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.[a] Ellis chose to base his measures on the hundredth part of a semitone, 1200√2, 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,[1] 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.[2][3]
History[edit]
Alexander John Ellis' paper On the Musical Scales of Various Nations,[1] 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 hundrecths [sic] of an equal semitone, or cents as they may be briefly called."[4]
Ellis defined the pitch of a musical note in his 1880 work History of Musical Pitch[5] 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".[6] 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."[7] 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".[8] Absolute and relative pitches were also defined based on these ratios.[8]
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.[9] 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."[10]
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,[11] Persia, Arabia, Syria and Scottish Highlands,[12] India,[13] Singapore,[14] Burmah[15] and Siam,;[16] (II. Pentatonic scales) South Pacific, [17] Western Africa,[18] Java,[19] China[20] and Japan.[21] 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".[22]
Piecewise linear approximation[edit]
As x increases from 0 to 1⁄12, the function 2x increases almost linearly from 1.00000 to 1.05946, 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 + 0.0005946 instead of the true exponential relation 2c⁄1200. The rounded error is zero when is 0 or 100, and is only about 0.72 cents high at =50 (whose correct value of 21⁄24 ≅ 1.02930 is approximated by 1 + 0.0005946 × 50 ≅ 1.02973). This error is well below anything humanly audible, making this piecewise linear approximation adequate for most practical purposes.
Other representations of intervals by logarithms[edit]
Octave[edit]
The representation of musical intervals by logarithms is almost as old as logarithms themselves. Logarithms had been invented by Lord Napier in 1614.[30] 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.[31] In this base, the octave is represented by 1, the semitone by 1/12, etc.
Heptamerides[edit]
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.[b] 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.[32]
Savart[edit]
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.[33] This value often is rounded to 1/301 or to 1/300 octave.[34][35]
Prony[edit]
Early in the 19th century, Gaspard de Prony proposed a logarithmic unit of base , where the unit corresponds to a semitone in equal temperament.[36] 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,[37] the interval that separated them from a theoretical pitch of 370 Hz, taken as point of reference.[38]
![{\displaystyle {\sqrt[{12}]{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc835f27425fb3140e1f75a5faa35b1e8b9efc35)
Centitones[edit]
A centitone (also Iring) is a musical interval (21⁄600, ) equal to two cents (22⁄1200)[39][40] proposed as a unit of measurement (ⓘ) 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).[41] 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).[42][43]
For example: Equal tempered perfect fifth = 700 cents = 175.6 savarts = 583.3 millioctaves = 350 centitones.[44]
![{\displaystyle {\sqrt[{600}]{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea43c0bc8f62081f2f0b830dd7da893a9d97a9a9)