Musical temperament
Encyclopedia
In musical tuning
, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation
in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament
system. Temperament is especially important for keyboard instruments, which typically allow a player to play only the pitches assigned to the various keys, and lack any way to alter pitch of a note in performance. Historically, the use of just intonation
, pythagorean tuning
and meantone temperament
meant that such instruments could sound "in tune" in one key, or some keys, but would then have more dissonance in other keys.
The development of well temperament
allowed fixed-pitch instruments to play reasonably well in all of the keys. The famous "Well-Tempered Clavier" takes full advantage of this breakthrough, with pieces written for all 24 major and minor keys. However, while unpleasant intervals (such as the wolf interval
) were avoided, the sizes of intervals were still not consistent between keys, and so each key still had its own character. This variation led in the 18th century to an increase in the use of equal temperament
, in which the frequency ratio between each pair of adjacent notes on the keyboard was made equal, allowing music to be transposed between keys without changing the relationship between notes.
between two pitches corresponds to a whole number
ratio
between their frequencies
. For instance, 660 Hz / 440 Hz constitutes a fifth, and 880 Hz / 440 Hz an octave. Such intervals (termed "just") have a stability, or purity to their sound, when played simultaneously (assuming they are played using timbres with harmonic partials). If one of those pitches is adjusted slightly to deviate from the just interval, a trained ear can detect this change by the presence of beats, which are explained below.
, the most commonly used tuning system was Pythagorean tuning
. Pythagorean tuning was a system of just intonation which tuned every note in a scale from a progression of pure perfect fifth
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
. The major third
of Pythagorean tuning differed from a just major third by an amount known as Syntonic comma
, which musicians of the time found annoying.
Their solution, laid out by Pietro Aron
in the early 16th century, and referred to as meantone temperament
(or quarter-comma meantone
temperament), 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
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
(see below), which is restricted because both have a broken circle of fifths
. A series of 12 just fifths as in Pythagorean tuning does not return to the original pitch, but rather differs by a Pythagorean comma
, 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
, as its intense beating was likened to a "howling".) 53 equal temperament
provides a solution for the Pythagorean tuning, and 31 equal temperament
for the Meantone.
to a different key
(a very common means of expression throughout the common practice period
of music) without discarding many of the tones used in the previous key, thus for every key to which the musician wishes to modulate, the instrument must provide a few more string
s, fret
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
, which alters the fifth to temper out the Syntonic comma, 12-TET tempers out the Pythagorean comma
, thus creating a cycle of fifths that repeats itself exactly after 12 steps. This allowed the intervals of Tertian harmony
, 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
). This freedom of modulation also allowed substantial use of more distant harmonic relationships, such as the Neapolitan chord
, which became very important to Romantic
composers in the 19th century.
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 pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
, 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 frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
system. Temperament is especially important for keyboard instruments, which typically allow a player to play only the pitches assigned to the various keys, and lack any way to alter pitch of a note in performance. Historically, the use of just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
, pythagorean tuning
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
and 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 stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
meant that such instruments could sound "in tune" in one key, or some keys, but would then have more dissonance in other keys.
The development of well temperament
Well temperament
Well temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...
allowed fixed-pitch instruments to play reasonably well in all of the keys. The famous "Well-Tempered Clavier" takes full advantage of this breakthrough, with pieces written for all 24 major and minor keys. However, while unpleasant intervals (such as the wolf interval
Wolf interval
In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament...
) were avoided, the sizes of intervals were still not consistent between keys, and so each key still had its own character. This variation led in the 18th century to an increase in the use of equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
, in which the frequency ratio between each pair of adjacent notes on the keyboard was made equal, allowing music to be transposed between keys without changing the relationship between notes.
What temperament is
In just intonation, every intervalInterval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
between two pitches corresponds to a whole number
Whole number
Whole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts.Whole numbers may refer to:*natural numbers in sense — the positive integers...
ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
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...
. For instance, 660 Hz / 440 Hz constitutes a fifth, and 880 Hz / 440 Hz an octave. Such intervals (termed "just") have a stability, or purity to their sound, when played simultaneously (assuming they are played using timbres with harmonic partials). If one of those pitches is adjusted slightly to deviate from the just interval, a trained ear can detect this change by the presence of beats, which are explained below.
