Pitch perception

Pitch is the perceived fundamental frequency of a sound. While the actual fundamental frequency can be precisely determined through physical measurement, it may differ from the perceived pitch because of overtones, or partials, in the sound. The human auditory perception system may also have trouble distinguishing frequency differences between notes under certain circumstances. According to ANSI acoustical terminology, it is the auditory attribute of sound according to which sounds can be ordered on a scale from low to high.

Perception of pitch
The note A above middle C played on a piano is perceived to be of the same pitch as a pure tone of 440 Hz. However, a slight change in frequency need not lead to a perceived change in pitch. The just noticeable difference (the threshold at which a change in pitch is perceived) is about five cents (that is, about five hundredths of a semitone), but varies over the range of hearing and is more precise when the two pitches are played simultaneously. Like other human stimuli, the perception of pitch also can be explained by the Weber-Fechner law.

Pitch may depend on the amplitude of the sound, especially at low frequencies. For instance, a low bass note will sound lower in pitch if it is louder. Like other senses, the relative perception of pitch can be fooled, resulting in "audio illusions". There are several of these, such as the tritone paradox, but most notably the Shepard scale, where a continuous or discrete sequence of specially formed tones can be made to sound as if the sequence continues ascending or descending forever.

A special type of pitch often occurs in free nature when the sound of a sound source reaches the ear of an observer directly and also after being reflected against a sound-reflecting surface. This phenomenon is called Repetition Pitch, because the addition of a true repetition of the original sound to itself is the basic prerequisite.

Standardized pitch (A440)
The A above middle C is nowadays set at (often written as "A = 440 Hz" or sometimes "A440"), although this has not always been the case (see "History of pitch standards in Western music").

Concert pitch
Since some instruments in an orchestra use different key signatures (because of transposition), "concert pitch" describes a particular pitch in absolute terms, regardless of notation.

Labeling pitches
Pitches are often labeled using scientific pitch notation or some combination of a letter and a number representing a fundamental frequency. For example, one might refer to the A above middle C as "A4" or "A440." However, there are two problems with this practice. First, in standard Western equal-temperament, the notion of pitch is insensitive to spelling: the description "G4 double sharp" refers to the same pitch as "A4." Second, human pitch perception is logarithmic with respect to fundamental frequency: the perceived distance between the pitches "A220" and "A440" is the same as the perceived distance between the pitches "A440" and "A880."

To avoid these problems, music theorists sometimes represent pitches using a numerical scale based on the logarithm of fundamental frequency. For example, one can adopt the widely used MIDI standard to map fundamental frequency $$ f $$ to a real number $$ p $$ as follows



p = 69 + 12\times\log_2 { \left(\frac {f}{440\; \mbox{Hz}} \right) } $$

This creates a linear pitch space in which octaves have size 12, semitones (the distance between adjacent keys on the piano keyboard) have size 1, and A440 is assigned the number 69. Distance in this space corresponds to musical distance as measured in psychological experiments and understood by musicians. The system is flexible enough to include "microtones" not found on standard piano keyboards. For example, the pitch halfway between C (60) and C♯ (61) can be labeled 60.5.

Varying pitch
Pitches may be described in various ways, including high or low, as discrete or indiscrete, pitch that changes with time (chirping) and the manner in which this change with time occurs: gliding; portamento; or vibrato, and as determinate or indeterminate. Musically the frequency of specific pitches is not as important as their relationships to other frequencies — the difference between two pitches can be expressed by a ratio or measured in cents. People with a sense of these relationships are said to have relative pitch while people who have a sense of the actual frequencies independent of other pitches are said to have "absolute pitch", or "perfect pitch".

Scales
The relative pitches of individual notes in a scale may be determined by one of a number of tuning systems. In the west, the twelve-note chromatic scale is the most common method of organization, with equal temperament now the most widely used method of tuning that scale. In it, the pitch ratio between any two successive notes of the scale is exactly the twelfth root of two (or about 1.05946). In well-tempered systems (as used in the time of Johann Sebastian Bach, for example), different methods of musical tuning were used. Almost all of these systems have one interval in common, the octave, where the pitch of one note is double the frequency of another. For example, if the A above middle C is 440 Hz, the A an octave above that will be.

