Pitch space

Sound system called the Tonvorrat used in a music culture and the organizational principles that are the basis of tonal relationships and determine the functions of individual tones or pitches. Sound systems are therefore a central subject of music theory or musicology and provide a means to compare different musical cultures together.

Many cultures, especially for indigenous peoples, but have themselves often no actual music theory. A sound system can be derived by the different music culture under investigation musicologist or ethnomusicologist by empirical observations of the practice of music and is possibly being used musical instruments here.

A most common feature sound systems, for example, the similarity or identity sensation of tones separated by an octave. In this interval to each other two tones in the frequency ratio of 2:1. They are considered directly related to perceived and can be interpreted as clayey or equivalent.

The modern sound system of Western music

Despite a variety of new experiments, the alternative to today dominant in Western music of the modern era sound system is the diatonic -chromatic - enharmonic sound system, as it appears in the general standard notation. Although its Tonvorrat principle can be extended, it is limited by the notation to 35 levels by additional to each of the seven diatonic naturals four variants by single and double increase or decrease. This results in the following Tonvorrat which more than satisfies the requirements of musical practice in general and is rarely fully utilized:

For a theoretical justification of this sound system, various design principles that are at the same time tuning systems, be used. The two most important are:

Especially with keyboard instruments, the practical implementation of the tonal system is fraught with problems. The Archi harpsichord around which can be understood with its 36 keys per octave in a rough approximation, as an attempt to complete mapping of the tonal system, proved to be practicable because of its unwieldy complexity as little. Therefore, efforts were made to reduce the number of keys on what is absolutely necessary. This reduction was made possible by the fact that many sounds ( eg, his and c, and the cis, dis and it, s and firm, and so on ) are very close together and called themselves only to a very small interval comma differ. Such closely spaced tones, can be collapsed by conscious small as possible upsets to a button so that enharmonic confusion are possible. Among the many tuning systems that have been devised to solve the problem, finally won by the now generally accepted zwölfstufig - equal -tempered tuning.

Since then, the diatonic -chromatic - enharmonic sound system is acoustically represented by a scale with twelve steps per octave ( chromatic scale ), which are separated by semitone fixed size at gleichstufiger mood.

The twelve tones are basically equal - each sound can be a key in Western music dominating major-minor system for root. The tonalities of major-minor system are in their structure but always the same; that is, recognizable as major or minor. The Western music distinguishes two Tongeschlechter with major and minor. The names in C major, F # major, A minor, B flat minor, etc. indicate on which sound from the sound system builds up the major or minor structure and what diatonic tones to the respective scale include. If you keep the same basic Tonfolgestruktur, a melody can be sung or played in principle on any root note. Tunes or entire pieces of music are thus transposed. The individual keys have different meanings but are sometimes on the key characteristics attributed to, but these assignments are subjective and historically variable and overall controversial.

The twelve-tone scale is a scale material, while C major or A minor, etc. use ladders. An exception is in western music atonality and tonality free - here cover the material and utility director.

Marked by signs tones are derivatives or alterations of the original diatonic scale degrees, and listed in the notation as either increased ( ♯ ) or decreased ( ♭ ) variants. This is also suggested to the appointments, modifications of the Note labels are attached by letters or letter combinations.

• For example, as is written as debased a ( ♭ a), as increased gis g ( ♯ g).

Both tones are physically identical but in the same stage mood. Now, because the music written grammar of the major-minor system or functional harmony by ♯ - ♭ and different - keys must be used to maintain the consistency in the notation Note labels that match the respective key.

• For example, in C-sharp major is listed, for example, as a second step, a dis, in D flat major, however it on.

This applies to diatonic, ie gamut own, as well tonart strange sounds that are introduced, for example in the case of modulations. The possibility of different notation is called enharmonic. True enharmonic confusion occur in Note pictures in which that offers the exchange of a ♯ - key in a ♭ - key, or vice versa due to better readability or Sing-/Spielbarkeit for the musicians. In pure and mean tone, however, for example, as and gis are actually physically different sounds. These are also known as split tones, since they share the whole step between g and a just in two different Halbtondistanzen. The difference between as and gis shall be considered as actual enharmonic because confusion is excluded as the basis of the equal temperament tuning. This enharmonic plays eg a cappella choirs, string quartets and orchestras good a decisive role - in particular, if it is intended to come as close as possible to the sound of music from the Middle Ages and classical music.

