An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In other words, the ratios of the frequencies of any adjacent pair of notes is the same, and, as pitch is perceived as the logarithm of frequency, equal perceived "distance" from every note to its nearest neighbor. In equal temperament tunings, the generating interval is found by dividing some larger desired interval the octave, into a number of smaller equal steps. In classical music and Western music in general, the most common tuning system since the 18th century has been twelve-tone equal temperament, which divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2; that resulting smallest interval, 1⁄12 the width of an octave, is called a half step. In modern times, 12TET is tuned relative to a standard pitch of 440 Hz, called A440, meaning one note, A, is tuned to 440 hertz and all other notes are defined as some multiple of semitones apart from it, either higher or lower in frequency.
The standard pitch has not always been 440 Hz. It has varied and risen over the past few hundred years. Other equal temperaments divide the octave differently. For example, some music has been written in 19-TET and 31-TET. Arabic music uses 24-TET as a notational convention. In Western countries the term equal temperament, without qualification means 12-TET. To avoid ambiguity between equal temperaments that divide the octave and those that divide some other interval, the term equal division of the octave, or EDO is preferred for the former. According to this naming system, 12-TET is called 12-EDO, 31-TET is called 31-EDO, so on. An example of an equal temperament that finds its smallest interval by dividing an interval other than the octave into equal parts is the equal-tempered version of the Bohlen–Pierce scale, which divides the just interval of an octave and a fifth, called a "tritave" or a "pseudo-octave" in that system, into 13 equal parts. Unfretted string ensembles, which can adjust the tuning of all notes except for open strings, vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for acoustic reasons.
Other instruments, such as some wind and fretted instruments only approximate equal temperament, where technical limitations prevent exact tunings. Some wind instruments that can and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles and vocal groups; the two figures credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu in 1584 and Simon Stevin in 1585. According to Fritz A. Kuttner, a critic of the theory, it is known that "Chu-Tsaiyu presented a precise and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that "Simon Stevin offered a mathematical definition of equal temperament plus a somewhat less precise computation of the corresponding numerical values in 1585 or later." The developments occurred independently. Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu and provides textual quotations as evidence. Zhu Zaiyu is quoted as saying. I establish one foot as the number from which the others are to be extracted, using proportions I extract them.
Altogether one has to find the exact figures for the pitch-pipers in twelve operations." Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications." Kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament and that neither of the two should be treated as inventors. The origin of the Chinese pentatonic scale is traditionally ascribed to the mythical Ling Lun, his writings discussed the equal division of the scale in the 27th century BC. However, evidence of the origins of writing in this period in China is limited to rudimentary inscriptions on oracle bones and pottery. A complete set of bronze chime bells, among many musical instruments found in the tomb of the Marquis Yi of Zeng, covers five full 7-note octaves in the key of C Major, including 12 note semi-tones in the middle of the range. An approximation for equal temperament was described by He Chengtian, a mathematician of Southern and Northern Dynasties around 400 AD.
He came out with the earliest recorded approximate numerical sequence in relation to equal temperament in history: 900 849 802 758 715 677 638 601 570 536 509.5 479 450. There was a seven-equal temperament or hepta-equal temperament practice in Chinese tradition. Zhu Zaiyu, a prince of the Ming court, spent thirty years on research based on the equal temperament idea postulated by his father, he described his new pitch theory in his Fusion of Music and Calendar 律暦融通 published in 1580. This was followed by the publication of a detailed account of the new theory of the equal temperament with a precise numerical specification for 12-TET in his 5,000-page work Complete Compendium of Music and Pitch in 1584. An extended account is given by Joseph Needham. Zhu obtained his result mathematically by dividing the length of string and pipe successively by 12√2 ≈ 1.059463, for pipe length by 24√2, such that after twelve divisions the le
In classical music from Western culture, a third is a musical interval encompassing three staff positions, the major third is a third spanning four semitones. Along with the minor third, the major third is one of two occurring thirds, it is qualified as major because it is the larger of the two: the major third spans four semitones, the minor third three. For example, the interval from C to E is a major third, as the note E lies four semitones above C, there are three staff positions from C to E. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones; the major third may be derived from the harmonic series as the interval between the fourth and fifth harmonics. The major scale is so named because of the presence of this interval between its tonic and mediant scale degrees; the major chord takes its name from the presence of this interval built on the chord's root. A major third is different in different musical tunings: in just intonation corresponds to a pitch ratio of 5:4 or 386.31 cents.
