Sound system

Tone system refers to the tone supply used in a musical culture and the principles of order that underlie tone relationships and determine the functions of individual tones or tone levels . Sound systems are thus a central subject of music theory and musicology and represent a means of comparing different musical cultures with one another.

Many cultures, especially among indigenous peoples, often have no actual music theory themselves. A sound system can be derived here by the music researcher or music ethnologist examining the respective musical culture through empirical observations of the music practiced and any musical instruments used.

A characteristic common to most sound systems is e.g. B. the sense of similarity or identity of tones an octave apart . At this interval there are two tones with a frequency ratio of 2: 1 to each other. They are perceived as directly related and can be understood as tonal or equivalent.

The modern tone system of western music

In spite of various new alternative experiments, the tone system that has dominated modern western music to this day is the diatonic - chromatic - enharmonic tone system , as it is in the commonly used musical notation. Although its tone supply can in principle be expanded as required, the notation limits it to 35 levels by adding four variants to each of the seven diatonic base tones by single and double increases or decreases. This results in the following set of notes, which as a rule more than meets the requirements of musical practice and is only rarely fully used:

increased twice	cisis	disis	eisis	fisis	gisis	aisis	hisis
simply increased	cis	dis	ice	f sharp	g sharp	ais	his
Root tone	c	d	e	f	G	a	H
simply humiliated	ces	of	it	fes	total	as	b
doubly humiliated	ceses	deses	eses	feses	geses	asas	heses

Various construction principles, which are tuning systems at the same time, can be used for the theoretical justification of this tone system. The two most important are:

The Pythagorean principle of the stratification of fifths, according to which the diatonic root tone series results from the sequence of fifths fcgdaeh and the remaining tones are obtained by adding further fifths up and down. The Pythagorean tuning is based on this .
The principle of the layering of thirds, which arose in connection with the emergence of the major-minor system . To derive the levels of the tone system, major and minor thirds are alternately lined up, which together each form a major or minor triad. The diatonic root tones now result e.g. B. from the layering faceghd, the remaining notes by adding further thirds upwards (d-fis-a-cis-e-gis-b-ds ...) and down (fdbg-es-c-a-flat ...) . In contrast to the Pythagorean tuning, here, in addition to the fifths, the thirds are also intended as acoustically pure intervals. The pure mood is based on this .

In the case of keyboard instruments in particular, the practical implementation of the tone system is associated with problems with regard to pure intonation . The Archicembalo, for example, which with its 36 keys per octave can be understood as a good representation of the tone system of the pure tuning, proved to be impractical due to its unwieldy complexity. Therefore, efforts were made to reduce the number of keys to what was absolutely necessary. This reduction was made possible by the fact that many tones (e.g. his and c, c sharp and des, d flat and es, e and fes and so on) are very close to each other and only differ by a very small interval, called a comma , distinguish. Such closely adjacent tones can be put together on one key by deliberately keeping the smallest possible detuning, so that enharmonic mix-ups are possible. Among the many tuning systems that were devised to solve the problem, the twelve-level, equally tempered tuning that is common today finally prevailed.

Since then, the diatonic-chromatic-enharmonic tone system has been represented acoustically by a scale with twelve steps per octave ( chromatic scale ), which are separated by semitone intervals of a fixed size when the tuning is equal .

The twelve tones are in principle equally important - in the major - minor system that dominates Western music, each tone can become the fundamental tone of a key . The keys of the major-minor system are always the same in their structure; d. H. recognizable as major or minor. Western music distinguishes between two key genders, major and minor . The terms C major, F sharp major, A minor, B minor etc. indicate on which tone from the tone system the major or minor structure is based and which diatonic tones belong to the respective scale. If the basic tone sequence structure is retained, a melody can in principle be sung or played on any fundamental tone. Melodies or entire pieces of music can thus be transposed . The individual keys are sometimes assigned different meanings via the key characteristics , but these assignments are subjectively and historically variable and overall controversial.

