Английская Википедия:Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each. Typically, cents are used to express small intervals, to check intonation, or to compare the sizes of comparable intervals in different tuning systems. For humans, a single cent is too small to be perceived between successive notes.
Cents, as described by Alexander John Ellis, follow a tradition of measuring intervals by logarithms that began with Juan Caramuel y Lobkowitz in the 17th century.Шаблон:Efn Ellis chose to base his measures on the hundredth part of a semitone, Шаблон:Radic, at Robert Holford Macdowell Bosanquet's suggestion. Making extensive measurements of musical instruments from around the world, Ellis used cents to report and compare the scales employed,Шаблон:Sfn and further described and utilized the system in his 1875 edition of Hermann von Helmholtz's On the Sensations of Tone. It has become the standard method of representing and comparing musical pitches and intervals.Шаблон:SfnШаблон:Sfn
History
Alexander John Ellis' paper On the Musical Scales of Various Nations,Шаблон:Sfn published by the Journal of the Society of Arts in 1885, officially introduced the cent system to be used in exploring, by comparing and contrasting, musical scales of various nations. The cent system had already been defined in his History of Musical Pitch, where Ellis writes: "If we supposed that, between each pair of adjacent notes, forming an equal semitone [...], 99 other notes were interposed, making exactly equal intervals with each other, we should divide the octave into 1200 equal Шаблон:Sic of an equal semitone, or cents as they may be briefly called."Шаблон:Sfn
Ellis defined the pitch of a musical note in his 1880 work History of Musical PitchШаблон:Sfn to be "the number of double or complete vibrations, backwards and forwards, made in each second by a particle of air while the note is heard".Шаблон:Sfn He later defined musical pitch to be "the pitch, or V [for "double vibrations"] of any named musical note which determines the pitch of all the other notes in a particular system of tunings."Шаблон:Sfn He notes that these notes, when sounded in succession, form the scale of the instrument, and an interval between any two notes is measured by "the ratio of the smaller pitch number to the larger, or by the fraction formed by dividing the larger by the smaller".Шаблон:Sfn Absolute and relative pitches were also defined based on these ratios.Шаблон:Sfn
Ellis noted that "the object of the tuner is to make the interval [...] between any two notes answering to any two adjacent finger keys throughout the instrument precisely the same. The result is called equal temperament or tuning, and is the system at present used throughout Europe.Шаблон:Sfn He further gives calculations to approximate the measure of a ratio in cents, adding that "it is, as a general rule, unnecessary to go beyond the nearest whole number of cents."Шаблон:Sfn
Ellis presents applications of the cent system in this paper on musical scales of various nations, which include: (I. Heptatonic scales) Ancient Greece and Modern Europe,Шаблон:Sfn Persia, Arabia, Syria and Scottish Highlands,Шаблон:Sfn India,Шаблон:Sfn Singapore,Шаблон:Sfn BurmahШаблон:Sfn and Siam,;Шаблон:Sfn (II. Pentatonic scales) South Pacific, Шаблон:Sfn Western Africa,Шаблон:Sfn Java,Шаблон:Sfn ChinaШаблон:Sfn and Japan.Шаблон:Sfn And he reaches the conclusion that "the Musical Scale is not one, not 'natural,' nor even founded necessarily on the laws of the constitution of musical sound, so beautifully worked out by Helmholtz, but very diverse, very artificial, and very capricious".Шаблон:Sfn
Use
A cent is a unit of measure for the ratio between two frequencies. An equally tempered semitone (the interval between two adjacent piano keys) spans 100 cents by definition. An octave—two notes that have a frequency ratio of 2:1—spans twelve semitones and therefore 1200 cents. The ratio of frequencies one cent apart is precisely equal to Шаблон:Nowrap, the 1200th root of 2, which is approximately Шаблон:Val. Thus, raising a frequency by one cent corresponds to multiplying the original frequency by this constant value. Raising a frequency by 1200 cents doubles the frequency, resulting in its octave.