The older concept of a ditone made a dissonantly wide major third with the ratio 81:64. The septimal major third is 9:7, the undecimal major third is 14:11, the tridecimal major third is 13:10. A helpful way to recognize a major third is to hum the first two notes of "Kumbaya" or of "When the Saints Go Marching In". A descending major third is heard at the starts of "Goodnight, Ladies" and "Swing Low, Sweet Chariot". In equal temperament three major thirds in a row are equal to an octave; this is sometimes called the "circle of thirds". In just intonation, three 5:4 major thirds are less than an octave. For example, three 5:4 major thirds from C is B♯; the difference between this just-tuned B♯ and C, like that between G♯ and A♭, is called a diesis, about 41 cents. The major third is classed as an imperfect consonance and is considered one of the most consonant intervals after the unison, perfect fifth, perfect fourth. In the common practice period, thirds were considered interesting and dynamic consonances along with their inverses the sixths, but in medieval times they were considered dissonances unusable in a stable final sonority.
A diminished fourth is enharmonically equivalent to a major third. For example, B–D♯ is a major third. B–E♭ occurs in the C harmonic minor scale; the major third is used in guitar tunings. For the standard tuning, only the interval between the 3rd and 2nd strings is a major third. In an alternative tuning, the major-thirds tuning, each of the intervals are major thirds. Decade, compound just major third Ear training List of meantone intervals
Music notation or musical notation is any system used to visually represent aurally perceived music played with instruments or sung by the human voice through the use of written, printed, or otherwise-produced symbols. Types and methods of notation have varied between cultures and throughout history, much information about ancient music notation is fragmentary. In the same time period, such as in the 2010s, different styles of music and different cultures use different music notation methods; the symbols used include ancient symbols and modern symbols made upon any media such as symbols cut into stone, made in clay tablets, made using a pen on papyrus or parchment or manuscript paper. Although many ancient cultures used symbols to represent melodies and rhythms, none of them were comprehensive, this has limited today's understanding of their music; the seeds of what would become modern western notation were sown in medieval Europe, starting with the Catholic Church's goal for ecclesiastical uniformity.
The church began notating plainchant melodies so that the same chants could be used throughout the church. Music notation developed further in the Baroque music eras. In the classical period and the Romantic music era, notation continued to develop as new musical instrument technologies were developed. In the contemporary classical music of the 20th and 21st century, music notation has continued to develop, with the introduction of graphical notation by some modern composers and the use, since the 1980s, of computer-based score writer programs for notating music. Music notation has been adapted to many kinds of music, including classical music, popular music, traditional music; the earliest form of musical notation can be found in a cuneiform tablet, created at Nippur, in Babylonia, in about 1400 BC. The tablet represents fragmentary instructions for performing music, that the music was composed in harmonies of thirds, that it was written using a diatonic scale. A tablet from about 1250 BC shows a more developed form of notation.
Although the interpretation of the notation system is still controversial, it is clear that the notation indicates the names of strings on a lyre, the tuning of, described in other tablets. Although they are fragmentary, these tablets represent the earliest notated melodies found anywhere in the world. Ancient Greek musical notation was in use from at least the 6th century BC until the 4th century AD; the notation consists of symbols placed above text syllables. An example of a complete composition is the Seikilos epitaph, variously dated between the 2nd century BC to the 1st century AD. Three hymns by Mesomedes of Crete exist in manuscript; the Delphic Hymns, dated to the 2nd century BC use this notation, but they are not preserved. Ancient Greek notation appears to have fallen out of use around the time of the Decline of the Western Roman Empire. Byzantine music has survived as music for court ceremonies, including vocal religious music, it is not known if it is based on the monodic modal singing and instrumental music of Ancient Greece.