Derivation and history

After the octave the Western tonal system characterize foremost the fifth and the fourth. On Pythagoras of Samos and the Pythagoreans founded the philosophical belief goes back to the simplest, integer division ratios reflect the harmonious proportions. This applies as for the geometry, the architecture as well as for physical vibrational states such as tones. The frequency ratios of the octave ( frequency ratio 1:2 ), the fifth and fourth ( frequency ratios 2:3 and 3:4 ) be in exact order 1:2:3:4. This number sequence - in a sense the " world formula " of the Pythagoreans; called Tetraktys - give this meeting the gradations of human Konsonanzempfindens two or more tones again. Physical and mathematical principles to meet human Sinnesperzeption. Based on the Pythagorean doctrine of the first tone of a sound system so derives from the octave. The derivation of the other tones of the system is done by Quint stratification. This is based on that the fifth is the first consonant, the scale a, ie distinct diatonic scale degrees, can be derived. Sounds in the distance a fifth are in the first grade, etc. etc. etc. at a distance of two fifths in the second degree as can be seen by the circle of fifths. The fifth and their complementary interval, the fourth, constitute itself as pure and fixed sizes than 702 cents or 498 cents.

• Example of the construction of a diatonic scale on C (C - D - E - F - G - A - h):

However, the original construction of the Western tonal system first takes only consideration of the Pythagorean tuning, which is also known as fifths. The specific interval in the example is the Pythagorean major third ( 64:81, 408 cents), which to the syntonic comma ( 22 cents ) is greater than the pure major third ( 4:5, 386 cents). The bottom stratification is true to some extent as the first theory to explanation of the Western tonal system, since, in accordance with the Pythagorean ideal, with it, the Tonvorrat the pentatonic scale and heptatonic can best be derived. Pentatonic scales and heptatonic apply, in development, in that order, as the Urtonleitern the Western tonal system. Already the sound system of ancient Greece underwent a development of a five-stage ( pentatonic ) to a seven-point ( heptatonically ) system. The pentatonic scale prevails, for example up to the present time in well-preserved folk music traditions. Basically, these scales are considered diatonic scales. Enter only the pitches again, arising from the stratification of four or six fifths. example:

• F - c - g - d - a (4 fifths ) gives the halbtonlose pentatonic scale

• F - c - g - d - a - e - h (6 fifths ) gives the heptatonic scale

The music of the Middle Ages used basically a heptatonisches system, but knew it already hierarchies, so that individual pitches experienced only minor importance. The Gregorian system with the church modes or modes arranged in the basic Tonvorrat scales with different structure and therefore different character. With the polyphony, but especially since the late Middle Ages and early modern musical practice added the possibility of altered scale degrees. This stopped in the notation the sign ♯ and ♭ feeder. But Altered pitches initially did not change even the sound system. They had only sonic character and acted as "only" as an intermediate tones or leading notes in closing formulas and cadences or when changing the Hexachordes (see: Musica ficta ).

In the wake of the highly complex polyphony in the 15th century, but especially with the rapidly increasing development of keyboard instruments in the Baroque period and the establishment of the major-minor system, but the mood was a serious problem. The sound system had expanded due to the musical and compositional practice. In terms of mood problems occur, this could no longer explain alone the Pythagorean fifths stratification. Other explanations for the Western tonal system are, for example, the decomposition of the octave into major and minor thirds. Especially in the setting of thirds but ignites the problem of mood. The fifth is divided into major and minor third ( 4:5 and 5:6 ). The major third is accordingly but for one large and one small whole tone ( 8:9 and 9:10 ), the difference of which 21.51 cents, called the syntonic comma is. As shown above, but the bottom progression determines the greater whole tone ( 8:9, 204 cents) as the authentic. Now there is again an octave of six whole tones - but six major whole tones exceed the Oktavrahmen to 23.46 cents, called the Pythagorean comma. Furthermore, the Western tonal system can also be explained by the overtone series that specifies all twelve tones and their relationships to each other. But even here there are difficulties with the mood.