If one knows the frequencies <math>f_1</math> and <math>f_2</math> of two notes, the number of cents <math>c</math> measuring the interval from <math>f_1</math> to <math>f_2</math> is:
- <math>c = 1200 \cdot \log_2 \left( \frac{f_2}{f_1} \right)</math>
Likewise, if one knows <math>f_1</math> and the number of cents <math>c</math> in the interval from <math>f_1</math> to <math>f_2</math>, then <math>f_2</math> equals:
- <math>f_2 = f_1 \times 2 ^ \frac{c}{1200} \, .</math>
Comparison of major third in just and equal temperament
The major third in just intonation has a frequency ratio 5:4 or ~386 cents, but in equal temperament is 400 cents. This 14 cent difference is about a seventh of a half step and large enough to be audible.
Piecewise linear approximation
As x increases from 0 to Шаблон:Frac, the function 2x increases almost linearly from Шаблон:Val to Шаблон:Val, allowing for a piecewise linear approximation. Thus, although cents represent a logarithmic scale, small intervals (under 100 cents) can be loosely approximated with the linear relation 1 + Шаблон:Val <math>c</math> instead of the true exponential relation 2Шаблон:Sup. The rounded error is zero when <math>c</math> is 0 or 100, and is only about 0.72 cents high at <math>c</math>=50 (whose correct value of 2Шаблон:Sup ≅ Шаблон:Val is approximated by 1 + Шаблон:Val × 50 ≅ 1.02973). This error is well below anything humanly audible, making this piecewise linear approximation adequate for most practical purposes.
Human perception
It is difficult to establish how many cents are perceptible to humans; this precision varies greatly from person to person. One author stated that humans can distinguish a difference in pitch of about 5–6 cents.Шаблон:Sfn The threshold of what is perceptible, technically known as the just noticeable difference (JND), also varies as a function of the frequency, the amplitude and the timbre. In one study, changes in tone quality reduced student musicians' ability to recognize, as out-of-tune, pitches that deviated from their appropriate values by ±12 cents.Шаблон:Sfn It has also been established that increased tonal context enables listeners to judge pitch more accurately.Шаблон:Sfn "While intervals of less than a few cents are imperceptible to the human ear in a melodic context, in harmony very small changes can cause large changes in beats and roughness of chords."Шаблон:Sfn
When listening to pitches with vibrato, there is evidence that humans perceive the mean frequency as the center of the pitch.Шаблон:Sfn One study of modern performances of Schubert's Ave Maria found that vibrato span typically ranged between ±34 cents and ±123 cents with a mean of ±71 cents and noted higher variation in Verdi's opera arias.Шаблон:Sfn
Normal adults are able to recognize pitch differences of as small as 25 cents very reliably. Adults with amusia, however, have trouble recognizing differences of less than 100 cents and sometimes have trouble with these or larger intervals.Шаблон:Sfn
Other representations of intervals by logarithms
Octave
The representation of musical intervals by logarithms is almost as old as logarithms themselves. Logarithms had been invented by Lord Napier in 1614.[1] As early as 1647, Juan Caramuel y Lobkowitz (1606-1682) in a letter to Athanasius Kircher described the usage of base-2 logarithms in music.[2] In this base, the octave is represented by 1, the semitone by 1/12, etc.
Heptamerides
Joseph Sauveur, in his Principes d'acoustique et de musique of 1701, proposed the usage of base-10 logarithms, probably because tables were available. He made use of logarithms computed with three decimals. The base-10 logarithm of 2 is equal to approximately 0.301, which Sauveur multiplies by 1000 to obtain 301 units in the octave. In order to work on more manageable units, he suggests to take 7/301 to obtain units of 1/43 octave.Шаблон:Efn The octave therefore is divided in 43 parts, named "merides", themselves divided in 7 parts, the "heptamerides". Sauveur also imagined the possibility to further divide each heptameride in 10, but does not really make use of such microscopic units.[3]
Savart
Félix Savart (1791-1841) took over Sauveur's system, without limiting the number of decimals of the logarithm of 2, so that the value of his unit varies according to sources. With five decimals, the base-10 logarithm of 2 is 0.30103, giving 301.03 savarts in the octave.[4] This value often is rounded to 1/301 or to 1/300 octave.[5][6]
Prony
Early in the 19th century, Gaspard de Prony proposed a logarithmic unit of base <math>\sqrt[12]{2}</math>, where the unit corresponds to a semitone in equal temperament.[7] Alexander John Ellis in 1880 describes a large number of pitch standards that he noted or calculated, indicating in pronys with two decimals, i.e. with a precision to the 1/100 of a semitone,[8] the interval that separated them from a theoretical pitch of 370 Hz, taken as point of reference.[9]
Centitones
A centitone (also Iring) is a musical interval (2Шаблон:Sup, <math>\sqrt[600]{2}</math>) equal to two cents (2Шаблон:Sup)Шаблон:SfnШаблон:Sfn proposed as a unit of measurement (Шаблон:Audio) by Widogast Iring in Die reine Stimmung in der Musik (1898) as 600 steps per octave and later by Joseph Yasser in A Theory of Evolving Tonality (1932) as 100 steps per equal tempered whole tone.
