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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 basis.

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 or too low, or sharp or flat, respectively. While an instrument might be in tune in itself (relative tuning) but is not on A = 440 Hz when playing with a piano or other fixed-pitch instrument nobody will agree it is in tune as it fails to match the reference pitch. Some instruments get 'out of tune' with damage or age when they will no longer play true and have to be repaired.

Different methods of sound production require different methods of adjustment:

Tuning to a pitch with one's voice is called matching pitch and is the most basic skill learned in ear training.
Manually turning the pegs to increase or decrease the tension on the strings so as to control the pitch. Instruments such as the harp, piano, and harpsichord require a wrench to turn the tuning pegs.
Modifying the length or width of the tube of a wind instrument, brass instrument, pipe, bell, or similar instrument to adjust the pitch.

Some instruments produce a sound which contains irregular overtones harmonic series, and are known as inharmonic. This makes their tuning complicated, and usually compromised. The tuning of bells, for instance, is extremely involved.

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 beats 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.

Harmonics may be used to check the tuning of strings which are not tuned to the unison. For example, lightly touching the highest string of a cello at halfway down its length (at a node) while bowing produces the same pitch as doing the same one third of the way down its second highest string.

Open 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), 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, usually 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.

This table lists open strings on some common string instruments and their standard tunings.

violin, mandolin	G, D, A, E
viola, cello, tenor banjo, mandola, tenor guitar	C, G, D, A
double bass, bass guitar*	(B*,) E, A, D, G
guitar	E, A, D, G, B, E
ukulele	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)

Altered tunings

Unconventional tunings, or scordatura (It., from scordare, to mistune), were first used in the 16th century by Italian lutenists. It was primarily used to facilitate difficult passages, but was also used to alter timbral characteristics, reinforce tonalities through the use of open strings, and to extend the range of the instrument.

Violin scordatura was employed in the 17th and 18th centuries by Italian and German composers, namely, Biagio Marini, Antonio Vivaldi, Heinrich Ignaz Biber - who in the Rosary Sonatas prescribes a great variety of scordaturas, including crossing the middle strings - Johann Pachelbel and J.S. 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. The open D-string then sounds the tonic of the piece, E-flat. However, in modern performance it is often performed without scordatura.

Scordatura for the violin was also used in the 19th and 20th centuries in works by Paganini, Schumann, Saint-Saëns and 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.

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.

Tuning systems

A tuning system is the system used to define which tones, or pitches, 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,

a tone caused by a vibration twice the speed of another (the ratio of 1:2) forms the natural sounding octave
a tone caused by a vibration three times the speed of another (the ratio of 1:3, or 2:3 when octave-reduced) forms the natural sounding perfect fifth.

More complex musical effects can be created through other relationships.

Many cultures whose dominant instruments emit non-harmonic sounds use other tuning systems, in which other intervals sound more "natural.

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.

Using a tuning invariant isomorphic keyboard with a compatible synthesizer enables one to retain consistent fingering across many different tunings, facilitating their exploration.

Theoretical comparison

There are many techniques for theoretical comparison of tunings, usually utilizing mathematical tools such as those of linear algebra, topology and group theory. Techniques of interest include:

Comma, a measure of a tuning system's compromise between just intervals.
Modulatory space, geometrical analysis of transpositional possibilities.
Pitch space, geometrical analysis of tuning systems.
Regular temperament, a system's definition in terms of a small number of generating units.
Tonnetz, an arrangement of a tuning system as a lattice.

Systems for the twelve-note chromatic scale

It is impossible to tune the twelve-note chromatic scale so that all intervals are "perfect"; many different methods with their own various compromises have thus been put forward. The main ones are:

Just intonation

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), 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 less suitable system for use on keyboard instruments 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 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, a ratio of central importance to the School of Pythagoras in Ancient Greece. 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. As a concept it was further developed by Safi ad-Din al-Urmawi, who divided the octave into seventeen parts (limmas and commas) and used in the Turkish and Persian tone systems.

Meantone temperament

A system of tuning which averages out pairs of ratios used for the same interval (such as 9:8 and 10:9), thus making it possible to tune keyboard instruments. Next to the twelve-equal temperament, which some would not regard as a form of meantone, the best known form of this temperament is quarter-comma meantone, which tunes major thirds justly in the ratio of 5:4 and divides them into two whole tones of equal size. To do this, eleven perfect fifths in each octave are flattened by a quarter of a syntonic comma, with the remaining fifth being left very sharp (such an unacceptably out-of-tune fifth is known as a wolf interval). 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; examples include the 31-equal fifth and Lucy tuning.

