Pitch (music)

See also: Note (music)

Pitch is the perceived (as distinct from measured) frequency of a sound. While the actual frequencies contained in a sound can be measured, the perceived pitch of the sound is affected by the mix of frequencies it contains. For many musical instruments, these frequencies include a fundamental frequency (the lowest frequency present) and multiples of this basic underlying frequency called overtones, harmonics, or upper partial tones. A sound containing only one frequency, like that from a tuning fork, is called a pure tone or a simple tone (as distinguished from a compound tone with many frequencies). See the article Tone (music).

Other aspects of a sound also affect its pitch, namely, its loudness (amplitude), and its temporal evolution (sound envelope). The human auditory perception system also may have trouble distinguishing frequency differences between notes under certain circumstances.

According to ANSI acoustical terminology, pitch is the auditory attribute of sound according to which sounds can be ordered on a scale from low to high.

Perception of pitch

When tuned in the typical manner, the note A above middle C (often denoted as A4) when 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 the human response to other stimuli, the perception of pitch can be described using a logarithmic scale (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.

Labeling pitches

Scale

A scale is a selection of pitches used in composing music. Almost all scales throughout the world have one interval in common, the octave, where the pitch of one note is double the frequency of another. It appears the role of the octave is related to the response of the human ear:

...the more we increase the size of an interval above an octave, the weaker its identity becomes: the less the interval has any kind of interval quality other than being a very large leap. This means that the octave is a kind of critical range within which we can discern intervals clearly, and beyond which our ability to discern intervals declines.

Accordingly, a scale is an initial choice of a pure tone and a selection of additional pure tones within one octave (that is, tones extending from the beginning tone up to a pure tone with a frequency twice that of the starting tone). The scale continues outside the defining octave simply by repeated doubling or halving of frequencies within the defining octave. The musical interval in pitch between adjacent notes is called a mode.

In the West, twelve simple tones commonly are chosen, the basis of the chromatic scale, with equal temperament now the method of tuning that scale most widely used. In this tuning, each note is separated a semitone from its adjacent notes.

An older, parochial view of the matter is as follows:

From the infinite multitude of possible Tones, perceptible to the ear, the intuition of man (in civilized countries) has singled out a limited number (at first seven, and later – as now – twelve) which, with their reproduction or duplication in higher or lower registers, by the Octave proportion, represent the entire absolute tone-material of the art of music.

Apparently the Greeks used several scales, and circa 600 BC adopted the old enharmonic scale of five tones, a scale that persists in some music of China and with the Celts of Scotland and Ireland. The diatonic scale consists of seven tones. According to some, Arabic-Persian music uses twenty-four tones spaced approximately half a semitone (a quarter-tone) apart.

Tuning

See also: Tuning (music)

The relative pitches of individual notes in a scale may be determined by one of a number of tuning systems. With equal temperament, the pitch ratio between any two successive notes of the chromatic (twelve-tone) scale is exactly the twelfth root of two (or about 1.05946), an interval referred to as a semitone or half tone. The notes are labeled in ascending order using the first seven letters of the alphabet A–G, but the interval between notes in half-tones includes sharps and flats. The labels for the 12 notes separated by semitones are then: A, (A♯, B♭), B, C, (C♯, D♭), D, (D♯, E♭), E, F, (F♯, G♭), G, (G♯, A♭), A, and so on.

In well-tempered systems (as used in the time of Johann Sebastian Bach, for example), different methods of musical tuning were used.

A pure tone at pitch A (A4) above middle C corresponds to a frequency 440 Hz, the pitch of a pure tone A (A5) located an octave above that will have frequency 880 Hz.

Digital labels

Pitches are often labeled using so-called scientific pitch notation, that is a letter denoting a pitch with a number indicating the octave it belongs in. For example, the first A above middle C is A4. Another approach uses the fundamental frequency of the note instead of the octave number as in A440. However, when referring to standard Western equal-temperament, the description "G4 double sharp" refers to the same pitch as "A4", so different labels may refer to the same pitch. Also, human pitch perception is logarithmic with respect to frequency, and not linear as the labels "A220" and "A440" and "A880" seem to suggest, making this labeling somewhat misleading.

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 a frequency ƒ to a real number n as follows

$n=69+12\times \log _{2}{\left({\frac {f}{440\;{\mbox{Hz}}}}\right)}$

According to this formula, a frequency of 440 Hz is assigned the MIDI pitch number 69, and this frequency commonly is labeled as A4. Middle C is assigned the integer MIDI pitch number 60, and accordingly has the frequency 261.625... Hz, which is not an integer number of cycles/second.

The formula creates a linear pitch space called equal temperament, in which semitones (the distance between adjacent keys on the piano keyboard) have size 1, and octaves (the interval between notes with frequencies in the ratio 2:1) have size 12, corresponding to the twelve notes in the chromatic scale. Distance in this pitch space corresponds quite well to musical distance as measured in psychological experiments and as understood by musicians.