Greek theoretical categories played a key role to understand and transmit Byzantine music the tradition of Damascus had a strong impact on the pre-Islamic Near East comparable to Persian music. Unlike Western notation Byzantine neumes always indicate modal steps in relation to a clef or modal key; this key or the incipit of a common melody was enough to indicate a certain melodic model given within the echos. Despite ekphonetic notation further early melodic notation developed not earlier than between the 9th and the 10th century. Like the Greek alphabet notational signs are ordered left to right; the question of rhythm was based on cheironomia, well-known melodical phrases given by gestures of the choirleaders, which existed once as part of an oral tradition. Today the main difference between Western and Eastern neumes is that Eastern notation symbols are differential rather than absolute, i.e. they indicate pitch steps, the musicians know to deduce from the score and the note they are singing presently, which correct interval is meant.
These step symbols themselves, or better "phonic neumes", resemble brush strokes and are colloquially called gántzoi in Modern Greek. Notes as pitch classes or modal keys are represented in written form only between these neumes. In modern notation they serve as an optional reminder and modal and tempo directions have been added, if necessary. In Papadic notation medial signatures meant a temporary change into another echos; the so-called "great signs" were once related to cheironomic signs. Since Chrysanthos of Madytos there are seven standard note names used for "solfège" pá, vú, ghá, dhē, ké, zō, nē, while the older practice still used t
In music, an accidental is a note of a pitch, not a member of the scale or mode indicated by the most applied key signature. In musical notation, the sharp and natural symbols, among others, mark such notes—and those symbols are called accidentals. In the measure where it appears, an accidental sign raises or lowers the following note from its normal pitch, overriding sharps or flats in the key signature. A note is raised or lowered by a semitone, although microtonal music may use "fractional" accidental signs. There are occasionally double sharps or flats, which raise or lower the indicated note by a whole tone. Accidentals apply within the measure and octave in which they appear, unless canceled by another accidental sign, or tied into the following measure. If a note has an accidental and the note is repeated in a different octave within the same measure, the accidental does not apply to the same note of the different octave; the modern accidental signs derive from the two forms of the lower-case letter b used in Gregorian chant manuscripts to signify the two pitches of B, the only note that could be altered.
The "round" b became the flat sign, while the "square" b diverged into the natural signs. Sometimes the black keys on a musical keyboard are called accidentals, the white keys are called naturals. In most cases, a sharp raises the pitch of a note one semitone. A natural is used to cancel the effect of a sharp; this system of accidentals operates in conjunction with the key signature, whose effect continues throughout an entire piece, unless canceled by another key signature. An accidental can be used to cancel a previous accidental or reinstate the flats or sharps of the key signature Accidentals apply to subsequent notes on the same staff position for the remainder of the measure where they occur, unless explicitly changed by another accidental. Once a barline is passed, the effect of the accidental ends, except when a note affected by an accidental is tied to the same note across a barline. Subsequent notes at the same staff position in the second or bars are not affected by the accidental carried through with the tied note.
Under this system, the notes in the example above are: m. 1: G♮, G♯, G♯ m. 2: G♮, G♭, G♭ m. 3: G♭, G♯, G♮ Though this convention is still in use in tonal music, it may be cumbersome in music that features frequent accidentals, as is the case in atonal music. As a result, an alternative system of note-for-note accidentals has been adopted, with the aim of reducing the number of accidentals required to notate a bar; the system is as follows: An accidental carries through the bar affecting both the note it precedes and any following notes on the same line or space in the measure. Accidentals are not repeated on tied notes unless the tie goes from line to page to page. Accidentals are not repeated for repeated notes. If a sharp or flat pitch is followed directly by its natural form, a natural is used. Courtesy accidentals or naturals may be used to clarify ambiguities but are kept to a minimumBecause seven of the twelve notes of the chromatic equal-tempered scale are naturals this system can reduce the number of naturals required in a notated passage.
An accidental may change the note by more than a semitone: for example, if a G♯ is followed in the same measure by a G♭, the flat sign on the latter note means it is two semitones lower than if no accidental were present. Thus, the effect of the accidental must be understood in relation to the "natural" meaning of the note's staff position. In some atonal scores, an accidental is notated on every note, including natural notes and repeated pitches; this system was adopted for "the specific intellectual reason that a note with an accidental was not an inflected version of a natural note but a pitch of equal status." Double accidentals raise or lower the pitch of a note by two semitones, an innovation developed as early as 1615. This applies to the written note, ignoring key signature. An F with a double sharp applied raises it a whole step so it is enharmonically equivalent to a G. Usage varies on how to notate the situation in which a note with a double sharp is followed in the same measure by a note with a single sharp.