The twelve- note scale represents a material scale , while C major or A minor, etc. are scales of use . Atonality and free tonality are an exception in Western music - material and usage guides coincide here.

By sign marked tones are derivatives or alteration of the original diatonic pitches and in the notation (as either increased ♯ ) or decreased ( ♭ listed) variants. This is also indicated by the names, which are modifications of the tone designations through appended letters or letter combinations.

Example: as is written as a lowered a ( ♭ a), g sharp as a raised g ( ♯ g).

Both tones, however, are physically identical in the equal tuning. Because now the music written grammar of the major-minor system or functional harmony by ♯ - and ♭ different -Tonarten, the consistency must in the score to safeguard Note labels are used that match the respective key.

Example: In C sharp major you write e.g. As a second stage, a dis , in D flat major , however, one there .

This applies to diatonic, i.e. its own scale, as well as tones not belonging to the key. B. be introduced in the case of modulations . The possibility of different notation is called enharmonic mix-up . Real enharmonic mix-ups occur in music notations in which a change from a ♯ -key to a ♭ -key, or vice versa, makes it easier for the musician to read or sing / play.
In the pure and mid-tone mood, however, z. B. As and G sharp also actually physically different tones. These are also called split tones because they divide the whole step between g and a into two different semitone distances. The difference between A flat and G sharp then counts as the actual Enharmonik , because a mix-up is excluded, as is the case with the equal tuning. This Enharmonik plays z. For example, it plays a decisive role in a cappella choirs, string quartets or good orchestras - especially if the intention is to come as close as possible to the sound of music between the Middle Ages and the Classical .

Derivation and history

After the octave, the western tone system is shaped first and foremost by the fifth and fourth . The philosophically founded conviction that the simplest, integer division ratios reproduce the most harmonious proportions goes back to Pythagoras of Samos or the Pythagoreans . This applies, for example, to geometry , architecture and physical vibration states such as B. Sounds. The frequency ratios of the octave (frequency ratio 1: 2), the fifth and fourth (frequency ratios 2: 3 and 3: 4) are 1: 2: 3: 4 in exactly this order. This sequence of numbers - to a certain extent the "universal formula" of the Pythagoreans; called Tetraktys - aptly reproduces the gradations of human consonance perception of two or more tones. Basic physical and mathematical principles meet human sensory perception . Based on the Pythagorean doctrine, the first tone of a tone system is derived from the octave. The derivation of the other notes of the system is done by stratification of fifths . This is based on the fact that the fifth is the first consonance that a scale, i.e. H. distinct diatonic pitches, can be derived. Tones with a distance of a fifth are related in the first degree, in the distance of two fifths in the second degree, and so on, as can be seen from the circle of fifths . The fifth and its complementary interval, the fourth, constitute pure and have fixed values of 702 cents and 498 cents, respectively.

Example for the construction of a diatonic ladder on c (c - d - e - f - g - a - h):

Stratification of fifths: c - g - d - a - e - h. From the first fifth following c we get the tone g (1200 cents x log ₂ (3: 2) = 702 cents). The second fifth leads to the d (1404 cents - 1200 cents = 204 cents). So you continue with the other tones, always paying attention to how often the octave range is changed due to the stratification of fifths. The corresponding tone is then "back-octaved" into the correct octave range, i.e. the octaves are subtracted again. A fifth lower from c is F and an octave above F is f. The figure opposite shows how the note e, i.e. the third step ( third ) in a ladder on c, is obtained by layering fifths .

Extraction of the tone e in a ladder on c by means of stratification of fifths. There are 4 fifths (4 × 702 cents = 2808 cents) and two octaves (2 × 1200 cents = 2400 cents). The difference leads to the major third with 408 cents.

