Musical tuning
Encyclopedia
In music
, there are two common meanings for tuning:
) or too low (flat) in relation to a given reference pitch. While an instrument might be in tune relative to its own range of notes, it may not be considered 'in tune' if it does not match A = 440 Hz (or whatever reference pitch one might be using). Some instruments become 'out of tune' with damage or time and have to be readjusted or repaired.
Different methods of sound production require different methods of adjustment
:
Some instruments produce a sound which contains irregular overtones in the harmonic series
, and are known as inharmonic
.
Tuning may be done aurally by sounding two pitches and adjusting one of them to match or relate to the other. A tuning fork
or electronic tuning device may be used as a reference pitch, though in ensemble rehearsals often a piano
is used (as its pitch cannot be adjusted for each rehearsal). Symphony orchestras tend to tune to an A
provided by the principal oboist
.
Interference beat
s are used to objectively measure the accuracy of tuning. As the two pitches approach a harmonic relationship, the frequency of beating decreases. When tuning a unison or octave it is desired to reduce the beating frequency until it cannot be detected. For other intervals, this is dependent on the tuning system being used.
Harmonic
s may be used to facilitate tuning of strings which are not themselves tuned to the unison. For example, lightly touching the highest string of a cello at the middle (at a node
) while bowing produces the same pitch as doing the same one third of the way down its second-highest string. The resulting unison is more easily and quickly judged than the quality of the perfect fifth between the fundamentals of the two strings.
In music
, the term open string refers to the fundamental note of the unstopped, full string.
The strings of a guitar
are normally tuned to fourths
(excepting the G and B strings in standard tuning, which are tuned to a third), as are the strings of the bass guitar
and double bass
. Violin
, viola
, and cello
strings are tuned to fifths
. However, non-standard tunings (called scordatura
) exist to change the sound of the instrument or create other playing options.
To tune an instrument, often only one reference pitch is given. This reference is used to tune one string, to which the other strings are tuned in the desired intervals. On a guitar, often the lowest string is tuned to an E. From this, each successive string can be tuned by fingering the fifth fret of an already tuned string and comparing it with the next higher string played open. This works with the exception of the G string, which must be stopped at the fourth fret to sound B against the open B string above. Alternatively, each string can be tuned to its own reference tone.
This table lists open strings on some common string instruments and their standard tunings.
, Antonio Vivaldi
, Heinrich Ignaz Franz Biber (who in the Rosary Sonatas prescribes a great variety of scordaturas, including crossing the middle strings), Johann Pachelbel
and Johann Sebastian Bach
, whose Fifth Suite For Unaccompanied Cello
calls for the lowering of the A string to G. In Mozart
's Sinfonia Concertante
in E-flat major (K. 364), all the strings of the solo viola are raised one half-step, ostensibly to give the instrument a brighter tone so as not to be overshadowed by the solo violin.
Scordatura for the violin was also used in the 19th and 20th centuries in works by Niccolò Paganini
, Robert Schumann
, Camille Saint-Saëns
and Béla Bartók
. In Saint-Saëns' "Danse Macabre
", the high string of the violin is lower half a tone to the E so as to have the most accented note of the main theme sound on an open string. In Bartók's Contrasts, the violin is tuned G-D-A-E to facilitate the playing of tritones on open strings.
American folk violinists of the Appalachians and Ozarks often employ alternate tunings for dance songs and ballads. The most commonly used tuning is A-E-A-E. Likewise banjo players in this tradition employ many tunings in order to play melody in different keys - a common alternative is A-D-A-D-E for playing in D.
Many Folk guitar players also used different tunings from standard, such as D-A-D-G-A-D, which is very popular for Irish music.
A musical instrument which has had its pitch deliberately lowered during tuning is colloquially said to be "down-tuned". Common examples include the electric guitar and electric bass in contemporary heavy metal music
, whereby one or more strings are often tuned lower than concert pitch
. This is not to be confused with electronically changing the fundamental frequency
, which is referred to as pitch shifting.
, or pitch
es, to use when playing music
. In other words, it is the choice of number and spacing of frequency
values which are used.