Both just intonation and meantone temperament can be regarded as forms of regular temperament.
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 intervals. A well temperament system is usually named after whoever first came up with it.

Equal temperament

(a special case of mean-tone temperament), in which adjacent notes of the scale are all separated by logarithmically equal distances (100 cents) - (.ogg format, 96.9KB). This is the most common tuning system used in Western music, and is the standard system for tuning a piano. 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, 2^1/12, or ~1.05946309...

Tempered timbres

, in which the partials of a given timbre are adjusted to align with a tempered tuning's tones. This can be seen as a generalization of the relationship between the Harmonic Series and Just Intonation to a number of different pseudo-tonal tunings and timbres (Sethares 2004). Tempered timbres are a key component of Dynamic Tonality.

Tuning systems that are not produced with exclusively just intervals are usually referred to as temperaments.

Other scale systems

Slendro, a pentatonic scale used in Indonesian music.
Pelog, the other main gamelan scale, with three hemitonic pentatonic modes superimposed to form a scale with seven notes to the octave
43-tone scale, created by Harry Partch, an American composer who wrote musical and dramatic works in just intonation
Bohlen-Pierce scale
LucyTuning, a meantone system advocated by Charles Lucy, related to the number Pi and writings of John Harrison.
Alpha and beta scales of Wendy Carlos
Quarter tone scale, first presented by Mikha'il Mishaqah, used in the theory of Arabic music tone systems. From this the heptatonic scales consisting of minor, neutral, and major seconds of maqamat are chosen, this system was first promoted by al-Farabi using a 25 tone scale.
Thirteenth Sound
19 equal temperament
22 equal temperament
31 equal temperament
53 equal temperament
88 equal temperament
Schismatic temperament
Miracle temperament
Stretched tuning makes an octave represent slightly more than a doubling in frequency. It is usually applied to keyboard instruments with tines or thick strings, where the ratio of harmonic to fundamental can be slightly greater than a true integer ratio (most notably the piano, and some electric pianos).
Hexany

Comparisons and controversies among tunings

All musical tunings have advantages and disadvantages. Twelve tone equal temperament (12-TET) is the standard and most usual tuning system used in Western music today because it gives the advantage of modulation to any key without dramatically going out of tune, as all keys are equally and slightly out of tune. However, just intonation provides the advantage of being entirely in tune, with at least some, and possibly a great deal, loss of ease in modulation. The composer Terry Riley, said "Western music is fast because it's not in tune", meaning that its inherent beating forces motion. Twelve tone equal temperament also, currently, has an advantage over just intonation in that most musicians are trained in, and have instruments designed to play in equal temperament. Other tuning systems have other advantages and disadvantages and are chosen for various qualities.

The octave (or even other intervals, such as the so-called tritave, or twelfth) can advantageously be divided into a number of equal steps different from twelve. Popular choices for such an equal temperament include 19, 22, 31, 53 and 72 parts to an octave, each of these and the many other choices possible have their own distinct characteristics.

The two paragraphs above assume the use of harmonic timbres, in which the partials' placement follows a pattern of ratios of small whole numbers. Western music uses harmonic timbres almost exclusively, so their use is often assumed in discussions of tuning such as this. However, the timbres of the dominant instruments of some other cultures are non-harmonic, and sound most natural in tunings that do not follow ratios of small whole numbers (except perhaps the octave at 2:1). For instance, William Sethares shows that the tunings of Balinese gamelans are related to the inharmonic spectra or timbre of their metallophones and the harmonic spectra of stringed instruments such as the rebab, just as just intonation and twelve tone equal temperament are related to the spectra or timbre of harmonic instruments alone.

Some instruments, such as the violin, don't limit the musician to particular pitches, allowing to choose the tuning system "on the fly". Many performers on such instruments adjust the notes to be more in tune than the equal temperament system allows, perhaps even without realizing it.

Like the violin and other fretless stringed instruments, the pedal steel guitar places absolute control of pitch into the hands of the player. Most steel guitarists tune their instrument to just intonation. The steel guitar is unique among western instruments in its ability to create complex chords in just intonation in any key. Smooth, beatless chords are part of the steel guitar's characteristic sound.

Likewise, using a tuning invariant isomorphic keyboard to drive a Dynamic Tonality-compatible synthesizer, one can change the tuning "on the fly," adjusting the frequencies of the tuning's tones and of the timbres' partials to sound natural in any tuning across a wide tuning range.