This original construction of the Western tone system initially only takes into account the Pythagorean tuning , which is also known as the tuning of the fifth. The interval determined in the example opposite is the Pythagorean major third (64:81, 408 cents), which is greater than the pure major third (4: 5, 386 cents) by the syntonic comma (22 cents).
The stratification of fifths is, so to speak, the first theory to explain the western tone system, since, in accordance with the Pythagorean ideal, it is the best way to derive the tones of the pentatonic and heptatonic scales . Pentatonic and heptatonic scales are considered to be the original scales of the Western tonal system when they develop in that order . The tone system of ancient Greece already developed from a five-level (pentatonic) to a seven-level (heptatonic) system. The pentatonic ruled z. B. up to our time in well-preserved folk music traditions. Basically, these scales are considered diatonic scales. They only reproduce the pitches that result from the layering of four or six fifths. Example:

f - c - g - d - a (4 fifths) results in the semitone pentatonic scale

c - d - f - g - a

f - c - g - d - a - e - h (6 fifths) results in the heptatonic scale

c - d - e - f - g - a - h - c with two semitones.

The music of the Middle Ages basically used a heptatonic system, but already knew hierarchies in it , so that individual pitches were only less important. The Gregorian system with the church modes or modes arranged the basic tone stock in scales with different structures and consequently different characters.
With polyphony , but especially since the late Middle Ages and the early modern period , music practice added the possibility of altered pitches. Thus held in the notation the sign ♯ and ♭ feeder. Altered pitches did not initially change the tone system. They only had a tonal character and thus “only” functioned as intermediate tones or leading tones in final formulas or cadences or when changing the hexachord (see: Musica ficta ).

In the course of the highly complex polyphony since the 15th century, but especially with the rapidly increasing development of keyboard instruments in the Baroque era and the establishment of the major - minor system, tuning became a serious problem. The sound system had expanded as a result of musical and compositional practice. In terms of mood problems, this could no longer be explained solely through the Pythagorean stratification of fifths. Further explanatory models for the western tone system are z. B. the division of the octave into major and minor thirds. But it is precisely in the definition of the thirds that the problem of tuning ignites. The fifth is divided into a major and minor third (4: 5 and 5: 6). The major third consists of a major and a minor whole tone (8: 9 and 9:10), the difference between which is 21.51 cents, the so-called syntonic comma . As shown above, the progression of the fifths determines the major whole tone (8: 9, 204 cents) as the authentic one . Now an octave again consists of six whole tones - but six large whole tones exceed the octave frame by 23.46 cents, the so-called Pythagorean comma .
Furthermore, the occidental tone system can also be explained by the overtone series , which indicates all twelve tones and their relationships to one another. But here too there are difficulties with the mood.

In the course of time different tuning systems have been tested and developed, one of the main problems being that of the third.

The mean- tone tunings that have prevailed for a long time with many pure thirds approximate the pure tuning very well, but only (in the 1/4-point mean-tone tuning) in the keys of Bb, F, C, G, D and A- Major, as well as G, D, A, E, B and F sharp minor. The keys of the whole circle of fifths to make playable, the mean-moods were the well-tempered tunings finally to equal temperament tuning expanded so that the keys of the whole circle of fifths were playable. This was only made possible by bringing the pure thirds closer to the Pythagorean thirds (sharpening). In the case of equal tuning, the twelve fifths are adapted to the octave space with twelve tones, so that all twelve semitones have the same distance of 100 cents from one another. The original perfect fifth of 702 cents, on which the stratification of fifths was based, has been reduced somewhat in favor of equality with the fifth of 700 cents.

This rationalization of the occidental sound system, which followed the demands of the composers and especially the instrumentalists, goes back to the attempts of Andreas Werckmeister (between 1681 and 1691). Johann Sebastian Bach's work The Well-Tempered Clavier demonstrates how all ♯ and ♭ keys could now be played on a piano - at that time each with its own key characteristic .

Mathematical description

The material scales of tone systems can be described by a tone structure . Tone structures are based on tones and intervals . These clay structures can be described more precisely and briefly using mathematical formulas. The interval space is divided up differently depending on the system, whereby mathematically hardly any distinction is made between system and mood. Material scales and practical scales converge, whereby the purely mathematical-theoretical description mostly deviates more or less from the real tuning on the instrument, such as the tuning of an acoustic organ.