Due to the psychoacoustic
interaction of tones and timbres, various tone combinations will sound more or less "natural" when used in combination with various timbres. For example, using harmonic
timbres,
More complex musical effects can be created through other relationships.
The creation of a tuning system is complicated because musicians want to make music with more than just a few differing tones. As the number of tones is increased, conflicts arise in how each tone combines with every other. Finding a successful combination of tunings has been the cause of debate, and has led to the creation of many different tuning systems across the world. Each tuning system has its own characteristics, strengths and weaknesses.
, topology
and group theory
.
so that all interval
s are "perfect"; many different methods with their own various compromises have thus been put forward. The main ones are:
Tuning systems that are not produced with exclusively just intervals are usually referred to as temperaments
.
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, there are two common meanings for tuning:
- Tuning practice, the act of tuning an instrument or voice.
- Tuning systems, the various systems of pitchesPitch (music)Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
used to tune an instrument, and their theoretical bases.
Tuning practice
Tuning is the process of adjusting the pitch of one or many tones from musical instruments to establish typical intervals between these tones. Tuning is usually based on a fixed reference, such as A = 440 Hz. Out of tune refers to a pitch/tone that is either too high (sharpSharp (music)
In music, sharp, dièse , or diesis means higher in pitch and the sharp symbol raises a note by a half tone. Intonation may be flat, sharp, or both, successively or simultaneously...
) or too low (flat) in relation to a given reference pitch. While an instrument might be in tune relative to its own range of notes, it may not be considered 'in tune' if it does not match A = 440 Hz (or whatever reference pitch one might be using). Some instruments become 'out of tune' with damage or time and have to be readjusted or repaired.
Different methods of sound production require different methods of adjustment
Adjustment
Adjustment means regulating, adapting or settling in a variety of contexts:...
:
- Tuning to a pitch with one's voice is called matching pitch and is the most basic skill learned in ear trainingEar trainingEar training or aural skills is a skill by which musicians learn to identify, solely by hearing, pitches, intervals, melody, chords, rhythms, and other basic elements of music. The application of this skill is analogous to taking dictation in written/spoken language. Ear training may be...
. - Turning pegsTuning pegA tuning peg is used to hold a string in the pegbox of a stringed instrument. It may be made of ebony, rosewood, boxwood or other material. Some tuning pegs are ornamented with shell, metal, or plastic inlays, beads or rings....
to increase or decrease the tensionTension (mechanics)In physics, tension is the magnitude of the pulling force exerted by a string, cable, chain, or similar object on another object. It is the opposite of compression. As tension is the magnitude of a force, it is measured in newtons and is always measured parallel to the string on which it applies...
on strings so as to control the pitch. Instruments such as the harp, piano, and harpsichord require a wrench to turn the tuning pegs, while others such as the violinViolinThe violin is a string instrument, usually with four strings tuned in perfect fifths. It is the smallest, highest-pitched member of the violin family of string instruments, which includes the viola and cello....
can be tuned manually. - Modifying the length or width of the tube of a wind instrumentWind instrumentA wind instrument is a musical instrument that contains some type of resonator , in which a column of air is set into vibration by the player blowing into a mouthpiece set at the end of the resonator. The pitch of the vibration is determined by the length of the tube and by manual modifications of...
, brass instrumentBrass instrumentA brass instrument is a musical instrument whose sound is produced by sympathetic vibration of air in a tubular resonator in sympathy with the vibration of the player's lips...
, pipe, bellBell (instrument)A bell is a simple sound-making device. The bell is a percussion instrument and an idiophone. Its form is usually a hollow, cup-shaped object, which resonates upon being struck...
, or similar instrument to adjust the pitch.
Some instruments produce a sound which contains irregular overtones in the harmonic series
Harmonic series (music)
Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
, and are known as inharmonic
Inharmonicity
In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency....
.