Some publications use the single accidental for the latter note, whereas others use a combination of a natural and a sharp, with the natural being understood to apply to only the second sharp. The double accidental with respect to a specific key signature raises or lowers the notes containing a sharp or flat by a semitone. For example, when in the key of C♯ minor or E major, F, C, G, D contain a sharp. Adding a double accidental to F in this case only raises F♯ by one further semitone, creating G natural. Conversely, adding a double sharp to any other note not sharped or flatted in the key signature raises the note by two semitones with respect to the chromatic scale. For example, in the aforementioned key signature, any note, not F, C, G, D is raised by two semitones instead of one, so an A double sharp raises the note A natural to the enharmonic equivalent of B natural. In modern scores, a barline cancels an accidental —but publishers use a courtesy accident
Meantone temperament is a musical temperament, a tuning system, obtained by compromising the fifths in order to improve the thirds. Meantone temperaments are constructed the same way as Pythagorean tuning, as a stack of equal fifths, but in meantone each fifth is narrow compared to the perfect fifth of ratio 3:2. Equal temperament, obtained by making all semitones the same size, each equal to one-twelfth of an octave (with ratio the 12th root of 2 to one, narrows the fifths by about 2 cents or 1/12 of a Pythagorean comma, produces thirds that are only better than in Pythagorean tuning. Equal temperament is the same as 1/11 comma meantone tuning. Quarter-comma meantone, which tempers the fifths by 1/4 comma, is the best known type of meantone temperament, the term meantone temperament is used to refer to it specifically. Four ascending fifths tempered by 1/4 comma produce a perfect major third, one syntonic comma narrower than the Pythagorean third that would result from four perfect fifths.
Quarter-comma meantone has been practiced from the early 16th century to the end of the 19th. In third-comma meantone, the fifths are tempered by 1/3 comma, three descending fifths produce a perfect minor third one syntonic comma wider than the Pythagorean one that would result from three perfect fifths. Third-comma meantone can be approximated by a division of the octave in 19 equal steps; the name "meantone temperament" derives from the fact that all such temperaments have only one size of the tone, while just intonation produces a major tone and a minor one, differing by a syntonic comma. In any regular system the tone is reached after two fifths, while the major third is reached after four fifths: the tone therefore is half the major third; this is one sense. In the case of quarter-comma meantone, in addition, where the major third is made narrower by a syntonic comma, the tone is half a comma narrower than the major tone of just intonation, or half a comma wider than the minor tone: this is another sense in which the tone in quarter-tone temperament may be considered a mean tone, it explains why quarter-comma meantone is considered the meantone temperament properly speaking.
"Meantone" can receive the following equivalent definitions: The meantone is the geometric mean between the major whole tone and the minor whole tone. The meantone is the mean of its major third; the family of meantone temperaments share the common characteristic that they form a stack of identical fifths, the tone being the result of two fifths minus one octave, the major third of four fifths minus two octaves. Meantone temperaments are described by the fraction of the syntonic comma by which the fifths are tempered: quarter-comma meantone, the most common type, tempers the fifths by 1⁄4 of a syntonic comma, with the result that four fifths produce a just major third, a syntonic comma lower than a Pythagorean major third. A meantone temperament is a linear temperament, distinguished by the width of its generator, as shown in the central column of Figure 1. Notable meantone temperaments, discussed below, occupy a narrow portion of this tuning continuum, with fifths ranging from 695 to 699 cents.
While the term meantone temperament refers to the tempering of 5-limit musical intervals, temperaments that approximate 5-limit intervals well, such as Quarter-comma meantone, can approximate 7-limit intervals well, defining septimal meantone temperament. In Figure 1, the valid tuning ranges of 5-limit, 7-limit, 11-limit tunings are shown, can be seen to include many notable meantone tunings. Meantone temperaments can be specified in various ways: by what fraction of a syntonic comma the fifth is being flattened, what equal temperament has the meantone fifth in question, the width of the tempered perfect fifth in cents, or the ratio of the whole tone to the diatonic semitone; this last ratio was termed "R" by American composer and theoretician Easley Blackwood, but in effect has been in use for much longer than that. It is useful because it gives us an idea of the melodic qualities of the tuning, because if R is a rational number N/D, so is 3R + 1/5R + 2 or 3N + D/5N + 2D, the size of fifth in terms of logarithms base 2, which tells us what division of the octave we will have.