Acoustic physics
When a musical instrument with harmonic overtones is played, the ear hears a composite waveform that includes a fundamental frequency (e.g., 440 Hz) and those overtones (880 Hz, 1320 Hz, etc.) – a series of just intervals. The waveform of such a tone (as pictured on an oscilloscope) is characterized by a shape that is complex compared to a simple (sine) waveform, but which remains periodic. When two tones depart from exact integer ratios, the shape waveform becomes erratic – a phenomenon which may be described as destabilization. As the composite waveform becomes more erratic, the consonance of the interval also changes.Temperament in music
Tempering an interval involves the deliberate use of such minor adjustments (accepting the related destabilization) to enable musical possibilities that are impractical using just intonation. The most widely known example of this is the use of equal temperament to address problems of older temperaments, allowing for consistent tuning of keyboard and fretted instruments and enabling musical composition in, and modulation among, the various keys.Meantone temperament
Before Meantone temperament became widely used in the RenaissanceRenaissance music
Renaissance music is European music written during the Renaissance. Defining the beginning of the musical era is difficult, given that its defining characteristics were adopted only gradually; musicologists have placed its beginnings from as early as 1300 to as late as the 1470s.Literally meaning...
, 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 intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
. Pythagorean tuning was a system of just intonation which tuned every note in a scale from a progression of pure perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
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 theory, tertian describes any piece, chord, counterpoint etc. constructed from the interval of a third...
. The major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans 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 chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
, which musicians of the time found annoying.
Their solution, laid out by Pietro Aron
Pietro Aron
Pietro Aron, also known as Pietro Aaron , was an Italian music theorist and composer. He was born in Florence and probably died in Bergamo .-Biography:...
in the early 16th century, and referred to as 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 stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
(or quarter-comma meantone
Quarter-comma meantone
Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
temperament), 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 chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
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. Modulations articulate or create the structure or form of many pieces, as well as add interest...
(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 12 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
In musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...
, 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
In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament...
, as its intense beating was likened to a "howling".) 53 equal temperament
53 equal temperament
In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into 53 equal steps . Each step represents a frequency ratio of 21/53, or 22.6415 cents , an interval sometimes called the Holdrian comma.- History :Theoretical interest in this...
provides a solution for the Pythagorean tuning, and 31 equal temperament
31 equal temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered 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 modulateModulation (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. Modulations articulate or create the structure or form of many pieces, as well as add interest...
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 specific key, such as in the key of C major or in the key of F-sharp. Sometimes the terms "major" or "minor" are appended, as in the key of A minor or in the...
(a very common means of expression throughout the common practice period
Common practice period
The common practice period, in the history of Western art music , spanning the Baroque, Classical, and Romantic periods, lasted from c. 1600 to c. 1900.-General characteristics:...
of music) without discarding many of the tones used in the previous key, thus for every key to which the musician wishes to modulate, the instrument must provide a few more string
Strings (music)
A string is the vibrating element that produces sound in string instruments, such as the guitar, harp, piano, and members of the violin family. Strings are lengths of a flexible material kept under tension so that they may vibrate freely, but controllably. Strings may be "plain"...
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 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
Well temperament
Well temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...
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 stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
, which alters the fifth to temper out the Syntonic comma, 12-TET tempers out the Pythagorean comma
Pythagorean comma
In musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...
, thus creating a cycle of fifths that repeats itself exactly after 12 steps. This allowed the intervals of Tertian harmony
Tertian
In music theory, tertian describes any piece, chord, counterpoint etc. constructed from the interval of a third...
, 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. Modulations articulate or create the structure or form of many pieces, as well as add interest...
). 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
Romantic music or music in the Romantic Period is a musicological and artistic term referring to a particular period, theory, compositional practice, and canon in Western music history, from 1810 to 1900....
composers in the 19th century.