Tuning may be done aurally by sounding two pitches and adjusting one of them to match or relate to the other. A tuning fork
Tuning fork
A tuning fork is an acoustic resonator in the form of a two-pronged fork with the prongs formed from a U-shaped bar of elastic metal . It resonates at a specific constant pitch when set vibrating by striking it against a surface or with an object, and emits a pure musical tone after waiting a...
or electronic tuning device may be used as a reference pitch, though in ensemble rehearsals often a piano
Piano
The piano is a musical instrument played by means of a keyboard. It is one of the most popular instruments in the world. Widely used in classical and jazz music for solo performances, ensemble use, chamber music and accompaniment, the piano is also very popular as an aid to composing and rehearsal...
is used (as its pitch cannot be adjusted for each rehearsal). Symphony orchestras tend to tune to an A
A (musical note)
La or A is the sixth note of the solfège. "A" is generally used as a standard for tuning. When the orchestra tunes, the oboe plays an "A" and the rest of the instruments tune to match that pitch. Every string instrument in the orchestra has an A string, from which each player can tune the rest of...
provided by the principal oboist
Oboe
The oboe is a double reed musical instrument of the woodwind family. In English, prior to 1770, the instrument was called "hautbois" , "hoboy", or "French hoboy". The spelling "oboe" was adopted into English ca...
.
Interference beat
Beat (acoustics)
In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies....
s are used to objectively measure the accuracy of tuning. As the two pitches approach a harmonic relationship, the frequency of beating decreases. When tuning a unison or octave it is desired to reduce the beating frequency until it cannot be detected. For other intervals, this is dependent on the tuning system being used.
Harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
s may be used to facilitate tuning of strings which are not themselves tuned to the unison. For example, lightly touching the highest string of a cello at the middle (at a node
Node (physics)
A node is a point along a standing wave where the wave has minimal amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the...
) while bowing produces the same pitch as doing the same one third of the way down its second-highest string. The resulting unison is more easily and quickly judged than the quality of the perfect fifth between the fundamentals of the two strings.
Open strings
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, the term open string refers to the fundamental note of the unstopped, full string.
The strings of a guitar
Guitar
The guitar is a plucked string instrument, usually played with fingers or a pick. The guitar consists of a body with a rigid neck to which the strings, generally six in number, are attached. Guitars are traditionally constructed of various woods and strung with animal gut or, more recently, with...
are normally tuned to fourths
Perfect fourth
In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...
(excepting the G and B strings in standard tuning, which are tuned to a third), as are the strings of the bass guitar
Bass guitar
The bass guitar is a stringed instrument played primarily with the fingers or thumb , or by using a pick....
and double bass
Double bass
The double bass, also called the string bass, upright bass, standup bass or contrabass, is the largest and lowest-pitched bowed string instrument in the modern symphony orchestra, with strings usually tuned to E1, A1, D2 and G2...
. Violin
Violin
The violin is a string instrument, usually with four strings tuned in perfect fifths. It is the smallest, highest-pitched member of the violin family of string instruments, which includes the viola and cello....
, viola
Viola
The viola is a bowed string instrument. It is the middle voice of the violin family, between the violin and the cello.- Form :The viola is similar in material and construction to the violin. A full-size viola's body is between and longer than the body of a full-size violin , with an average...
, and cello
Cello
The cello is a bowed string instrument with four strings tuned in perfect fifths. It is a member of the violin family of musical instruments, which also includes the violin, viola, and double bass. Old forms of the instrument in the Baroque era are baryton and viol .A person who plays a cello is...
strings are tuned to fifths
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...
. However, non-standard tunings (called scordatura
Scordatura
A scordatura , also called cross-tuning, is an alternative tuning used for the open strings of a string instrument, in which the notes indicated in the score would represent the finger position as if played in regular tuning, while the actual pitch is altered...
) exist to change the sound of the instrument or create other playing options.
To tune an instrument, often only one reference pitch is given. This reference is used to tune one string, to which the other strings are tuned in the desired intervals. On a guitar, often the lowest string is tuned to an E. From this, each successive string can be tuned by fingering the fifth fret of an already tuned string and comparing it with the next higher string played open. This works with the exception of the G string, which must be stopped at the fourth fret to sound B against the open B string above. Alternatively, each string can be tuned to its own reference tone.
