The cents (from the Latin centum "hundred" ) serves as a logarithmic unit of measurement for musical intervals. The name comes from the fact that a gleichstufiger semitone is divided into 100 steps. Since an octave consists of twelve semitones, it corresponds to 1200 cents. The unit cents in DIN 13320 standardized ( see below) and corresponds to a frequency ratio of.
Using information in cents, various sound systems and moods easily be compared. Comparison means of the pitch of the unit has the advantage that it corresponds to the additive interval sense of hearing. It is therefore more practical than the details of frequency ratios, where a size comparison is not immediately possible.
The calculation is logarithmic. If the frequency ratio of the interval, then the interval is calculated as a multiple of one cent:
Mathematical Precision of the definition
The word denotes both cents a pitch interval with the frequency ratio ( proportion)
As well as a dimensionless auxiliary referring to this unit.
Namely applies to the logarithmic frequency interval cents:
For clarity, however, the auxiliary unit octave is usually not added, creating the following working definition equation:
The designation cents in 1875 by Alexander John Ellis ( 1814-1890 ) proposed in the appendix to his translation of Hermann von Helmholtz's theory of the sensations of tone as a unit for size comparison of intervals.
The Cent - unit is chosen so that noticeable frequency differences can be expressed with sufficient accuracy as integer multiples of cents. Roughly it can be assumed that the smallest detectable difference in frequency for pure tones in humans at frequencies above 1000 Hz is about three to six cents. Lower interval differences are no longer recognized in the succession - sound of tones. With the simultaneous sounding are by beat effects still much lower interval differences audible. For larger pitch intervals up interval sizes by beats of the harmonics can be determined very accurately, which are present mostly in musical tones used. At low sine tones at low volume, however, the discrimination threshold increases to about 100 cents, or a semitone.
The use of the measure cent of intervals in music theory
Intervals can be specified as multiples of one octave. It is a logarithmic measure of the frequency ratios. Cents is a subunit of the octave with the definition in 1200 cents = 1 octave (or 1 gleichstufiger semitone = 100 cents). This is a very accurate comparison of the interval size is possible. The Centmaß is proportional to the interval size, while the frequency ratio behaves exponentially.
Become intervals performed successively, one can add their sizes in cents ( while their frequency ratios should be multiplied ).
See: clay structure (mathematical description)
Impact on the musical practice
Especially for the representation of the subtle differences between intervals in the various mean-tone and well tempered tunings used to the unit cents.
To as many keys to make playable (with a twelve -point scale of the octave ), you have to accept upsets in perfect fifths and thirds.
In the Mean moods, there are deviations to about 8 cents, if only C major chords are used close.
Listen? / I
( Easy to hear beats )
Listen? / I
( No beats )
With 14 cents deviation one has come to terms, if you want to use on keyboard instruments all scales. This exploits that human hearing is " rightly hear the intervals".
Listen equally-? / I
( The interval sounds " rough": many beats )
Listen clean? / I
( No beats )
Even larger deviations such as the " wolf fifth " of the mean tone with heavily from C major remote keys will not be tolerated by musicians.
Tables of more or less pure thirds and fifths in different mood systems: See spirits.
Conversion of proportions in cents
Given the proportion ( frequency ratio ) of any interval. The logarithmic Intervallmaß then calculated according to the ( contents known since about 1650) definition formula:
This equation translates the multiplicative acoustic proportions in the additive logarithmic interval dimensions ( example above).
With we obtain:
After conversion of the binary logarithm into a logarithm of the equation creates a pocket calculator for convenient- equation:
Conversion of cents in proportions
The reverse conversion of cents in proportion ( frequency ratio ) is rarely needed. To calculate the proportion of any given interval in cents one solves the equation after by dividing both sides by 1200 cents and then entlogarithmiert:
With well-known calculation rules for powers results in the following approximation for the calculator:
In the triad intervals obtained following conversion:
Calculation of frequencies
The above factor is the proportion ( frequency ratio ) of a Tonunterschieds of one cent. The frequency calculation is therefore with this number as a base and the interval in cents in the exponent.
Examples of some tuning pitch as a ' used frequencies of 440 Hz:
- Increase by 1 cent:
- Increase of 100 cents:
- Reduction of 1 cent:
- Reduction of 100 cents:
Example from the theory of music
The note A 'has the frequency of 440 Hz tone c'' is a minor third above.
The tone c'' has therefore in pure tuning ( frequency ratio of the minor third: 6:5 ) is the frequency, in gleichstufiger mood but ( minor third = 3 semitones = 300 cents) the frequency.
According to DIN 13320 "Acoustics; Spectra and transmission curves; Concepts, representation "refers to cents a Frequenzmaßintervall, is the frequency ratio. The penny can be used as a unit; thus the Frequenzmaßintervall the frequencies f1 and f2 ( f2 > f1) are referred to as.
You can also assign the entire frequency range, a scale fixed cent values . To calculate this absolute cents 1 Hz = 0 Cent is set. Then result: 2 Hz = 1200 cents, 4 Hz = 2400 cents, etc., with the corresponding intermediate values .