If we multiply by 1200, we have the size of fifth in cents. In these terms, some notable meantone tunings are listed below; the second and fourth column are corresponding approximations to the first column. The third column shows how close the second column's approximation is to the actual size of the fifth interval in the given meantone tuning from the first column. Neither the just fifth nor the quarter-comma meantone fifth is a rational fraction of the octave, but several tunings exist which approximate the fifth by such an interval. Equal temperaments useful as meantone tunings include 19-ET, 50-ET, 31-ET, 43-ET, 55-ET; the farther the tuning gets away from quarter-comma meantone, the less related the tuning is to harmonic timbres, which can be overcome by tempering the timbre to match the tuning. A whole number of just perfect fifths
In music, there are two common meanings for tuning: Tuning practice, the act of tuning an instrument or voice. Tuning systems, the various systems of pitches used to tune an instrument, their theoretical bases. Tuning is the process of adjusting the pitch of one or many tones from musical instruments to establish typical intervals between these tones. Tuning is based on a fixed reference, such as A = 440 Hz; the term "out of tune" refers to a pitch/tone, either too high or too low in relation to a given reference pitch. While an instrument might be in tune relative to its own range of notes, it may not be considered'in tune' if it does not match the chosen reference pitch; some instruments become'out of tune' with temperature, damage, or just time, must be readjusted or repaired. Different methods of sound production require different methods of adjustment: Tuning to a pitch with one's voice is called matching pitch and is the most basic skill learned in ear training. Turning pegs to decrease the tension on strings so as to control the pitch.
Instruments such as the harp and harpsichord require a wrench to turn the tuning pegs, while others such as the violin can be tuned manually. Modifying the length or width of the tube of a wind instrument, brass instrument, bell, or similar instrument to adjust the pitch; the sounds of some instruments such as cymbals are inharmonic—they have irregular overtones not conforming to the harmonic series. Tuning may be done aurally by sounding two pitches and adjusting one of them to match or relate to the other. A tuning fork or electronic tuning device may be used as a reference pitch, though in ensemble rehearsals a piano is used. Symphony orchestras and concert bands tune to an A440 or a B♭ provided by the principal oboist or clarinetist, who tune to the keyboard if part of the performance; when only strings are used the principal string has sounded the tuning pitch, but some orchestras have used an electronic tone machine for tuning. Interference beats are used to objectively measure the accuracy of tuning.
As the two pitches approach a harmonic relationship, the frequency of beating decreases. When tuning a unison or octave it is desired to reduce the beating frequency until it cannot be detected. For other intervals, this is dependent on the tuning system being used. Harmonics may be used to facilitate tuning of strings. For example touching the highest string of a cello at the middle while bowing produces the same pitch as doing the same a third of the way down its second-highest string; the resulting unison is more and judged than the quality of the perfect fifth between the fundamentals of the two strings. In music, the term open string refers to the fundamental note of the full string; the strings of a guitar are tuned to fourths, as are the strings of the bass guitar and double bass. Violin and cello strings are tuned to fifths. However, non-standard tunings exist to change the sound of the instrument or create other playing options. To tune an instrument only one reference pitch is given; this reference is used to tune one string, to which the other strings are tuned in the desired intervals.
On a guitar the lowest string is tuned to an E. From this, each successive string can be tuned by fingering the fifth fret of an tuned string and comparing it with the next higher string played open; this works with the exception of the G string, which must be stopped at the fourth fret to sound B against the open B string above. Alternatively, each string can be tuned to its own reference tone. Note that while the guitar and other modern stringed instruments with fixed frets are tuned in equal temperament, string instruments without frets, such as those of the violin family, are not; the violin and cello are tuned to beatless just perfect fifths and ensembles such as string quartets and orchestras tend to play in fifths based Pythagorean tuning or to compensate and play in equal temperament, such as when playing with other instruments such as the piano. For example, the cello, tuned down from A220, has three more strings and the just perfect fifth is about two cents off from the equal tempered perfect fifth, making its lowest string, C-, about six cents more flat than the equal tempered C.