This table lists open strings on some common string instruments and their standard tunings.
| violin Violin The violin is a string instrument, usually with four strings tuned in perfect fifths. It is the smallest, highest-pitched member of the violin family of string instruments, which includes the viola and cello.... , mandolin Mandolin A mandolin is a musical instrument in the lute family . It descends from the mandore, a soprano member of the lute family. The mandolin soundboard comes in many shapes—but generally round or teardrop-shaped, sometimes with scrolls or other projections. A mandolin may have f-holes, or a single... , Irish tenor banjo |
G, D, A, E |
| viola Viola The viola is a bowed string instrument. It is the middle voice of the violin family, between the violin and the cello.- Form :The viola is similar in material and construction to the violin. A full-size viola's body is between and longer than the body of a full-size violin , with an average... , cello Cello The cello is a bowed string instrument with four strings tuned in perfect fifths. It is a member of the violin family of musical instruments, which also includes the violin, viola, and double bass. Old forms of the instrument in the Baroque era are baryton and viol .A person who plays a cello is... , tenor banjo, mandola Mandola The mandola or tenor mandola is a fretted, stringed musical instrument. It is to the mandolin what the viola is to the violin: the four double courses of strings tuned in fifths to the same pitches as the viola , a fifth lower than a mandolin... , mandocello Mandocello The mandocello is a plucked string instrument of the mandolin family. It has eight strings in four paired courses, tuned in 5ths like a mandolin, but is larger, and tuned CC-GG-dd-aa . It is to the mandolin what the cello is to the violin.-Construction:Mandocello construction is similar to the... , tenor guitar Tenor guitar 1932 Martin 0-18 T Sunburst Tenor Guitar|thumb|rightThe tenor guitar or four-string guitar is a slightly smaller, four-string relative of the steel-string acoustic guitar or electric guitar. The instrument was developed so that players of the four-string tenor banjo could double on the guitar... |
C, G, D, A |
| double bass Double bass The double bass, also called the string bass, upright bass, standup bass or contrabass, is the largest and lowest-pitched bowed string instrument in the modern symphony orchestra, with strings usually tuned to E1, A1, D2 and G2... , mando-bass, bass guitar Bass guitar The bass guitar is a stringed instrument played primarily with the fingers or thumb , or by using a pick.... * |
(B*,) E, A, D, G |
| guitar Guitar The guitar is a plucked string instrument, usually played with fingers or a pick. The guitar consists of a body with a rigid neck to which the strings, generally six in number, are attached. Guitars are traditionally constructed of various woods and strung with animal gut or, more recently, with... |
E, A, D, G, B, E |
| ukulele Ukulele The ukulele, ; from ; it is a subset of the guitar family of instruments, generally with four nylon or gut strings or four courses of strings.... |
G, C, E, A (the G string is higher than the C and E, and two half steps below the A string, known as reentrant tuning Reentrant tuning A reentrant tuning is a tuning of a stringed instrument where the strings are not ordered from the lowest pitch to the highest pitch .... ) |
| 5-string banjo | G, D, G, B, D |
| cavaquinho Cavaquinho The cavaquinho is a small string instrument of the European guitar family with four wire or gut strings. It is also called machimbo, machim, machete , manchete or marchete, braguinha or braguinho, or cavaco.The most common tuning is D-G-B-D ; other tunings include D-A-B-E... |
D, G, B, D (standard Brazilian tuning) |
Altered tunings
Violin scordatura was employed in the 17th and 18th centuries by Italian and German composers, namely, Biagio MariniBiagio Marini
Biagio Marini was an Italian virtuoso violinist and composer of the first half of the seventeenth century.Marini was born in Brescia. His works were printed and influential throughout the European musical world...
, Antonio Vivaldi
Antonio Vivaldi
Antonio Lucio Vivaldi , nicknamed because of his red hair, was an Italian Baroque composer, priest, and virtuoso violinist, born in Venice. Vivaldi is recognized as one of the greatest Baroque composers, and his influence during his lifetime was widespread over Europe...