Although MIDI pitch numbers are based upon equal temperament and so are integers, a number n that is not an integer can be used to refer to "microtones" not found on standard piano keyboards. For example, the pitch halfway between C (MIDI pitch number 60) and C♯ (MIDI pitch number 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 as a ratio or 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".

Discrete pitches, rather than continuously variable pitches, are virtually universal, with exceptions including "tumbling strains" and "indeterminate-pitch chants". Gliding pitches are used in most cultures, but are related to the discrete pitches they reference or embellish.

Concert pitch and transposition

See also: Musical notation

(PD) Image: John R. Brews
A scale in concert pitch (top) and the scale transposed for the B♭ trumpet (bottom).

Different instruments, for example, the clarinet and the oboe, are designed with the same valve structure or fingerings, but different notes are produced on different instruments for the same fingering. The sheet music provided to the players corresponds to their fingering, not to the absolute pitch of the notes. Thus, music for one instrument has to be transposed to produce the same sounds as another. Instruments for which this situation arises are called transposing instruments.

To play together, the musicians could transpose their music in their heads, but instead the composer usually writes the music separately for each instrument, so when playing the same melody, all instruments will be playing the same notes. The pitch aimed for by all the players is called concert pitch, and the music provided to the player of the instrument is transposed so they can use the fingering normal to their instrument and hit the right note. For example, the lightly colored C-note in the lower staff of the figure is played as C♯ by the trumpeter in this key, which will produce a pitch two semitones lower (an interval of a major second) because it is a B♭ instrument (that is, a C sounds like B♭; hence the name). The notes in the chromatic scale by semitones are: A, (A♯, B♭), B, C, (C♯, D♭), D, (D♯, E♭), E, F, (F♯, G♭), G, (G♯, A♭), A, and so on. So the pitch actually heard, two semitones lower than C♯, is B, as required by the concert-pitch scale in the upper staff.

In short, some instruments in an orchestra use different key signatures because of transposition, but "concert pitch" describes the music in absolute terms. Standardization of International Concert Pitch sets the frequency of a simple tone at pitch A4 to be 440 Hz.

Pitch classes

In atonal music, twelve-tone music, and in musical set theory a "pitch" is a specific frequency while a pitch class is all the octaves of a frequency. A pitch class set is a collection of three or more pitches, and pitch class set theory studies characteristics of differently chosen sets. The idea is to generalize classical notions of how different possible combinations of pitches function together.

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). Among the conventions of music set theory are:

1. Octave equivalence: no distinction is made concerning the register in which a pitch occurs. Humans hear octave related pitches as having the same color or chroma.
2. Enharmonic equivalence: all spellings of a given pitch class are equal.
3. Pitch classes are represented with numbers, not letters
4. Sets are not represented on a staff; rather, the members of a set are grouped within brackets and separated by commas, for example, [0, 1, 2].

This formalization is introduced to systematize understanding of various symmetries involved in music, ultimately to clarify harmony, consonance, and dissonance by analytical, not necessarily aural means. There has been resistance to the approach. A meeting between European and American music theorists in 1999 included this exchange:

"You guys are discussing methods of analyzing twentieth-century music. Why don't you talk about pitch-class sets?"

(Comment by an American participant)

"We do not talk about pitch-class sets, because we do not hear them! "

(Response from a European participant)

An extended discussion of the history of this approach and issues involved is found in Schuijer.

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 set 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—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 380 Hz, while the organs played by Johann Sebastian Bach in Hamburg, Leipzig and Weimar were pitched at A = 480 Hz, 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 = 422.5 Hz, while a later one from 1780 is pitched at A = 409 Hz, 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 400 to 450 Hz.

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. (In string instruments, this is not all acoustic illusion: when tuned up, they actually sound objectively brighter because the higher string tension results in larger amplitudes for the harmonics.) 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 =423.2 Hz, while one of eleven years later from the same opera house gives A = 435 Hz. At La Scala in Milan, the A above middle C rose as high as 451 Hz.

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 258.65 Hz. An alternative pitch standard known as philosophical or scientific pitch, which fixed middle C at exactly 256 Hz (that is, 2⁸ Hz), and resulted in the A above it being tuned to approximately 430.54 Hz, gained some popularity due to its mathematical convenience (the frequencies of all the C’s 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 standardization 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 a standard known as International 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 initial standard was A = 439 Hz, 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. The difference between this standard and the French diapason normal at 435 Hz is further confused by the temperature dependence of the pitch of tuning forks and musical instruments.

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 continental European and American orchestras (the Boston symphony being the best-known example), 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, an even-tempered semitone lower (rounded to the nearest integer Hz) than A–440. (An exact even-tempered semitone lower than A=440 Hz would be 440/2^1/12=415.3047 Hz.) At least in principle, this allows for playing along with modern fixed-pitch instruments if their parts are transposed down a semitone.

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.

References