Historic European sound systems

Historical tone systems cannot always be described with the number of steps in an octave and the ordering principle of the tone relationships, because tone names that are the same in every octave were first introduced by Guido von Arezzo around 1025. For the time before, the framework for the description may need to be broader and usually extends over more than an octave. This is z. B. in the ancient Greek Systema Téleion , which spans two octaves. That the music theory in ancient Greece was already very far advanced, show u. a. extant writings of Aristoxenus , which were written around 320 BC. In them the Greek sound system is described mathematically for the first time.

A different tone order was created for the music of the Middle Ages , namely the Gregorian system with hexachords and the church modes that are based on it. Medieval Christian liturgical music with Gregorian chant became 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 large number of composers have been grappling with the question of a tone system for their own works. So composers tried to leave the standardized system of twelve tones and to divide the semitone step into smaller intervals. Here one speaks of microtonality . Ferruccio Busoni z. B. had a third-tone harmonium built; however, no “microtonal” works have come down to us. Essential compositional positions since about Busoni are:

Charles Ives (Danbury, Connecticut 1874 - New York City 1954) was one of the most innovative composers of his time. It was performed very rarely during his lifetime. His preoccupation with new types of tuning systems was part of his wide-ranging compositional experiments. He used quarter tones in 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) uses 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 extensive relationships with musicians such as Leopold Stokowski in the USA. Since the 1920s he advocated new tuning systems and had special pianos built in Mexico, which are systematically in the 1/3, 1/4 etc. to 1/16 tone system with almost the same number of keys. He published various papers on his theory of Sonido 13 , e.g. B. 1934: La revolución musical del Sonido 13 . In Mexico he was celebrated like a national hero.
Alois Hába (Wisowitz, Moravia 1893 - Prague 1973) was a Franz Schreker student and wrote in various tempered systems, namely in the quarter and sixth tone system. He published works such as Mein Weg zur Viertel- und Sixth-tone music (1986) or New Harmony of the diatonic, chromatic, quarter, third, sixth and twelfth tone system (1927).

Ivan Wyschnegradsky (St. Petersburg 1893 - Paris 1979) wrote mainly in the quarter-tone system for two pianos (also used within orchestral works), but also for string quartet. There are also works in the 1/6 or 1/12 tone system, for example Deux pièces opus 44 (1958): Poème, pour piano à micro-intervalles de Julian Carrillo en 1/6 de ton - Etude, pour piano à microintervalles de Julian Carrillo en 1/12 de ton . A work in the 31-tone system for the organ by Adriaan Fokker is striking: Étude Ultrachromatique 1959. In 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 pure tuning : Just Intonation . At the same time, he constructed instruments such as the oversized kithara (a string instrument, in two versions) or the chromelodeon on the basis of a harmonium (also executed twice). With these and many other instruments, especially percussion instruments, he performed opera-like oratorios at US universities in which the instruments act like main characters, e.g. B. Delusion of the Fury . Its tuning system avoids any tempering of intervals and is highly individual due to pure thirds, fifths, minor sevenths up to the 11th natural note. He published the book Genesis of a Music in 1949.

Giacinto Scelsi (La Spezia, Italy 1905 - Rome 1988) had his piano improvisations recorded microtonally by employed composers for various ensembles. His approach avoids any systematization and lives from the constant flexing of the pitches towards microclusters. In these clusters, beats play an essential role. Stylistically, he often meets with Carrillo, but rather came to a dissolution of the concrete pitch through his turn to Far Eastern thinking.

Lou Harrison (Portland, Oregon 1917 - Lafayette, Indiana 2003) was a student of the American innovator Henry Cowell and Arnold Schönberg. He was influenced by Indonesian gamelan music. Harry Partch's book Genesis of a Music gave him the impetusto discover Just Intonation himself. Most of his works are written in Just Intonation . He stood up for Charles Ives and Harry Partch, with whom he was friends.