Over time, as different tuning systems have been tested and developed, one of the main problems, which represented the third.

The long period mean- prevailing moods with many pure thirds approaching the pure atmosphere terrific, but only ( in the mean- 1/4-Komma mood ) in the keys of B, F, C, G, D and A major, and g, d, a, e, h, and F sharp minor. To make the keys in the whole circle of fifths playable, the mean- moods of the well-tempered tunings were eventually extended to same -stage mood that the keys in the whole circle of fifths were playable. This was only made ​​possible by the pure thirds again approached the Pythagorean thirds ( sharpened ). In the equal temperament tuning the twelve fifths in the Oktavraum be adjusted with twelve tones, so that all twelve semitones of 100 cents each other have the same distance measure. So you have the originally perfect fifth of 702 cents, which was the fifth stratification based, in favor of equal temperament with the fifth of 700 cents somewhat reduced.

This rationalization of the Western tonal system, which followed the requirements of composers and especially the instrumentalists, going to the attempts Andreas Werckmeister's back ( 1681-1691 ). Johann Sebastian Bach's work The Well-Tempered Clavier demonstrates how now all ♯ - and ♭ - keys on a piano - then each with its own key characteristics - were playable.

Mathematical Description

→ Main article: clay structure (mathematical description)

The material scales of sound systems can be described by a clay structure. Clay structures by building on tones and intervals. This clay structures can be described with mathematical formulas more precise and shorter. The interval space is divided according to the system, which is mathematically hardly distinguish between system and mood. Material scales and Gebauchstonleitern sew on it, with the purely mathematical theoretical description usually like the mood of an acoustic organ, deviates more or less from the real mood on the instrument.

Historical European sound systems

Historical sound systems can not always be with the number of stages in a octave and the organizing principle of tonal relationships describe, because note names that are the same in every octave, were only introduced by Guido of Arezzo by about 1025. For the time before the framework for describing may need to be broadened and extends usually over more than one octave. This is for example the ancient Greek Systema teleion clear that spans two octaves. The fact that the music theory in ancient Greece was already very advanced, show, inter alia, preserved writings of Aristoxenus, which were written about 320 BC. In them, the Greek sound system is first described mathematically.

For the music of the Middle Ages a different Tonordnung was created, namely the Gregorian system with Hexachorden and the church modes based on it. The medieval Christian liturgical music with the Gregorian chant was the essential basis for the further development of European art music.

Alternative sound systems in modern compositions

Since the beginning of the 20th century, a variety of composers sets with the question for a sound system for their own works. So composers were the standardized system of twelve tones to leave and to divide the semitone into smaller intervals. This is called micro- tonality. Ferruccio Busoni, for example, a third tone harmonium was built; but there are " microtonal " no works narrated by him. Significant compositional positions for about Busoni are:

• Charles Ives ( Danbury, Connecticut 1874 - New York City 1954) was one of the most innovative composers of his time. He was listed very rarely during his lifetime. His employment with novel mood systems was part of his wide-ranging compositional experiments. He placed fourth tones in the Symphony No.. 4 ( 1910-16 ) and in the Three Quarter Tone Piano Pieces ( 1923-24 ). The Universe Symphony ( 1911-16, unfinished, work on it until 1954 ) used an extreme Pythagorean system of perfect fifths in combination with additional quarter tones.

• Julián Carrillo ( Ahualulco, Mexico 1875 - Mexico City 1965) was a composer, conductor and violinist (Studies in Leipzig, among others ) with wide-ranging relations also with musicians such as Leopold Stokowski in the USA. Since the 20s, he championed new tuning systems and had special pianos in Mexico to build the very systematically in 1/3-, 1/4-, etc. to 1/16-Tonsystem stand at approximately beibehaltener number of keys. He published several writings on his theory of Sonido 13, eg 1934: La revolución musical del Sonido 13, he was hailed as a national hero in Mexico.