Iring noticed that the Grad/Werckmeister (1.96 cents, 12 per Pythagorean comma) and the schisma (1.95 cents) are nearly the same (≈ 614 steps per octave) and both may be approximated by 600 steps per octave (2 cents).[10] Yasser promoted the decitone, centitone, and millitone (10, 100, and 1000 steps per whole tone = 60, 600, and 6000 steps per octave = 20, 2, and 0.2 cents).Шаблон:SfnШаблон:Sfn
For example: Equal tempered perfect fifth = 700 cents = 175.6 savarts = 583.3 millioctaves = 350 centitones.Шаблон:Sfn
| Centitones | Cents |
|---|---|
| 1 centitone | 2 cents |
| 0.5 centitone | 1 cent |
| 2Шаблон:Sup | 2Шаблон:Sup |
| 50 per semitone | 100 per semitone |
| 100 per whole tone | 200 per whole tone |
Sound files
The following audio files play various intervals. In each case the first note played is middle C. The next note is sharper than C by the assigned value in cents. Finally, the two notes are played simultaneously.
Note that the JND for pitch difference is 5–6 cents. Played separately, the notes may not show an audible difference, but when they are played together, beating may be heard (for example if middle C and a note 10 cents higher are played). At any particular instant, the two waveforms reinforce or cancel each other more or less, depending on their instantaneous phase relationship. A piano tuner may verify tuning accuracy by timing the beats when two strings are sounded at once.
Шаблон:Audio, beat frequency = 0.16 Hz
Шаблон:Audio, beat frequency = 1.53 Hz
Шаблон:Audio, beat frequency = 3.81 Hz
Шаблон:Listen Шаблон:Listen Шаблон:Listen
See also
References
Footnotes
Citations
Sources
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite thesis
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Citation
- Шаблон:Cite journal
- Шаблон:Cite book
External links
- Cent conversion: Whole number ratio to cent Шаблон:Webarchive [rounded to whole number]
- Cent conversion: Online utility with several functions
Шаблон:Musical tuning Шаблон:Intervals
- ↑ Ernest William Hobson (1914), John Napier and the invention of logarithms, 1614, Cambridge, The University Press
- ↑ Ramon Ceñal, "Juan Caramuel, su epistolario con Athanasio Kircher, S.J.", Revista de Filosofia XII/44, Madrid 1954, p. 134 ss.
- ↑ Joseph Sauveur, Principes d'acoustique et de musique ou Système général des intervalles des sons, Minkoff Reprint, Geneva, 1973; see online Mémoires de l'Académie royale des sciences, 1700, Acoustique; 1701 Acoustique.
- ↑ Émile Leipp, Acoustique et musique : Données physiques et technologiques, problèmes de l'audition des sons musicaux, principes de fonctionnement et signification acoustique des principaux archétypes d'instruments de musique, les musiques expérimentales, l'acoustique des salles, Masson, 1989, 4th edition, p. 16.
- ↑ "Ordinary savart", 1/301 octave, and "modified savart", 1/300 octave. Herbert Arthur Klein, The Science of Measurement. A Historical Survey, New York, 1974, p. 605
- ↑ Alexander Wood, The Physics of Music, London, 1944, ²2007, p. 53-54.
- ↑ Gaspard de Prony, Instruction élémentaire sur les moyens de calculer les intervalles musicaux, Paris, 1832. Online: [1].
- ↑ The precision is the same as with cents, but Ellis had not yet devised this unit.
- ↑ Alexander John Ellis, "On the History of Musical Pitch," Journal of the Society of Arts, 1880, reprinted in Studies in the History of Musical Pitch, Frits Knuf, Amsterdam, 1968, p. 11-62.
- ↑ Ошибка цитирования Неверный тег
<ref>; для сносокhuygens-fokkerне указан текст
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