Equal temperament scales
- 12 tone equal temperamentEqual temperamentAn equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
- 15 tone equal temperament15 equal temperamentIn music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the octave into 15 equal steps . Each step represents a frequency ratio of 21/15, or 80 cents...
- 17 tone equal temperament17 equal temperamentIn music, 17 tone equal temperament is the tempered scale derived by dividing the octave into 17 equal steps . Each step represents a frequency ratio of 21/17, or 70.6 cents . Alexander J. Ellis refers to a tuning of seventeen tones based on perfect fourths and fifths as the Arabic scale...
- 19 tone equal temperament19 equal temperamentIn music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
- 22 tone equal temperament22 equal temperamentIn music, 22 equal temperament, called 22-tet, 22-edo, or 22-et, is the tempered scale derived by dividing the octave into 22 equal steps . Each step represents a frequency ratio of 21/22, or 54.55 cents ....
- 24 tone equal temperamentQuarter toneA quarter tone , is a pitch halfway between the usual notes of a chromatic scale, an interval about half as wide as a semitone, which is half a whole tone....
- 31 tone equal temperament31 equal temperamentIn music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
- 34 tone equal temperament34 equal temperamentIn musical theory, 34 equal temperament, also referred to as 34-tet, 34-edo or 34-et, is the tempered tuning derived by dividing the octave into 34 equal-sized steps...
- 41 tone equal temperament41 equal temperamentIn music, 41 equal temperament, often abbreviated 41-tET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally-sized steps . Each step represents a frequency ratio of 21/41, or 29.27 cents , an interval close in size to the septimal comma. 41-ET can be seen as a...
- 53 tone equal temperament53 equal temperamentIn music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into 53 equal steps . Each step represents a frequency ratio of 21/53, or 22.6415 cents , an interval sometimes called the Holdrian comma.- History :Theoretical interest in this...
- 72 tone equal temperament72 equal temperamentIn music, 72 equal temperament, called twelfth-tone, 72-tet, 72-edo, or 72-et, is the tempered scale derived by dividing the octave into twelfth-tones, or in other words 72 equal steps...
See also
- Piano tuningPiano tuningPiano 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 in tune. The meaning of the term in tune in the context of piano tuning is not simply a particular fixed set of pitches...
- CommaComma (music)In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word "comma" used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning...
- Regular temperamentRegular temperamentRegular temperament is any tempered 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
- Whole-tone scale
- Pythagorean intervalPythagorean intervalIn musical tuning theory, a Pythagorean interval is a musical interval with frequency ratio equal to a power of two divided by a power of three, or vice versa...
- Mathematics of musical scales
- Schismatic temperamentSchismatic temperamentThe schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 to a unison. It is also called the schismic temperament or Helmholtz temperament.-Comparison with other tunings:...
Articles
Books
- Music: a Mathematical Offering (mathematical perspective with two chapters on temperament) by Dave Benson
- Tuning And Temperament A Historical Survey (1951) by J. Murray Barbour
- Essay on Musical Temperament (part 2) by Prof. Fisher (Yale College)
- "Temperament" from A supplement to Mr. Chambers's cyclopædia (1753)
- Theory and practice of just intonation (1850) by Thomas Perronet Thompson
- Elements of musical composition: comprehending the rules of thorough bass and the theory of tuning (1812) by William Crotch
- An essay on temperament (1832) by J. Jousse
- Essay on musical intervals, harmonics, and the temperament of the musical scale, &c (1835) by Wesley Stoker B. Woolhouse
- Harmonics, or The philosophy of musical sounds (1759) by Robert Smith (1689-1768)
- Modern organ tuning : the how and why? by Hermann Smith (1824-1910)
- Piano Tuning: A Simple and Accurate Method for Amateurs by Jerry Cree Fischer
- The organ viewed from within : a practical handbook on the mechanism of the organ, with a chapter on tuning by John Broadhouse
- Construction, Tuning and Care of the Piano-forte (1887) by Edward Quincy Norton
- Regulation and Repair of Piano and Player Mechanism, Together with Tuning as Science and Art (1909) by William Braid White
- Modern piano tuning and allied arts (1917) by William Braid White (1878-1959)