• Alois Haba ( Wisowitz, Moravia 1893 - Prague 1973) was a Franz Schreker students and wrote in various temperate systems, especially in the quarter-and sixth-tone system. He published writings such as My way to quarter-and sixth-tone music ( 1986) and New Harmony of the diatonic, chromatic, quarter-, third -, sixth - and twelfth - tone system (1927 ).

• Ivan Wyschnegradsky ( St.Petersburg 1893 - Paris 1979) wrote mostly in the quarter-tone system for Two Pianos ( also used within orchestral works ), but also for string quartet. Likewise, there are works in 1/6- or 1/12-Tonsystem, Deux pièces example, opus 44 (1958 ): Poème, pour piano à micro - interval de Julian Carrillo s sixth de ton - Etude pour piano à micro interval de Julian Carrillo en 1/12 de ton. Striking is a work in 31- tone system for the organ of Adriaan Fokker. Étude Ultrachromatique 1959 1932 he published Manuel d'harmonie à quarts de ton.
• Harry Partch ( Oakland, California 1901 - San Diego 1974) built his own tuning system from 43 tones per octave in just intonation: Just Intonation. He constructed parallel to instruments such as the oversized lyre ( a stringed instrument in two versions ) or the Chromelodeon on the basis of a harmonium ( two times running). With these and many other instruments, especially percussion instruments, he led U.S. universities like opera oratorios, in which the instruments act as main characters, for example in Delusion of the Fury. His tuning system avoids any tempering of intervals and is highly individual due to pure thirds, fifths, sevenths small up to 11 natural clay. He published in 1949 the book Genesis of a Music.

• Giacinto Scelsi (La Spezia, Italy 1905 - Rome 1988) left his piano improvisations by hired composers write microtonal for various occupations. His approach avoids any systematization and lives of the constant bending of the pitch to micro clusters. In these clusters beats play an important role. Stylistically, he often meets Carrillo, but was rather about his pursuit of a Far Eastern thinking to a resolution of the actual pitch.

• Lou Harrison ( Portland, Oregon 1917 - Lafayette, Indiana, 2003 ) was a student of American Innovators Henry Cowell and Arnold Schoenberg. He was influenced by Indonesian gamelan music. Harry Partch's Genesis of a Music gave him the impetus to discover for themselves just intonation. The majority of his work is written in Just Intonation. He fought for Charles Ives and Harry Partch, with whom he was friends.

• Ben Johnston ( born in Macon, Georgia 1926) was a staff and supporters of Harry Partch. Some of Partch homemade instruments were developed at the University in Urbana, Illinois, at Johnston 1951-83 taught. There are also important performances of Partch's works took place. Johnston developed a notation system for Just Intonation, which he uses for conventional instruments. He is known primarily for his string quartets, which have been partly recorded by the Kronos Quartet. In the String Quartet No.. 9 he expanded Partch's 11 -limit to 31 partial tone.

• James Tenney (Silver City, New Mexico 1934 - Valencia, California 2006) was both a composer and theorist, including an employee of Harry Partch. He was associated with important musical innovators in the U.S. as Edgar Varèse and John Cage. He was interested in just intonation, but also more complex metrics like that of Conlon Nancarrow. His books include: META / Hodos: A Phenomenology of 20th Century Musical Materials and on Approach to the Study of Form (1961 ) and META Meta / Hodos (1975 ) (both published in 1988 together ) and A History of ' Consonance ' and ' Dissonance ' (1988).

• Gérard Grisey ( Belfort 1946 - Paris 1998) was one of the most important members of the group L' Itinéraire in Paris, among other things, Hugues Dufourt, Tristan Murail and Michael Levinas belonged, formed in 1973 from among former students of Olivier Messiaen. In Germany, the group is often referred to as spectralists. Originally stands with them the partials spectrum in the center. Gradually, however, further principles of harmonious construction can be used as FM ( frequency modulation) or the distortion of spectra ( elongation or compression ). The late works of Grisey go with this material very freely about: Vortex temporum or Quatre Chants. From 1986 until his death he was professor of composition at the Conservatoire national supérieur de musique de Paris.