, Heinrich Ignaz Franz Biber (who in the Rosary Sonatas prescribes a great variety of scordaturas, including crossing the middle strings), Johann Pachelbel
Johann Pachelbel
Johann Pachelbel was a German Baroque composer, organist and teacher, who brought the south German organ tradition to its peak. He composed a large body of sacred and secular music, and his contributions to the development of the chorale prelude and fugue have earned him a place among the most...
and Johann Sebastian Bach
Johann Sebastian Bach
Johann Sebastian Bach was a German composer, organist, harpsichordist, violist, and violinist whose sacred and secular works for choir, orchestra, and solo instruments drew together the strands of the Baroque period and brought it to its ultimate maturity...
, whose Fifth Suite For Unaccompanied Cello
Cello Suites (Bach)
The Six Suites for Unaccompanied Cello by Johann Sebastian Bach are some of the most performed and recognizable solo compositions ever written for cello...
calls for the lowering of the A string to G. In Mozart
Wolfgang Amadeus Mozart
Wolfgang Amadeus Mozart , baptismal name Johannes Chrysostomus Wolfgangus Theophilus Mozart , was a prolific and influential composer of the Classical era. He composed over 600 works, many acknowledged as pinnacles of symphonic, concertante, chamber, piano, operatic, and choral music...
's Sinfonia Concertante
Sinfonia Concertante for Violin, Viola and Orchestra
The Sinfonia Concertante for Violin, Viola and Orchestra in E-flat major, K. 364 , was written by Wolfgang Amadeus Mozart.At the time of its composition in 1779, Mozart was on a tour of Europe that included Mannheim and Paris...
in E-flat major (K. 364), all the strings of the solo viola are raised one half-step, ostensibly to give the instrument a brighter tone so as not to be overshadowed by the solo violin.
Scordatura for the violin was also used in the 19th and 20th centuries in works by Niccolò Paganini
Niccolò Paganini
Niccolò Paganini was an Italian violinist, violist, guitarist, and composer. He was one of the most celebrated violin virtuosi of his time, and left his mark as one of the pillars of modern violin technique...
, Robert Schumann
Robert Schumann
Robert Schumann, sometimes known as Robert Alexander Schumann, was a German composer, aesthete and influential music critic. He is regarded as one of the greatest and most representative composers of the Romantic era....
, Camille Saint-Saëns
Camille Saint-Saëns
Charles-Camille Saint-Saëns was a French Late-Romantic composer, organist, conductor, and pianist. He is known especially for The Carnival of the Animals, Danse macabre, Samson and Delilah, Piano Concerto No. 2, Cello Concerto No. 1, Havanaise, Introduction and Rondo Capriccioso, and his Symphony...
and Béla Bartók
Béla Bartók
Béla Viktor János Bartók was a Hungarian composer and pianist. He is considered one of the most important composers of the 20th century and is regarded, along with Liszt, as Hungary's greatest composer...
. In Saint-Saëns' "Danse Macabre
Danse Macabre (Saint-Saëns)
Danse macabre, Op. 40, is a tone poem for orchestra, written in 1874 by French composer Camille Saint-Saëns. It started out in 1872 as an art song for voice and piano with a French text by the poet Henri Cazalis, which is based in an old French superstition...
", the high string of the violin is lower half a tone to the E so as to have the most accented note of the main theme sound on an open string. In Bartók's Contrasts, the violin is tuned G-D-A-E to facilitate the playing of tritones on open strings.
American folk violinists of the Appalachians and Ozarks often employ alternate tunings for dance songs and ballads. The most commonly used tuning is A-E-A-E. Likewise banjo players in this tradition employ many tunings in order to play melody in different keys - a common alternative is A-D-A-D-E for playing in D.
Many Folk guitar players also used different tunings from standard, such as D-A-D-G-A-D, which is very popular for Irish music.
A musical instrument which has had its pitch deliberately lowered during tuning is colloquially said to be "down-tuned". Common examples include the electric guitar and electric bass in contemporary heavy metal music
Heavy metal music
Heavy metal is a genre of rock music that developed in the late 1960s and early 1970s, largely in the Midlands of the United Kingdom and the United States...
, whereby one or more strings are often tuned lower than concert pitch
Concert pitch
Concert pitch refers to the pitch reference to which a group of musical instruments are tuned for a performance. Concert pitch may vary from ensemble to ensemble, and has varied widely over musical history...