Ben Johnston (born Macon, Georgia in 1926) was a collaborator and promoter of Harry Partch. Some of Partch's homemade instruments were developed at the University of Urbana, Illinois, where Johnston taught from 1951-83. Important performances of Partch's works also took place there. Johnston developed a notation system for just intonation , which he uses for conventional instruments. He is best known for his string quartets, some of which were recorded by the Kronos Quartet. In the String Quartet No. 9 he extends Partch's 11-limit up to the 31st partial.

James Tenney (Silver City, New Mexico 1934 - Valencia, California 2006) was a composer as well as a theorist, including a collaborator with Harry Partch. He was connected to important musical innovators in the USA such as Edgar Varèse and John Cage. He was interested in pure mood, but also in complex metrics like that of Conlon Nancarrow. His books include: META / HODOS: A Phenomenology of 20th-Century Musical Materials and an Approach to the Study of Form (1961) and META Meta / Hodos (1975) (both published in 1988) 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, to which u. a. Hugues Dufourt, Tristan Murail and Michael Levinas belonged to Olivier Messiaen's former students, founded in 1973. In Germany the group is often referred to as spectralists . They originally focused on the partial tone spectrum. Gradually, however, other principles of harmonic construction are used, such as FM (frequency modulation) or the distortion of spectra (expansion or compression). Grisey's late works use this material very freely: Vortex temporum or Quatre Chants . From 1986 until his death he was composition professor at the Conservatoire national supérieur de musique de Paris .

literature

Willibald Gurlitt , Hans Heinrich Eggebrecht (Ed.): Riemann Music Lexicon (subject part) . B. Schott's Sons, Mainz 1967, p. 970 f .
Marc Honegger, Günther Massenkeil (ed.): The great lexicon of music. Volume 8: Štich - Zylis-Gara. Updated special edition. Herder, Freiburg im Breisgau a. a. 1987, ISBN 3-451-20948-9 , p. 148 f.
Ernst Karmann: The mathematics of euphony - a brief introduction to the structure of our sound system. In: Funkschau . 1975, No. 1, pp. 39-42.
Otto Abraham: Studies About The Tone System And The Music Of The Japanese ... 2012, ISBN 978-1-277-92972-0 .
Otto Bachr: The tone system of our music . 2008, ISBN 978-0-559-03146-5 .

