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In musical tuning
Musical tuning

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 Pitch used to tune an instrument, and their theoretical basis....
, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation
Just intonation

In music, just intonation is any musical tuning in which the frequency of notes are related by ratios of whole numbers. Any interval tuned in this way is called a just interval; in other words, the two notes are members of the same harmonic series ....
 in order to meet other requirements of the system.

In just intonation, every interval
Interval (music)

In music theory, the term interval describes the relationship between the pitch of two notes.Intervals may be described as:*vertical if the two notes sound simultaneously...
 between two pitches corresponds to a whole number
Whole number

The term whole number is used by various authors to mean either:*the nonnegative integer *the positive integer *all integer ...
 ratio
Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
 between their frequencies
Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
. Such just intervals have a stability, or purity to their sound. If one of those pitches is adjusted slightly, that stability decreases, and slow changes in the timbre
Timbre

In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices or musical instruments....
 of the interval's sound begin to appear: an effect known as beating
Beat (acoustics)

In acoustics, a beat is an interference between two sounds of slightly different frequency, perceived as periodic variations in volume whose rate is the difference between the two frequencies....
.






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In musical tuning
Musical tuning

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 Pitch used to tune an instrument, and their theoretical basis....
, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation
Just intonation

In music, just intonation is any musical tuning in which the frequency of notes are related by ratios of whole numbers. Any interval tuned in this way is called a just interval; in other words, the two notes are members of the same harmonic series ....
 in order to meet other requirements of the system.

In just intonation, every interval
Interval (music)

In music theory, the term interval describes the relationship between the pitch of two notes.Intervals may be described as:*vertical if the two notes sound simultaneously...
 between two pitches corresponds to a whole number
Whole number

The term whole number is used by various authors to mean either:*the nonnegative integer *the positive integer *all integer ...
 ratio
Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
 between their frequencies
Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
. Such just intervals have a stability, or purity to their sound. If one of those pitches is adjusted slightly, that stability decreases, and slow changes in the timbre
Timbre

In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices or musical instruments....
 of the interval's sound begin to appear: an effect known as beating
Beat (acoustics)

In acoustics, a beat is an interference between two sounds of slightly different frequency, perceived as periodic variations in volume whose rate is the difference between the two frequencies....
. As the adjustment becomes more severe, the beating becomes faster. To intentionally choose an interval with beating as a substitute for a just interval is the act of tempering that interval. These adjustments can make different musical possibilities available to the musician that would be impractical in just intonation. The actual measure of these adjustments are usually called commas
Comma (music)

In music theory, a comma is a small or very small interval between two enharmonic notes tuned in different ways. For example, an A flat tuned as a major third below C in just intonation, and a G sharp tuned as a major third above E, will not be exactly the same note....
.

Meantone temperament


Before Meantone temperament became widely used in the Renaissance
Renaissance music

Renaissance music is European music written during the Renaissance, approximately 1400 - 1600. Dates of classical music eras, given the lack of abrupt shifts in musical thinking during the 15th century....
, the most commonly used tuning system was Pythagorean tuning
Pythagorean tuning

Pythagorean tuning is a system of musical tuning in which the frequency relationships of all interval are based on the ratio sesquialterum. Its name comes from medieval texts which attribute its discovery to Pythagoras, but its use has been documented as long ago as 3500 B.C....
. Pythagorean tuning was a system of just intonation which tuned every note in a scale from a progression of pure perfect fifth
Perfect fifth

The perfect fifth is the musical interval between a note and the note seven semitones above it on the musical scale. For example, the note G lies a perfect fifth above C; D is a perfect fifth above G, C is a perfect fifth above F, and so on....
s. This was quite suitable for much of the harmonic practice until then (See: Quartal harmony), but in the Renaissance, musicians wished to make much more use of Tertian harmony
Tertian

In music or music theory, tertian is the quality of a chord constructed from thirds, and other things constructed from thirds such as counterpoint....
. The major third
Major third

A major third is one of two commonly occurring musical intervals that span three diatonic scale degrees, the other being the minor third. It is denoted 'major' because it is the larger of the two: the major third is a leap of four semitones, the minor third three....
 of Pythagorean tuning differed from a just major third by an amount known as Syntonic comma
Syntonic comma

In music theory, the syntonic comma , also known as the comma of Didymus the Musician or Ptolemy comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 Cent s....
, which musicians of the time found annoyingly impure.

Their solution, laid out by Pietro Aron
Pietro Aron

Pietro Aron, also known as Pietro Aaron , was an Italy music theorist and composer. He was born in Florence and probably died in Bergamo ....
 in the early 16th century, was to temper the interval of a perfect fifth slightly narrower than in just intonation, and then proceed much like Pythagorean tuning, but using this tempered fifth instead of the just one. With the correct amount of tempering, the Syntonic comma
Syntonic comma

In music theory, the syntonic comma , also known as the comma of Didymus the Musician or Ptolemy comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 Cent s....
 is removed from its major thirds, making them just. This compromise, however, leaves all fifths in this tuning system with a slight beating. However, because a sequence of four fifths makes up one third, this beating effect on the fifths is only one quarter as strong as the beating effect on the thirds of Pythagorean tuning, which is why it was considered a very acceptable compromise by Renaissance musicians.