. This is not to be confused with electronically changing the fundamental frequency
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
, which is referred to as pitch shifting.
Tuning systems
A tuning system is the system used to define which tonesSemitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
, or pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
es, to use when playing music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
. In other words, it is the choice of number and spacing of frequency
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...
values which are used.
Due to the psychoacoustic
Psychoacoustics
Psychoacoustics is the scientific study of sound perception. More specifically, it is the branch of science studying the psychological and physiological responses associated with sound...
interaction of tones and timbres, various tone combinations will sound more or less "natural" when used in combination with various timbres. For example, using harmonic
Harmonic series (music)
Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
timbres,
- a tone caused by a vibration twice the speed of another (the ratio of 1:2) forms the natural sounding octaveOctaveIn music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
- a tone caused by a vibration three times the speed of another (the ratio of 1:3) forms the natural sounding perfect twelfth, or perfect fifthPerfect fifthIn 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...
(ratio of 2:3) when octave-reduced
More complex musical effects can be created through other relationships.
The creation of a tuning system is complicated because musicians want to make music with more than just a few differing tones. As the number of tones is increased, conflicts arise in how each tone combines with every other. Finding a successful combination of tunings has been the cause of debate, and has led to the creation of many different tuning systems across the world. Each tuning system has its own characteristics, strengths and weaknesses.
Theoretical comparison
There are many techniques for theoretical comparison of tunings, usually utilizing mathematical tools such as those of linear algebraLinear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
.
Systems for the twelve-note chromatic scale
It is impossible to tune the twelve-note chromatic scaleChromatic scale
The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...
so that all interval
Interval (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...
s are "perfect"; many different methods with their own various compromises have thus been put forward. The main ones are:
- Just intonationJust intonationIn 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 Just Intonation the frequencies of the scale notes are related to one another by simple numeric ratios, a common example of this being 1:1, 9:8, 5:4, 4:3, 3:2, 5:3, 15:8, 2:1 to define the ratios for the 7 notes in a C major scale. In theory a variety of approaches are possible, such as basing the tuning of pitches on the harmonic series (music)Harmonic series (music)Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
, which are all whole number multiples of a single tone. In practice however this quickly leads to potential for confusion depending on context, especially in the larger system of 12 chromatic notes used in the West. For instance, a major second may end up either in the ratio 9:8 or 10:9. For this reason, just intonation may be a less suitable system for use on keyboard instrumentKeyboard instrumentA keyboard instrument is a musical instrument which is played using a musical keyboard. The most common of these is the piano. Other widely used keyboard instruments include organs of various types as well as other mechanical, electromechanical and electronic instruments...
s or other instruments where the pitch of individual notes is not flexible. (On fretted instruments like guitars and lutes, multiple frets for one interval can be practical.)- Pythagorean tuningPythagorean tuningPythagorean 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
- A Pythagorean tuning is technically a type of just intonation, in which the frequency ratios of the notes are all derived from the number ratio 3:2. Using this approach for example, the 12 notes of the Western chromatic scale would be tuned to the following ratios: 1:1, 256:243, 9:8, 32:27, 81:64, 4:3, 729:512, 3:2, 128:81, 27:16, 16:9, 243:128, 2:1. Also called "3-limit" because there are no prime factors other than 2 and 3, this Pythagorean system was of primary importance in Western musical development in the Medieval and Renaissance periods. See also: Shí-èr-lǜ.
- Meantone temperamentMeantone temperamentMeantone 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...
- Meantone temperament
- A system of tuning which averages out pairs of ratios used for the same interval (such as 9:8 and 10:9). The best known form of this temperament is quarter-comma meantoneQuarter-comma meantoneQuarter-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...
, which tunes major thirds justly in the ratio of 5:4 and divides them into two whole tones of equal size. However, the fifth may be flattened to a greater or lesser degree than this and the tuning system will retain the essential qualities of meantone temperament; historical examples include 1/3- and 2/7th comma meantone .- Well temperamentWell temperamentWell 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...
