Semantic system
The semantic system is based on a microtonal musical scale tuned in just intonation, developed by Alain Daniélou.
For Daniélou, the subtleties of the intervals of music of oral traditions cannot be expressed using the equal temperament tuning system of 12 notes per octave, which has been the prevalent system in Western culture for around two centuries. This "artificial" musical scale was developed as a compromise, to standardise musical instruments by reducing the number of notes they could play, but it also reduced the possibilities of expression for both composers and musicians.
Daniélou draws attention to the fact that a musical culture that adopts a system of equal temperament thereby sacrifices the possibility of expressing all but the most general significations inherent in a musical language. »
After many years spent researching and leading experiments in the world of Indian modal music, Daniélou published a book entitled in which he proposes one of the most elaborated microtonal scales of just intonation.
According to him, the human ear is able to identify and classify pitches by using binary, ternary and quinary frequency ratios as a reference point. This theory gives rise to the unequal division of the octave into 53 notes, with frequency ratios composed solely of products of powers of the prime numbers 2, 3 and 5.
Underlying theory
Alain Daniélou and ethnomusicology
was an ethnomusicologist who, after studying singing with Charles Panzéra and composition with Max d'Ollone, settled in India. He dedicated his work to the study of Hindu music and religion. Following a long collaboration with the University of Santiniketan in Bengal, Tagore offered him the position of head of the music department, which was in charge of broadcasting the poet's songs. He settled in Varanasi in 1935, where he was appointed director of the department of musicology of Banaras Hindu University in 1949. He was director of the Adyar Library and Research Centre in Madras from 1954 to 1956. He was a member of the French Institute of Pondicherry from 1957 to 1958, of the École française d'Extrême-Orient in 1959 and the Unesco International Music Council in 1960. Danielou founded an International Institute for Comparative Music Studies first in Berlin, then in Venice in 1969, and was director of both of them.He also created in 1961 the Unesco Collection of Traditional Music of the World, for which he was responsible for twenty years.
Alongside personalities such as the violinist Yehudi Menuhin and the sitar player Ravi Shankar, to whom he was close, he played a decisive role in the recognition of classical Indian music not as traditional folk music, which it had been considered as until then, but as a truly savant art, just as much as Western music.
Semantic system
For the last two centuries, Western musicians have been using imperfect musical intervals: those of the equal temperament of 12 notes per octave. While they have been used in the composition of a considerable amount of music, these intervals were a mathematical compromise that enabled the development of a certain category of acoustic, then electronic instruments, that some feel do not account for the finesse of our perceptual system.Historically, it was the philosopher and mathematician Leibniz who developed, in the 17th century, the theory of "subconscious calculation", according to which music was defined as "the pleasure the human soul experiences from counting without being aware that it is counting".
The Pythagorean Jean-Philippe Rameau followed a similar route when he established a connection between our perception of musical intervals and mathematics, and stated that according to him melody stems from harmony, through which it can "allow us to hear the numerical ratios enshrined within the universe".
More recently, a large number of composers such as Harry Patch, Harrison, Terry Riley, La Monte Young, Ben Johnston, Wendy Carlos, David B. Doty and Robert Rich have employed a variety of microtonal scales in just intonation.
In a similar approach to those of Leibniz and Rameau, Daniélou was deeply invested in the study of musical intervals, having studied Indian music and its subtleties for a large part of his life. He developed a musical scale of 53 notes, only using ratios of the prime factors 2, 3 and 5, which according to "shed a whole new light on intervals".
Five-limit tuning
The semantic system includes the notion of five-limit tuning —which refers to the fact that among all the whole numbers that form its ratios, it only uses only products of prime numbers up to five.However, because of their remarkable micro-coincidences, harmonic 7 and harmonics 17 and 19, if only to mention these three, are naturally present in various configurations, in particular within the Indian shrutis, and these intervals are therefore also part of the semantic system.
22 Indian ''shrutis''
The 22 shrutis represent the basic set of intervals required to perform of all the Indian modes, of both northern and southern India. Their frequency ratios are often expressed in the form of fractions of five-limit tuning, i.e. those that only use prime numbers 2, 3 or 5. Daniélou's just intonation system offers an extension of the 22 Indian shrutis, allowing it to include every one of them.Syntonic comma
Still known as the pramana shruti, the syntonic comma is the smallest of the intervals that separate the Indian shrutis. Its ratio is 81/80 and the scale of 22 shrutis includes 10 of them. Whilst the comma has been suppressed in the different historical Western temperaments and in our present-day equal temperament, it is of great importance in Indian music, and in all just intonation systems, since it expresses, for each chromatic degree, the subtle emotional polarities of harmonics 3 and 5.These 12 commas are larger than the other commas by around a third of a comma, and are found on the borders of the different chromatic notes of the semantic-53 scale. In five-limit tuning, their ratio is complex, measuring 20000 / 19683, or 3125 / 3072. In seven-limit tuning, they can be more simply defined as the septimal comma, with a ratio of 64/63.