This table lists open strings on their standard tunings. Violin scordatura was employed in the 17th and 18th centuries by Italian and German composers, Biagio Marini, Antonio Vivaldi, Heinrich Ignaz Franz Biber, Johann Pachelbel and Johann Sebastian Bach, whose Fifth Suite For Unaccompanied Cello calls for the lowering of the A string to G. In Mozart's Sinfonia Concertante in E-flat major, all the strings of the solo viola are raised one half-step, ostensibly to give the instrument a brighter tone so the solo violin does not overshadow it. Scordatura for the violin was used in the 19th and 20th centuries in works by Niccolò Paganini, Robert Schumann, Camille Saint-Saëns and Béla Bartók. In Saint-Saëns' "Danse Macabre", the high string of the violin is lower half a tone to the E♭ so as to have the most accented note of the main theme sound on an open string. In Bartók's Contrasts, the violin is tuned G♯-D-A-E♭ to facilitate the playing of tritones on open strings. American folk violinists of the Appalachians and Ozarks employ alternate tunings for dance songs and
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. Cents are used to express small intervals, or to compare the sizes of comparable intervals in different tuning systems, in fact the interval of one cent is too small to be heard between successive notes. Alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquet's suggestion. Ellis made extensive measurements of musical instruments from around the world, using cents extensively to report and compare the scales employed, further described and employed the system in his 1875 edition of Hermann von Helmholtz's On the Sensations of Tone, it has become the standard method of comparing musical pitches and intervals. Like a decibel's relation to intensity, a cent is a ratio between two close frequencies. For the ratio to remain constant over the frequency spectrum, the frequency range encompassed by a cent must be proportional to the two frequencies.
An tempered semitone spans 100 cents by definition. An octave—two notes that have a frequency ratio of 2:1—spans twelve semitones and therefore 1200 cents. Since a frequency raised by one cent is multiplied by this constant cent value, 1200 cents doubles a frequency, the ratio of frequencies one cent apart is equal to 21⁄1200 = 1200√2, the 1200th root of 2, 1.0005777895. If one knows the frequencies a and b of two notes, the number of cents measuring the interval from a to b may be calculated by the following formula: n = 1200 ⋅ log 2 Likewise, if one knows a note a and the number n of cents in the interval from a to b b may be calculated by: b = a × 2 n 1200 To compare different tuning systems, convert the various interval sizes into cents. For example, in just intonation the major third is represented by the frequency ratio 5:4. Applying the formula at the top shows that this is about 386 cents; the equivalent interval on the equal-tempered piano would be 400 cents. The difference, 14 cents, is about a seventh of a half step audible.
As x increases from 0 to 1⁄12, the function 2x increases linearly from 1.00000 to 1.05946. The exponential cent scale can therefore be approximated as a piecewise linear function, numerically correct at semitones; that is, n cents for n from 0 to 100 may be approximated as 1 + 0.0005946n instead of 2n⁄1200. The rounded error is zero when n is 0 or 100, is about 0.72 cents high when n is 50, where the 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, it is difficult to establish. One author stated; the threshold of what is perceptible, technically known as the just noticeable difference 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, it has been established that increased tonal context enables listeners to judge pitch more accurately.
Free, online web sites for self-testing are available. "While intervals of less than a few cents are imperceptible to the human ear in a melodic context, in harmony small changes can cause large changes in beats and roughness of chords."When listening to pitches with vibrato, there is evidence that humans perceive the mean frequency as the center of the pitch. One study of modern performances of Schubert's Ave Maria found that vibrato span ranged between ±34 cents and ±123 cents with a mean of ±71 cents and noted higher variation in Verdi's opera arias. Normal adults are able to recognize pitch differences of as small as 25 cents reliably. Adults with amusia, have trouble recognizing differences of less than 100 cents and sometimes have trouble with these or larger intervals. A centitone is a musical interval equal to two cents proposed as a unit of measurement by Widogast Iring in Die reine Stimmung in der Musik as 600 steps per octave and by Joseph Yasser in A Theory of Evolving Tonality as 100 steps per equal tempered whole tone.
Iring noticed that the Grad/Werckmeister and the schisma are nearly the same and both may be approximated by 600 steps per octave. Yasser promoted the decitone and millitone. For example: Equal tempered perfect fifth = 700 cents = 175.6 savarts = 583.3 millioctaves = 350 centitones. 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. 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 hea