- Well temperament
- Any one of a number of systems where the ratios between intervals are unequal, but approximate to ratios used in just intonation. Unlike meantone temperament, the amount of divergence from just ratios varies according to the exact notes being tuned, so that C-E will probably be tuned closer to a 5:4 ratio than, say, D-F. Because of this, well temperaments have no wolf intervalWolf intervalIn 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...
s.- 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...
- Equal temperament
- A special case of meantone temperament (extended eleventh-comma), in which adjacent notes of the scale are all separated by logarithmLogarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
ically equal distances (100 centsCent (music)The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
): A harmonized C major scale in equal temperament (.ogg format, 96.9KB). This is the most common tuning system used in Western music, and is the standard system for tuning a pianoPiano 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...
. Since this scale divides an octave into twelve equal-ratio steps and an octave has a frequency ratio of two, the frequency ratio between adjacent notes is then the twelfth root of two, 21/12, or ~1.05946309.... However, the octave can be divided into other than 12 equal divisions, some of which may be more harmonically pleasing as far as thirds and sixths are concerned, such as 19 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...
(extended third-comma meantone), 31 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...
(extended quarter-comma meantone) and 53 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...
(extended Pythagorean tuning).- Syntonic temperamentSyntonic temperamentThe syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
- Syntonic temperament
- A tuning system which subsumes nearly all of the above tuning systems. For example, of the 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...
s, "equal temperament" is the syntonic tuning in which the tempered perfect fifth (P5) is 700 cents wide; 1/4-comma meantone is the syntonic tuning in which the P5 is 696.6 cents wide; Pythagorean tuning is the syntonic tuning in which the P5 is 702 cents wide; 5-equal is the syntonic tuning in which the P5 is 720 cents wide; and 7-equal is the tuning in which the P5 is 686 cents wide. All of these syntonic tunings have identical fingering on an isomorphic keyboardIsomorphic keyboardAn isomorphic keyboard is a musical input device consisting of a two-dimensional array of note-controlling elements on which any given sequence and/or combination of musical intervals has the “same shape” on the keyboard wherever it occurs – within a key, across keys, across octaves, and across...
. So do many irregular tunings such as well temperaments and Just Intonation tunings.- Tempered timbres
- A timbre's partials (also known as harmonics or overtones) can be tempered such that each of the timbre's partials aligns with a note of a given tempered tuning. This alignment of tuning and timbre is a key component in the perception of consonance, of which one notable example is the alignment between the partials of a harmonic timbre and a Just Intonation tuning. Hence, using tempered timbres, one can achieve a degree of consonance, in any tempered tuning, that is comparable to the consonance achieved by the combination of Just Intonation tuning and harmonic timbres. Tempering timbres in real time, to match a tuning that can change smoothly in real time, using the tuning-invariant fingering of an isomorphic keyboardIsomorphic keyboardAn isomorphic keyboard is a musical input device consisting of a two-dimensional array of note-controlling elements on which any given sequence and/or combination of musical intervals has the “same shape” on the keyboard wherever it occurs – within a key, across keys, across octaves, and across...
, is a central component of Dynamic TonalityDynamic tonalityDynamic tonality is tonal music which uses real-time changes in tuning and timbre to perform new musical effects such as polyphonic tuning bends, new chord progressions, and temperament modulations, with the option of consonance. The performance of dynamic tonality requires an isomorphic keyboard...
(ibid., Milne et al., 2009).
Tuning systems that are not produced with exclusively just intervals are usually referred to as temperaments
Musical temperament
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...
.
Other scale systems
- Natural overtone scale, a scale derived from the harmonic seriesHarmonic series (music)Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
. - SlendroSlendroSlendro is a pentatonic scale, one of the two most common scales used in Indonesian gamelan music, the other being pélog.-Tuning:...
, a pentatonic scalePentatonic scaleA pentatonic scale is a musical scale with five notes per octave in contrast to a heptatonic scale such as the major scale and minor scale...
used in IndonesiaIndonesiaIndonesia , officially the Republic of Indonesia , is a country in Southeast Asia and Oceania. Indonesia is an archipelago comprising approximately 13,000 islands. It has 33 provinces with over 238 million people, and is the world's fourth most populous country. Indonesia is a republic, with an...
n music. - PelogPelogPelog is one of the two essential scales of gamelan music native to Bali and Java, in Indonesia. The other scale commonly used is called slendro. Pelog has seven notes, but many gamelan ensembles only have keys for five of the pitches...