Remarks

↑ “The major-minor system is the system of this musical culture, even if it is applicable to many folk music cultures (including occidental pop music). The concept of the chord has been understood since the 17th century as a directly given unit of different, synchronously sounding pitches, which in their basic position are formed by harmonic layering of thirds . The resulting major and minor chords are the basis of the chordal resp. (third) harmonic tone system (which is therefore also called the major-minor system ). “Judith Debbeler: Harmony and perspective: the emergence of the modern occidental art music system Volume 4 of Aesthetica theatralia . 2007, ISBN 978-3-940388-02-5 ( online in Google book search).
↑ ^a ^b “This mean or common equilibrium temperature was first established at the end of the 17th century, defended by D'Allambert and Lambert, but especially appreciated by FW Murpurg ( FW Marpurg ). It is indispensable for keyboard instruments and for the greater part of wind instruments; but practical music distinguishes on all instruments, where the production of the correct tone is left to the skill of the player, between raised and lowered neighboring main tones, which does not happen at medium temperature; therefore, practical requirements and the common tone system differ here. The Mr. Vf. Gives many, weighty reasons for the incompatibility of the sound system used in acoustics with the theoretical and practical requirements. This tone system only corresponds to instruments with bound intonation, the scope of which is limited to the chromatic scale, but no longer those with free intonation, which is required by the enharmonic scale. ” Zeitschrift für die Osterreichischen Gymnasien . 1862, p. 563 ( online in Google Book Search).
↑ “The Gregorian tone system can only be found completely pure in Gregorian chant. According to the characteristics peculiar to it, it is only suitable for unanimous sentences, so it could no longer be used in polyphonic sentences. One started with the formation of polyphonic sentences from the same as the existing one, but had to make various changes, forced by the laws of harmony. The principles that were valid for the unanimous sentence were no longer suitable for two or more votes and had to give way to other principles according to which the votes were combined. A new tonal system was formed for the polyphonic movement, which of course was not fully developed right away, but developed gradually until it reached its perfection in our modern one . Austrian Cecilia Association, Volume 4 . 1871, p. 11 ( online in Google Book Search).
↑ "If it were proven what Father Martini 10, but only from Europeans, claims that every uninstructed person who has only enough voice to produce a melodious series, he says, as soon as he has it from himself forms, and not imitates, always based on the pure fourth; so the tone system of the Greeks would be taken directly from the instructions of our vocal organ itself. Those three intervals, the fourth, fifth and octave (Diatessaron, Diapente, Diapason), as the basic conditions of the tonal system in general, were strictly required and unalterable: differences in fourths or fifths were not accepted at all. They were therefore called the fixed intervals. Those, on the other hand, into which the fourth was further broken down, allowed for differences in the proportions, from which the different tone sexes arose. That is why they were called the changeable intervals; and because the gender could be most clearly recognized only from the relation of the fourth to the tone closest to it, this had received the name of the character tone. If the interval, which was given from the fourth to the fifth and is one tone, is repeated twice backwards from the fourth, then the interval of a small semitone remains up to the root: because what this was lacking in the whole tone was slightly larger. This was the most original division of the fourth, and gave the oldest sex, which, because it offered three whole notes in succession in its octave, was called the diatonon. If one moved the character tone back so far that each of the two intervals up to the fundamental tone formed a semitone; so one got the second sex, which got the came, chroma. Finally, if one left out the original character tone of the diatonon entirely, so that there was only one interval from the semitone to the fourth, and on the other hand, divided the interval tone from that to the fundamental into two, each to a quarter: thus came the youngest sex, which the Greeks enharmony, TI also called it harmony. However, the Greeks did not stop at these differences in scale. The semitone found in the first division gave another larger one; this according to one took a controversy among the quarter-tones, and even in the whole they won it by the skidding through the character sound ditone shared unequally. This caused again deviations in the sexes, which were called Chroen Il. ”Hans Christian Genelli: The theater of Athens, with regard to architecture, scenery and art of representation in general . 1818, p. 110 ( online in Google Book Search).

Individual evidence

↑ Liberty Manik: The Arabic Tone System in the Middle Ages. [With tab.] 1969 ( online in the Google book search).
^ Willibald Gurlitt , Hans Heinrich Eggebrecht (ed.): Riemann Music Lexicon (subject part) . B. Schott's Sons, Mainz 1967, p. 970 .
↑ Marc Honegger, Günther Massenkeil (ed.): The great lexicon of music. Volume 8: Štich - Zylis-Gara. Updated special edition. Herder, Freiburg im Breisgau a. a. 1987, ISBN 3-451-20948-9 , p. 148 f.

Web links

Wiktionary: Sound system - explanations of meanings, word origins, synonyms, translations

Online tutorial on the occidental sound system (musikanalyse.net)
Sound systems in comparison

[Judith_Debbeler-4] “The major-minor system is the system of this musical culture, even if it is applicable to many folk music cultures (including occidental pop music). The concept of the chord has been understood since the 17th century as a directly given unit of different, synchronously sounding pitches, which in their basic position are formed by harmonic layering of thirds . The resulting major and minor chords are the basis of the chordal resp. (third) harmonic tone system (which is therefore also called the major-minor system ). “Judith Debbeler: Harmony and perspective: the emergence of the modern occidental art music system Volume 4 of Aesthetica theatralia . 2007, ISBN 978-3-940388-02-5 ( online in Google book search).