Other musical meanings of pitch
In atonal, twelve tone, or musical set theory a "pitch" is a specific frequency while a pitch class is all the octaves of a frequency. Pitches are named with integers because of octave and enharmonic equivalency (for example, C♯ and D♭ are the same pitch, while C4 and C5 are functionally the same, one octave apart).

Discrete pitches, rather than continuously variable pitches, are virtually universal, with exceptions including "tumbling strains" (Sachs & Kunst, 1962) and "indeterminate-pitch chants" (Malm, 1967). Gliding pitches are used in most cultures, but are related to the discrete pitches they reference or embellish. (Burns, 1999)

History of pitch standards in Western music
Historically, various standards have been used to fix the pitch of notes at certain frequencies. Various systems of musical tuning have also been used to determine the relative frequency of notes in a scale.

Pre-19th Century
Until the 19th century there was no concerted effort to standardize musical pitch, and the levels across Europe varied widely. Pitches did not just vary from place to place, or over time&mdash;pitch levels could vary even within the same city. The pitch used for an English cathedral organ in the 17th century for example, could be as much as five semitones lower than that used for a domestic keyboard instrument in the same city.

Even within one church, the pitch used could vary over time because of the way organs were tuned. Generally, the end of an organ pipe would be hammered inwards to a cone, or flared outwards, to raise or lower the pitch. When the pipe ends became frayed by this constant process they were all trimmed down, thus raising the overall pitch of the organ.

Some idea of the variance in pitches can be gained by examining old pitchpipes, organ pipes and other sources. For example, an English pitchpipe from 1720 plays the A above middle C at, while the organs played by Johann Sebastian Bach in Hamburg, Leipzig and Weimar were pitched at A = , a difference of around four semitones. In other words, the A produced by the 1720 pitchpipe would have been at the same frequency as the F on one of Bach's organs.

From the early 18th century, pitch could be also controlled with the use of tuning forks (invented in 1711), although again there was variation. For example, a tuning fork associated with Handel, dating from 1740, is pitched at A =, while a later one from 1780 is pitched at A = , almost a semitone lower. Nonetheless, there was a tendency towards the end of the 18th century for the frequency of the A above middle C to be in the range of to.

The frequencies quoted here are based on modern measurements and would not have been precisely known to musicians of the day. Although Mersenne had made a rough determination of sound frequencies as early as the 1600s, such measurements did not become scientifically accurate until the 19th century, beginning with the work of German physicist Johann Scheibler in the 1830s. The unit hertz (Hz), replacing cycles per second (cps), was not introduced until the twentieth century.

Pitch inflation
During historical periods when instrumental music rose in prominence (relative to the voice), there was a continuous tendency for pitch levels to rise. This "pitch inflation" seemed largely due to instrumentalists competing with each other, each attempting to produce a brighter, more "brilliant", sound than that of their rivals. This tendency was also prevalent with wind instrument manufacturers, who crafted their instruments to generally play at a higher pitch than those made by the same craftsmen years earlier.

It should be noted too that pitch inflation is a problem only where musical compositions are fixed by notation. The combination of numerous wind instruments and notated music has therefore restricted pitch inflation almost entirely to the Western tradition.

On at least two occasions, pitch inflation has become so severe that reform became needed. At the beginning of the 17th century, Michael Praetorius reported in his encyclopedic Syntagma musicum that pitch levels had become so high that singers were experiencing severe throat strain and lutenists and viol players were complaining of snapped strings. The standard voice ranges he cites show that the pitch level of his time, at least in the part of Germany where he lived, was at least a minor third higher than today's. Solutions to this problem were sporadic and local, but generally involved the establishment of separate standards for voice and organ ("Chorton") and for chamber ensembles ("Kammerton"). Where the two were combined, as for example in a cantata, the singers and instrumentalists might perform from music written in different keys. This system kept pitch inflation at bay for some two centuries.

The advent of the orchestra as an independent (as opposed to accompanying) ensemble brought pitch inflation to the fore again. The rise in pitch at this time can be seen reflected in tuning forks. An 1815 tuning fork from the Dresden opera house gives A =, while one of eleven years later from the same opera house gives A =. At La Scala in Milan, the A above middle C rose as high as.