, the other main gamelan scale. - 43-tone scaleHarry Partch's 43-tone scaleThe 43-tone scale is a just intonation scale with 43 pitches in each octave, invented and used by Harry Partch.The first of Partch's "four concepts" is "The scale of musical intervals begins with absolute consonance and gradually progresses into an infinitude of dissonance, the consonance of the...
, created by Harry PartchHarry PartchHarry Partch was an American composer and instrument creator. He was one of the first twentieth-century composers to work extensively and systematically with microtonal scales, writing much of his music for custom-made instruments that he built himself, tuned in 11-limit just intonation.-Early...
, an American composerComposerA composer is a person who creates music, either by musical notation or oral tradition, for interpretation and performance, or through direct manipulation of sonic material through electronic media...
. - Bohlen–Pierce scale
- AlphaAlpha scaleThe α scale is a non-octave-repeating musical scale which splits the minor third into two equal parts, or four equal parts of approximately 78 cents each . This totals approximately 15.39 steps per octave...
, betaBeta scaleThe β scale is a non-octave-repeating musical scale which splits the perfect fourth into two equal parts, or eight equal parts of approximately 64 cents each . This totals approximately 18.75 steps per octave...
, deltaDelta scaleThe δ scale is a non-octave repeating musical scale. It may be regarded as the beta scale's reciprocal since it is, "as far 'down' the circle from α as β is 'up.'" As such it would split the minor second into eight equal parts of approximately 14 cents each...
, and gamma scaleGamma scaleThe γ scale is a non-octave repeating musical scale which splits the neutral third into two equal parts, or ten equal parts of 35 cents each...
s of Wendy CarlosWendy CarlosWendy Carlos is an American composer and electronic musician. Carlos first came to notice in the late 1960s with recordings made on the Moog synthesizer, then a relatively new and unknown instrument; most notable were LPs of synthesized Bach and the soundtrack for Stanley Kubrick's film A...
. - Quarter tone scale.
- Thirteenth Sound
- 19 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 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 ....
- 31 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...
- 53 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...
- 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:...
- Miracle temperamentMiracle temperamentIn music, miracle temperament is a regular temperament discovered by George Secor which has as a generator an interval, called the secor, that serves as both the 15:14 and 16:15 semitones. Because 15:14 and 16:15 are equated, their ratio 225:224 \left is tempered out, and two secors give an 8:7...
- HexanyHexanyIn music theory, the hexany is a six-note just intonation scale, with the notes placed on the vertices of an octahedron, equivalently the faces of a cube...
Further reading
- J. Murray Barbour (?). Tuning and Temperament: A Historical Survey. ISBN 0-486-43406-0.
External links
- Alternate Tunings For Bass
- Comparison among three different temperaments: Just intonationJust intonationIn 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...
, meantoneMeantone temperamentMeantone 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...
and 12-equalEqual 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...
, featuring Pachelbel's canon. - The Alternate Tunings Mailing List
- Schmidt-Jones, Catherine. (November 22, 2005). Tuning Systems. ConnexionsConnexionsConnexions is a global repository of educational content provided by Rice University. The entire collection is available free of charge, and students and learners alike can explore all the content they desire....
- History of tuning by Kyle Gann
- Article on the history of absolute pitch to today's A=440 standard
- Understanding Temperaments, a concise, but comprehensive explanation of temperaments.
- Pythagorean tuning, detailed definition and explanation.
- Temperament resources on the Web, lots more links
- Albert Gräf: Musical scale rationalization — a graph-theoretic approach (the paper is number 45 on the list, and is in .pdf format)
- HyperPhysics: Temperament and Musical Scales
- The Keyboard Tuning of Domenico Scarlatti
- Bach Tuning
- Stopper Scale — equal temperament based on the 19th root of 3 (1988)
- Let's try Musical Tuning and Temperament on your computer!
- Barbieri, Patrizio. Enharmonic instruments and music, 1470-1900. (2008) Latina, Il Levante Libreria Editrice