Musical tuning

"Open string" redirects here. For the concept in string theory, see String (physics).

Two differently tuned thirds: Just major third  Play (help·info).

And the slightly wider: Pythagorean major third  Play (help·info).

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

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 (sharp) 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:

• Tuning to a pitch with one's voice is called matching pitch and is the most basic skill learned in ear training.
• Turning pegs to increase or decrease the tension 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 violin can be tuned manually.
• 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 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 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 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.

Open strings

Main article: Stringed instrument tunings

The pitches of open strings on a violin

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.

violin, mandolin, Irish tenor banjo	G, D, A, E
viola, cello, tenor banjo, mandola, mandocello, tenor guitar	C, G, D, A
double bass, mando-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)
5-string banjo	G, D, G, B, D
cavaquinho	D, G, B, D (standard Brazilian tuning)

Altered tunings

Main article: scordatura

Violin scordatura was employed in the 17th and 18th centuries by Italian and German composers, namely, Biagio Marini, 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.

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) forms the natural sounding perfect twelfth, or perfect fifth (ratio of 2:3) when octave-reduced

More complex musical effects can be created through other relationships.^[1]

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 algebra, topology and group theory.

Systems for the twelve-note chromatic scale

Comparison of tunings: I IV Play just (help·info),  Play Pythagorean (help·info),  Play meantone (help·info) (quarter-comma),  Play well temperament (help·info) (Werckmeister), and  Play equal temperament (help·info)

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 a 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. 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.

• 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 meantone, 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 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.

• Equal temperament

A special case of meantone temperament (extended eleventh-comma), in which adjacent notes of the scale are all separated by logarithmically equal distances (100 cents): 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 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.... 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 temperament (extended third-comma meantone), 31 equal temperament (extended quarter-comma meantone) and 53 equal temperament (extended Pythagorean tuning).

• Syntonic temperament

A tuning system which subsumes nearly all of the above tuning systems.^[2] For example, of the regular temperaments, "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 keyboard.^[3] So do many irregular tunings such as well temperaments and Just Intonation tunings.^[4]

• 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,^[5] 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 keyboard, is a central component of Dynamic Tonality (ibid., Milne et al., 2009).

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

Other scale systems

• Natural overtone scale, a scale derived from the harmonic series.
• Slendro, a pentatonic scale used in Indonesian music.
• Pelog, the other main gamelan scale.
• 43-tone scale, created by Harry Partch, an American composer.
• Bohlen–Pierce scale
• Alpha, beta, delta, and gamma scales of Wendy Carlos.
• Quarter tone scale.
• Thirteenth Sound
• 19 equal temperament
• 22 equal temperament
• 31 equal temperament
• 53 equal temperament
• Schismatic temperament
• Miracle temperament
• Hexany

See also

Music portal

• 3rd Bridge
• Beauty in the Beast
• Chinese musicology
• Electronic tuner
• Ethnomusicology
• Florence Foster Jenkins
• Mathematics of musical scales
• Microtonal music
• Microtuner
• MIDI
• MIDI tuning standard
• Musical theory
• Open chord
• Physics of music
• Pseudo-octave
• Psychoacoustics
• Standard tuning
• Stretched tuning
• String (physics)
• Xenharmonic

References

1. ^ W. A. Mathieu (1997) Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression. Inner Traditions
2. ^ Milne, A., Sethares, W.A. and Plamondon, J., Invariant Fingerings Across a Tuning Continuum, Computer Music Journal, Winter 2007, Vol. 31, No. 4, Pages 15-32.
3. ^ Milne, A., Sethares, W.A. and Plamondon, J., Tuning Continua and Keyboard Layouts, Journal of Mathematics and Music, Spring 2008.
4. ^ Milne, A., Sethares, W.A., Tiedje, S., Prechtl, A., and Plamondon, J., Spectral Tools for Dynamic Tonality and Audio Morphing, Computer Music Journal, Spring 2009 (in press).
5. ^ Sethares, W.A. (1993), Local consonance and the relationship between timbre and scale. Journal of the Acoustical Society of America, 94(1): 1218. (A non-technical version of the article is available at [1])

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 intonation, meantone and 12-equal, featuring Pachelbel's canon.
• The Alternate Tunings Mailing List
• Schmidt-Jones, Catherine. (November 22, 2005). Tuning Systems. Connexions
• 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