19th and 20th century standards
The most vocal opponents of the upward tendency in pitch were singers, who complained that it was putting a strain on their voices. Largely due to their protests, the French government passed a law on February 16, 1859 which set the A above middle C at 435 Hz. This was the first attempt to standardize pitch on such a scale, and was known as the diapason normal. It became quite a popular pitch standard outside of France as well, and has also been known at various times as French pitch, continental pitch or international pitch (the last of these not to be confused with the 1939 "international standard pitch" described below).

The diapason normal resulted in middle C being tuned at approximately. An alternative pitch standard known as philosophical or scientific pitch, which fixed middle C at exactly (that is, 28 Hz), and resulted in the A above it being tuned to approximately, gained some popularity due to its mathematical convenience (the frequencies of all the Cs being a power of two). This never received the same official recognition as A = 435 Hz, however, and was not as widely used.

British attempts at standardisation in the 19th century gave rise to the so-called old philharmonic pitch standard of about A = 452 Hz (different sources quote slightly different values), replaced in 1896 by the considerably "deflated" new philharmonic pitch at A = 439 Hz. The high pitch was maintained by Sir Michael Costa for the Crystal Palace Handel Festivals, causing the withdrawal of the principal tenor Sims Reeves in 1877, though at singers' insistence the Birmingham Festival pitch was lowered (and the organ retuned) at that time. At the Queen's Hall in London, the establishment of the diapason normal for the Promenade Concerts in 1895 (and retuning of the organ to A = 439 at 15 °C (59 °F), to be in tune with A = 435.5 in a heated hall) caused the Royal Philharmonic Society and others (including the Bach Choir, and the Felix Mottl and Artur Nikisch concerts) to adopt the continental pitch thereafter. In 1939, an international conference recommended that the A above middle C be tuned to 440 Hz, now known as concert pitch. This standard was taken up by the International Organization for Standardization in 1955 (and was reaffirmed by them in 1975) as ISO 16. The difference between this and the diapason normal is due to confusion over which temperature the French standard should be measured at. The initial standard was A =, but this was superseded by A = 440 Hz after complaints that 439 Hz was difficult to reproduce in a laboratory owing to 439 being a prime number.

Despite such confusion, A = 440 Hz is arguably the most common tuning used around the world. Many, though certainly not all, prominent orchestras in the United States and United Kingdom adhere to this standard as concert pitch. In other countries, however, higher pitches have become the norm: A = 442 Hz is common in certain American and continental European orchestras, while A = 445 Hz is heard in Germany, Austria, and China.

In practice, as orchestras still tune to a note given out by the oboe, rather than to an electronic tuning device (which would be more reliable), and as the oboist may not have used such a device to tune in the first place, there is still some variance in the exact pitch used. Solo instruments such as the piano (which an orchestra may tune to if they are playing together) are also not universally tuned to A = 440 Hz. Overall, it is thought that the general trend since the middle of the 20th century has been for standard pitch to rise, though it has been rising far more slowly than it has in the past.

Many modern ensembles which specialize in the performance of Baroque music have agreed on a standard of A=415 Hz, about a semitone lower than A–440.

Changing the pitch of a vibrating string
There are three ways to change the pitch of a vibrating string. String instruments are tuned by varying the strings' tension because adjusting length or mass per unit length is impractical.

Length
Pitch can be adjusted by varying the length of the string. A longer string will result in a lower pitch, while a shorter string will result in a higher pitch. The frequency is inversely proportional to the length:



f \propto \frac{1}{l} $$

A string twice as long will produce a tone of half the frequency (one octave lower).

Tension
Pitch can be adjusted by varying the tension of the string. A string with less tension (looser) will result in a lower pitch, while a string with greater tension (tighter) will result in a higher pitch. The frequency is proportional to the square root of the tension:



f \propto \sqrt{T} $$

Density
The pitch of a string can also be varied by changing the density of the string. The frequency is inversely proportional to the square root of the density:



f \propto {1 \over \sqrt{\rho}} $$

A string that is more dense will produce a lower pitch.