Pythagorean tuning also had a second problem, which Meantone temperament does not solve, which is the problem of modulation
Modulation (music)

In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature....
 (see below), which is restricted because both have a broken circle of fifths
Circle of fifths

In music theory, the circle of fifths shows the relationships among the twelve tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys....
. A series of 12 just fifths as in Pythagorean tuning does not return to the original pitch, but rather differs by a Pythagorean comma
Pythagorean comma

The Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the Microtonal music Pythagorean interval defined as the difference between a Pythagorean apotome and a Limma, e.g....
, which makes that tonal area of the system more or less unusable. In meantone temperament, this effect is even more pronounced (the fifth over the break in the circle is known as the Wolf interval
Wolf interval

When the twelve notes within the octave are tuned using meantone temperament, one of the perfect fifth will be much sharper than the rest. If the meantone fifths are tuned from E to G, the anomalous interval will be between G and E....
, as its intense beating was likened to a "howling".) 53 equal temperament
53 equal temperament

In music, 53 equal temperament, called 53-TET, 53-equal division of the octave, or 53-ET, is the Temperament scale derived by dividing the octave into fifty-three equally large steps....
 provides a solution for the Pythagorean tuning, and 31 equal temperament
31 equal temperament

In music, 31 equal temperament , which can be abbreviated 31-TET, 31-equal division of the octave, 31-ET, is the Temperament scale derived by dividing the octave into 31 equal-sized steps....
 for the Meantone.

Well temperament and Equal temperament


Just intonation has the problem that it cannot modulate
Modulation (music)

In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature....
 to a different key
Key (music)

In music theory, the term key is used in many different and sometimes contradictory ways. A common use is to speak of music as being "in" a certain key, such as in the key of C or in the key of F-sharp....
 (a very common means of expression throughout the Common practice period
Common practice period

The common practice period, in the history of European art music , spanning the Baroque Music, Classical music era, and Romantic Music periods, lasted from about 1600 until about 1900....
 of music) without discarding many of the tones used in the previous key, thus for every key the musician wishes to modulate to, the instrument must provide a few more string
Strings (music)

A string is the Vibrating string that is the source of vibration in string instruments, such as the guitar, harp, piano, and members of the violin family....
s, fret
Fret

A fret is a raised portion on the neck of a stringed instrument, that extends generally across the full width of the neck. On most modern western culture instruments, frets are metal strips inserted into the fingerboard....
s, or holes for him or her to use. When building an instrument, this can be very impractical.

Well temperament is the name given to a variety of different systems of temperament that were employed to solve this problem, in which some keys are more in tune than others, but all can be used. This phenomenon gives rise to infinite shades of key-colors, which are lost in the modern standard version: 12 tone equal temperament (12-TET). Unlike Meantone temperament
Meantone temperament

Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a chain of perfect fifths, but in a meantone, each fifth is narrowed by the same amount in order to make the other intervals, like the major third, closer to their ideal just intonat...
, which alters the fifth to temper out the Syntonic comma, 12-TET tempers out the Pythagorean comma
Pythagorean comma

The Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the Microtonal music Pythagorean interval defined as the difference between a Pythagorean apotome and a Limma, e.g....
, thus creating a cycle of fifths that repeats itself exactly after 12 steps. This allowed the intervals of Tertian harmony
Tertian

In music or music theory, tertian is the quality of a chord constructed from thirds, and other things constructed from thirds such as counterpoint....
, thirds and fifths, to be fairly close to their just counterparts (the fifths almost imperceptibly beating, the thirds a little milder than the Syntonic beating of Pythagorean tuning), while permitting the freedom to modulate to any key and by various means (e.g. common-tone and enharmonic modulation, see modulation
Modulation (music)

In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature....
). This freedom of modulation also allowed substantial use of more distant harmonic relationships, such as the Neapolitan chord
Neapolitan chord

In music theory, a Neapolitan chord is a major chord built on the lowered second scale degree. It most commonly occurs in first inversion so that it is notated either as ♭II6 or N6 and normally referred to as a Neapolitan sixth chord....
, which became very important to Romantic
Romantic music

In music, romanticism is a term, often considered misleading, and concept derived from literature traditionally defined by attributes including, "interest in nature, medieval chivalry, mysticism, [and] remoteness [ Social alienation and Solitude]"....
 composers in the 19th century.

Alternate equal temperament scales

  • 19 tone equal temperament
  • 22 tone equal temperament
  • 24 equal temperament
  • 31 tone equal temperament
  • 53 tone equal temperament
  • 72 tone equal temperament
  • 88 equal temperament


See also


  • Just intonation
    Just intonation

    In music, just intonation is any musical tuning in which the frequency of notes are related by ratios of whole numbers. Any interval tuned in this way is called a just interval; in other words, the two notes are members of the same harmonic series ....
  • Meantone temperament
    Meantone temperament

    Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a chain of perfect fifths, but in a meantone, each fifth is narrowed by the same amount in order to make the other intervals, like the major third, closer to their ideal just intonat...
  • Well temperament
    Well temperament

    Well temperament is a type of Temperament musical tuning described in twentieth-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of Johann Sebastian Bach famous composition, Well-Tempered Clavier....
  • Equal temperament
    Equal temperament

    Equal temperament is a musical temperament, or a system of Musical tuning in which every pair of adjacent notes has an identical frequency ratios....
  • Piano tuning
    Piano tuning

    Piano tuning is the act of making minute adjustments to the tensions of the strings of a piano to properly align the intervals between their tones so that the instrument is Musical tuning....
  • Comma
    Comma (music)

    In music theory, a comma is a small or very small interval between two enharmonic notes tuned in different ways. For example, an A flat tuned as a major third below C in just intonation, and a G sharp tuned as a major third above E, will not be exactly the same note....
  • Regular temperament
    Regular temperament

    Regular temperament is any Temperament system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios....
  • List of meantone intervals
    List of meantone intervals

    The following is a list of intervals of meantone temperament. These intervals constitute the standard vocabulary of intervals for the Western common practice era....
  • Whole-tone scale
  • Pythagorean interval
    Pythagorean interval

    The intervals of Pythagorean tuning are just intonation involving only powers of two and three.The fundamental intervals are the superparticular number 2/1, 3/2, and 4/3....


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