Quarter tones
In everyday language, these notes are located between two semitones and they are essentially heard in Arab and Greek music throughout Europe and Eastern countries, in Turkey, Persia, as well as in Africa and in Asia. They were also used in tempered scales by certain European microtonal composers during the 19th century.In traditional music, quarter tones result above all from more or less equal divisions of minor thirds, fourths or fifths, rather than of semitones themselves. Contrary to what can often be read, there are no quarter tones amongst the Indian shrutis. Their extension in the Semantic scale does however include a significant number of quarter tones, resulting mostly from the product of a comma and a disjunction, i.e. 7 kleismas. Since disjunctions are 12 in number, there are thus 24 of this type, with ratios most commonly of 250/243 in 5-limit tuning, or of 36/25 in 7-limit tuning.
Schisma
The 5-limit just intonation schisma is a micro-coincidence of approximately an eleventh of a comma, found for example between different versions of the first shruti in certain evening and morning ragas: for instance, it is clear that in the Todi raga, the harmonic path taken to reach the minor second is that of ratio 256/243, whilst in the harmonic context of the Marva raga, it is 135/128. The Todi harmony is extremely minor, whilst the Marva raga has an extremely major harmony, yet their difference in pitch is, by the standards of current musical practice, insignificant.Two different notes of the same schisma are considered by Indians as one same shruti, and are played with one same key on each version of the Semantic keyboard.
For this reason, in 5-limit tuning, many notes on the Semantic have an undefined ratio between two different possible expressions. For its current interval selections, in-depth studies of the Semantic system have enabled its developers to obtain the utmost precision in its deviations, so that for each of its notes, the ratios proposed are those most coherent with the system as a whole.
Semantic kleismas
Though never found between two successive notes of the Semantic 53-note scale, the kleisma, a coincidence of around a third of a comma, is nevertheless omnipresent within the Semantic system. The kleisma is the natural difference between the last note of a series of 6 minor thirds 6/5 and the third harmonic of the starting note. Its ratio in 5-limit is therefore 15 625 / 15 552.However, there are several simpler ratios for different kleismas of around one third of a comma, that prove more appropriate for dividing the syntonic comma 81/80 into three harmonic intervals: for example the septimal kleisma 225/224, or the 17-limit kleisma 256/255. One relatively simple harmonic division of the syntonic comma 81/80 is for example 16000 : 16065 : 16128 : 16200, which combines three different kleismas: 3213/3200; 256/25; 225/224.
In the Semantic 53-note scale, the kleisma is in reality the difference between a disjunction and a comma, and we invariably find the difference of a kleisma between two intervals comprising the same total number of comma + disjunctions, but different by their number of disjunctions, depending on their position in the scale.
With its perfectly balanced distribution of commas / disjunctions, for the same sum of commas + disjunctions, each interval of the Semantic-53 scale can only have one possible kleismic variation: the Semantic-53 scale interval table indicates the kleismic alternative of each of its intervals, with their ratios in 5-limit and 7-limit versions.
Finally, 41 commas + 12 disjunctions separate the 53 notes of the Semantic scale, generating together a total of 105 intervals, which are part of a global structure of 171 kleismas per octave. If we approach them from the angle of whole numbers of kleismas, the 171st of the octave is therefore the simplest logarithmic unit allowing us to measure the intervals of the Semantic system.
Given that the notes of the Semantic scale were generated from a series of fifths, we can determine the kleismic values of each of the intervals of the system by multiplying the value in kleismas of the fourths or the fifths by whole numbers.
A fifth comprises 100 kleismas and its octave complement, a fourth comprises 71.
Therefore, two fifths, for example, reach beyond the octave by one major tone, which comprises two times 100 kleismas minus one octave = 29 kleismas.
Inversely, 16/9, which is the product of two fourths, comprises two times 71 = 142 kleismas.
The major third of the schismatic temperament used in the Semantic system is the equivalent of a series of 8 4ths: 8 times 71 – 3 times 171 = 55 kleismas.
A perfect major sixth can be obtained by adding a fourth and a major third: 71 + 55 = 126 kleismas, etc.
The values of the Indian shrutis are as follows:
- a syntonic comma = 3 kleismas;
- a lagu = 10 kleismas;
- a limma = 13 kleismas.