# Pitch perception

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Pitch is an aspect of auditory perception and 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 440 Hz

(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 880 Hz .

## 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, Csharp and Dflat 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)

## 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.