Microinterval Comma Diësis Limma Apotome Ditone Tritone Wolf fifth Natural septime
units
Cent Millioctave Octave Savart

definition

The cent is defined by:

100 cents = 1  equal semitone

Since an octave comprises twelve semitones, the following also applies:

1200 cents = 1 octave

The cent is standardized in DIN  13320 ( see below ).

application

From the additive structure of the interval sizes it follows:

2 octaves = 2400 cents

3 octaves = 3600 cents etc.

As is known, for example, 12 equal fifths are ≈ 7 octaves, so 1 equal fifth comprises 700 cents (in pure tuning, on the other hand - see below - 702 cents.)

Since this corresponds to the additive interval perception of the hearing ( auditory event ), the comparison of pitches , tone systems and moods using the unit cent is more practical than information on frequency ratios, where a size comparison is not directly possible.

For the frequency ratio (higher frequency divided by lower) of the interval 1 cent applies: ${\displaystyle p}$

${\displaystyle p^{1200}=2}$

since 2 is just the frequency ratio of the octave. It follows:

${\displaystyle p={\sqrt[{1200}]{2}}\approx 1{,}000\,577\,789\,5}$

Emergence

The designation cent was proposed in 1875 by Alexander John Ellis (1814–1890) in the appendix to his translation of Hermann von Helmholtz ' theory of tone sensations as a unit for comparing the size of intervals.

The cent unit is chosen so that perceptible pitch differences can be expressed with sufficient accuracy as integer multiples of cents. Roughly it can be assumed that the smallest recognizable frequency difference for successive sine tones in humans at frequencies above 1000 Hz is around three to six cents; When the sound is heard at the same time , significantly smaller interval differences are audible due to beat effects.

With larger pitch intervals, interval sizes can be determined very precisely by beating the harmonic overtones , which are usually present in tones used musically. On the other hand, in the case of deep sine tones with a lower perceived volume (despite a high sound pressure level ), the discrimination threshold rises to over 100 cents, i.e. more than a semitone.

Use in music theory

The size of intervals is measured using the unit octave and its sub-unit cents. The octave and cents are proportional to the size of the interval. The unit of measurement octave corresponds to the frequency ratio 2: 1.

example

interval	Frequency ratio (in pure tuning)	Size in cents
1 octave	2	1200 cents
2 octaves	4th	2400 cents
3 octaves	8th	3600 cents
...
k octaves	2 k	1200 k cents
log 2 (q) octaves	q	1200 log 2 (q) cents
minor third	6 ⁄ 5	1200 log 2 ( 6 ⁄ 5 ) cents ≈ 315.641 cents
major third	5 ⁄ 4	1200 log 2 ( 5 ⁄ 4 ) cents ≈ 386.314 cents
Fourth	4 ⁄ 3	1200 log 2 ( 4 ⁄ 3 ) cents ≈ 498.045 cents
Fifth	3 ⁄ 2	1200 log 2 ( 3 ⁄ 2 ) cents ≈ 701.955 cents

If intervals are carried out one after the other, their sizes can be added, while their frequency ratios (proportions) have to be multiplied .

Examples:

Fifth + fourth = 702 cents + 498 cents = 1200 cents = octave. (Frequency ratios: ³ / ₂ · ⁴ / ₃ = ² / ₁ ).

Minor third + major third = 316 cents + 386 cents = 702 cents = fifth. (Frequency relationships: ⁶ / ₅ · ⁵ / ₄ = ³ / ₂ ).

See also : tone structure (mathematical description)

Applications in musical practice

With the unit of cents, the subtle differences in the intervals in the various mean-tone and well-tempered moods can be represented well, e.g. B. the slight detuning against perfect fifths and thirds that have to be accepted in order to make as many keys as possible (with a twelve-step scale of the octave) playable:

• In the mid-tone tunings, deviations of up to about 8 cents occur if only chords close to C major are used:

Example c'g '

perfect fifth 702 cents (No beats)

mean fifth 697 cents (Light beats)

• with up to 14 cents deviation has come to terms when on keyboard instruments and scales wants to use that are further away from C major. This makes use of the fact that the human ear "listens to the intervals":

Example a c sharp ' (first the third, then in the chord)

pure major third (220 Hz and 275 Hz) 386 cents (No beats)

equal major third (220 Hz and 277 Hz) 400 cents (many beats: the interval sounds rough )

• Musicians will not tolerate even greater deviations such as the wolf fifth of the mid-tone tuning in keys that are very far from C major.

Tables of the more or less pure thirds and fifths in different tuning systems: see tuning .

conversion

From proportions in cents

The proportion (frequency ratio) of any interval is given. The logarithmic interval measure is then calculated according to the definition formula (known in terms of content since approx. 1650): ${\displaystyle p={\frac {f_{2}}{f_{1}}}}$ ${\displaystyle i}$

${\displaystyle i=\log _{2}{p}\,\,{\text{Oktave}}}$

This equation translates the multiplicative acoustic proportions into the additive logarithmic interval measures ( example above ).