[Gymnasien-5] “This mean or common equilibrium temperature was first established at the end of the 17th century, defended by D'Allambert and Lambert, but especially appreciated by FW Murpurg ( FW Marpurg ). It is indispensable for keyboard instruments and for the greater part of wind instruments; but practical music distinguishes on all instruments, where the production of the correct tone is left to the skill of the player, between raised and lowered neighboring main tones, which does not happen at medium temperature; therefore, practical requirements and the common tone system differ here. The Mr. Vf. Gives many, weighty reasons for the incompatibility of the sound system used in acoustics with the theoretical and practical requirements. This tone system only corresponds to instruments with bound intonation, the scope of which is limited to the chromatic scale, but no longer those with free intonation, which is required by the enharmonic scale. ” Zeitschrift für die Osterreichischen Gymnasien . 1862, p. 563 ( online in Google Book Search).

[Cäcilien-Verein-6] “The Gregorian tone system can only be found completely pure in Gregorian chant. According to the characteristics peculiar to it, it is only suitable for unanimous sentences, so it could no longer be used in polyphonic sentences. One started with the formation of polyphonic sentences from the same as the existing one, but had to make various changes, forced by the laws of harmony. The principles that were valid for the unanimous sentence were no longer suitable for two or more votes and had to give way to other principles according to which the votes were combined. A new tonal system was formed for the polyphonic movement, which of course was not fully developed right away, but developed gradually until it reached its perfection in our modern one . Austrian Cecilia Association, Volume 4 . 1871, p. 11 ( online in Google Book Search).

[Genelli-7] "If it were proven what Father Martini 10, but only from Europeans, claims that every uninstructed person who has only enough voice to produce a melodious series, he says, as soon as he has it from himself forms, and not imitates, always based on the pure fourth; so the tone system of the Greeks would be taken directly from the instructions of our vocal organ itself. Those three intervals, the fourth, fifth and octave (Diatessaron, Diapente, Diapason), as the basic conditions of the tonal system in general, were strictly required and unalterable: differences in fourths or fifths were not accepted at all. They were therefore called the fixed intervals. Those, on the other hand, into which the fourth was further broken down, allowed for differences in the proportions, from which the different tone sexes arose. That is why they were called the changeable intervals; and because the gender could be most clearly recognized only from the relation of the fourth to the tone closest to it, this had received the name of the character tone. If the interval, which was given from the fourth to the fifth and is one tone, is repeated twice backwards from the fourth, then the interval of a small semitone remains up to the root: because what this was lacking in the whole tone was slightly larger. This was the most original division of the fourth, and gave the oldest sex, which, because it offered three whole notes in succession in its octave, was called the diatonon. If one moved the character tone back so far that each of the two intervals up to the fundamental tone formed a semitone; so one got the second sex, which got the came, chroma. Finally, if one left out the original character tone of the diatonon entirely, so that there was only one interval from the semitone to the fourth, and on the other hand, divided the interval tone from that to the fundamental into two, each to a quarter: thus came the youngest sex, which the Greeks enharmony, TI also called it harmony. However, the Greeks did not stop at these differences in scale. The semitone found in the first division gave another larger one; this according to one took a controversy among the quarter-tones, and even in the whole they won it by the skidding through the character sound ditone shared unequally. This caused again deviations in the sexes, which were called Chroen Il. ”Hans Christian Genelli: The theater of Athens, with regard to architecture, scenery and art of representation in general . 1818, p. 110 ( online in Google Book Search).

[Manik-1] Liberty Manik: The Arabic Tone System in the Middle Ages. [With tab.] 1969 ( online in the Google book search).

[Riemann-2] Willibald Gurlitt , Hans Heinrich Eggebrecht (ed.): Riemann Music Lexicon (subject part) . B. Schott's Sons, Mainz 1967, p. 970 .

[Herder-3] Marc Honegger, Günther Massenkeil (ed.): The great lexicon of music. Volume 8: Štich - Zylis-Gara. Updated special edition. Herder, Freiburg im Breisgau a. a. 1987, ISBN 3-451-20948-9 , p. 148 f.