With

${\displaystyle 1\,{\text{Oktave}}=1200\,\,{\text{Cent}}}$

we receive:

${\displaystyle \Rightarrow i=1200\cdot \log _{2}{p}\,\,\mathrm {Cent} }$

After converting the two-logarithm into a logarithm about creating a comfortable for calculator manageable equation: ${\displaystyle \log _{2}p={\frac {\lg p}{\lg 2}}}$

${\displaystyle {\begin{aligned}\Rightarrow i&=1200\cdot {\frac {\lg p}{\lg 2}}\,\,{\text{Cent}}\\&\approx 3986{,}3\cdot \lg p\,\,{\text{Cent}}\end{aligned}}}$

From cents to proportions

The reverse conversion of any interval specified in cents into proportion (frequency ratio) is required less often. To do this, you solve the equation   for by dividing both sides by 1200 cents and then raising to the power of 2 (this removes the logarithm on one side): ${\displaystyle i}$ ${\displaystyle p}$ ${\displaystyle i=1200\cdot \log _{2}{p}\;\mathrm {Cent} }$ ${\displaystyle p}$

${\displaystyle p=2^{\frac {i}{1200\,\mathrm {Cent} }}}$

With known calculation rules for powers , the following approximation results for the pocket calculator:

${\displaystyle p\approx 1{,}00057779^{\frac {i}{\mathrm {Cent} }}}$

The following conversion is obtained for the triad intervals:

Interval in cents ${\displaystyle i}$	proportion ${\displaystyle p}$	interval
316 cents	${\displaystyle 2^{\frac {316}{1200}}\approx 1{,}2={\tfrac {6}{5}}}$	pure minor third
386 cents	${\displaystyle 2^{\frac {386}{1200}}\approx 1{,}25={\tfrac {5}{4}}}$	pure major third
702 cents	${\displaystyle 2^{\frac {702}{1200}}\approx 1{,}5={\tfrac {3}{2}}}$	perfect fifth

In other intervals

1 cent = millioctaves ≈ 0.8333 millioctaves ${\displaystyle {\frac {1}{1,2}}}$ 

1 cent = Savart ≈ 0.2509 Savart ${\displaystyle {\frac {\log _{10}(2)}{1{,}2}}}$ 

Calculation of frequencies

The above factor is the proportion (frequency ratio) of a tone difference of one cent. The frequency is calculated with this number as the base and the interval in cents in the exponent. ${\displaystyle {\sqrt[{1200}]{2}}=2^{\frac {1}{1200}}}$

Examples of some  frequencies used as tuning tone a ', starting from 440 Hz:

• Increase by 100 cents: ${\displaystyle 440\,\mathrm {Hz} \cdot 2^{\frac {100}{1200}}\approx 466{,}164\,\mathrm {Hz} }$
• Increase by 1 cent: ${\displaystyle 440\,\mathrm {Hz} \cdot 2^{\frac {1}{1200}}\approx 440{,}254\,\mathrm {Hz} }$
• Reduction by 1 cent: ${\displaystyle 440\,\mathrm {Hz} \cdot 2^{\frac {-1}{1200}}\approx 439{,}746\,\mathrm {Hz} }$
• 100 cents reduction: ${\displaystyle 440\,\mathrm {Hz} \cdot 2^{\frac {-100}{1200}}\approx 415{,}305\,\mathrm {Hz} }$

Example from music theory

The tone  a ' has the frequency of 440 Hz. The tone c' ' is a minor third above.

The tone  c '' therefore has

• in pure tuning (frequency ratio 6: 5 of the minor third) the frequency ${\displaystyle 440\,\mathrm {Hz} \cdot {\tfrac {6}{5}}=528\,\mathrm {Hz} ,}$
• in equal tuning (minor third = 3 semitones = 300 cents) the frequency . ${\displaystyle 440\,\mathrm {Hz} \cdot 2^{\frac {300}{1200}}\approx 523{,}251\,\mathrm {Hz} }$

DIN standard

According to DIN  13320 “Acoustics; Spectra and transfer curves; Terms, representation ”,“ Cent ”denotes a frequency measurement interval whose frequency ratio is. The cent can be used like a unit; thus the frequency metric interval of frequencies f ₁ and f ₂  > f _{1 can be} referred to as . ${\displaystyle 2^{\frac {1}{1200}}}$ ${\displaystyle 1200\cdot \log _{2}\left({\frac {f_{2}}{f_{1}}}\right)\,\mathrm {Cent} }$

Absolute cent

A scale of fixed cent values ​​can also be assigned to the entire frequency range. That absolute cent is then a unit of measure of pitch, not interval size. 1 Hz = 0 cents is set. This results in: 2 Hz = 1200 cents, 4 Hz = 2400 cents etc. with the corresponding intermediate values.

See also

• Millioctave
• Savart

literature

• Hermann von Helmholtz : The theory of tone sensations as a physiological basis for the theory of music. Vieweg, Braunschweig 1863 (Unchanged reprint: Minerva-Verlag, Frankfurt am Main 1981, ISBN 3-8102-0715-2 , excerpt ).
• John R. Pierce : Sound. Music with the ears of physics. Spectrum - Akademischer Verlag, Heidelberg u. a. 1999, ISBN 3-8274-0544-0 .

Web links

• Interval conversion: frequency ratio to cents and cents to frequency (ratio)
• Conversion of cents into frequency ratio and back in Excel
• Joachim Mohr: The cent measure for intervals.

References and comments

1. ↑ ${\displaystyle 2^{k}=q\iff \log _{2}(q)=k}$
2. ↑ In normal cases it should be. If it is the other way around, the conversion result will be negative with the same absolute value. ${\displaystyle f_{2}\geq f_{1}}$
3. ↑ For example, the fifth has the frequency ratio . Their size is then calculated ${\displaystyle {\tfrac {3}{2}}}$

   ${\displaystyle {\text{Quinte}}=1200\cdot \log _{2}\left({\tfrac {3}{2}}\right){\text{ Cent}}=1200\cdot {\frac {\log({\tfrac {3}{2}})}{\log(2)}}{\text{ Cent}}\approx 702{\text{ Cent}}}$

4. ↑ https://www.beuth.de/de/norm/din-13320/515781 website for DIN 13320 at Beuth Verlag
5. ↑ Riemann Music Lexicon. Material part. Mainz 1967, p. 150.