The Semantic system

An electronic book and different musical pieces are in progress. They will detail the microtonal applications of the system presented by Alain Daniélou, and will guide the Semantic users through the instrument’s tunings, as well as several tools and analyses. Just Intonation workshops on the Semantic system are also conducted by Jacques Dudon.
Summed up here are some of this system’s characteristics, as well as some basic microtonal notions.

Just intonation is altogether the science and the art of consonances. Instead of tempered intervals, it uses a diversity of consonant intervals, the frequency ratios of which can be expressed in the form of fractions between whole numbers (ex. 3/2, 5/3, 64/45, etc.). It generates, among other acoustical effects, an overtone’s “fusion” and an enhancement of differential tones in coherence with the intervals.

The 5th harmonic limit proper to the Semantic system uses, among the whole numbers found in its ratios, only products of prime numbers up to 5. In other words, it only uses primes 2, 3, and 5, in accordance with the hypothesis formulated by Alain Daniélou in his book “Sémantique musicale” concerning our perceptions of musical intervals. Furthermore, thanks to remarkable micro-coincidences, the 7th natural overtone (14 occurrences of this interval in the Semantic-53 scale) and the 17th and 19th overtones, to name a few, are naturally present in several ways, notably among indian shrutis, as well as the Semantic system.

The 22 Indian shrutis are the framework of the necessary intervals needed for the expression of all indian modes (or ragas), either from North or South India. Their frequency ratios are commonly expressed in the form of 5-prime limit ratios, that is, only using prime numbers 2, 3 and 5. The intonation system of the Semantic Daniélou achieves an extension of the 22 Indian shrutis, which it contains entirely. You will find the 22 shrutis in the 4th column of the Semantic-53 interval table.

The syntonic comma or pramana shruti is the small interval between indian shrutis ; its ratio is 81/80 and there are 10 of them in the 22 shrutis scale. The syntonic comma, which was dissolved in the different western historical temperaments and the recent 12 tones equal temperament, is particularly essential in indian music as much as in any just intonation, as it expresses for each chromatic degree the subtle emotional polarities of consonances issued from harmonics 3 and 5.

Disjunctions are commas, larger of about one-third of a comma, in the number of twelve, found at the boundaries between the different chromatic notes of the Semantic-53 scale. In 5-limit their rather complex ratio, is either 20 000 / 19 683, or 3 125 / 3 072. In 7-limit, it comes down to the equivalent of the septimal comma 64/63.

Quartertones” are, in common language, notes situated close to the midpoints of semitones, essentially heard in arabian, greek, turkish and persian music, but also in Eastern Europe, Africa and Asia ; they were also used in a tempered way by a few western microtonal composers of the last XXth century. In traditional world music, quartertones are mostly issued from more or less equal divisions of the minor thirds, or forths or fifths, and very rarely of the semitones themselves. Contrary to what has often been written, there are no quartertones among indian shrutis. Extending the 22 shrutis, the Semantic scale, on the other hand, contains numerous quartertones, resulting typically of the product of a comma and a disjunction, that is 3 + 4 = 7 kleismas. Disjunctions being 12 in number in the Semantic-53, the scale then displays 24 quartertones of this type, whose most frequent ratios are 250/243 in 5-limit, or 36/35 in 7-limit.

The 5-limit schisma (32 805 / 32 768) is a micro-coincidence of about one eleventh of a comma (1.95372 cent) found, for example, between the two main 5-limit versions of the first shruti (the limma, or chromatic semitone) used in evening and morning Indian ragas : it is clear for example, that in the morning raga, Todi, harmonic paths leading to this minor second point to 256/243, while in the harmonic context of the sunset raga Marva they point to 135/128. In Todi, it is an extremely minor harmony. In Marva it is, on the contrary, extremely major, and yet the difference of pitch is negligible in common musical practice.
Two notes differing of one schisma are considered by Indians as the same shruti and are played by the same key in the Semantic.
Actually, in 5-limit, many of the Semantic pitches have their ratio indeterminate between two different expressions. In the present S-53 scale, a more thorough analysis of the Semantic system has made it possible to precisely state its least deviations. So for any of its notes, we can select the most coherent ratio within the whole system.

The kleisma, while never found between two successive notes of the Semantic-53 scale, is nevertheless an omnipresent coincidence in the Semantic system, of about one-third of a comma. It is the natural difference between the last note of a series of six minor thirds (6/5) and the 3rd harmonic of the initial note (that is, one pure fifth above the octave) ; its ratio in 5-limit is then 15 625 / 15 552, or 8.10728 cents.
More generally, in the Semantic scale the kleisma is the difference between one disjunction and a comma, that will be always be observed between intervals composed of one same total number of commas + disjunctions, but differing according to their position in the scale, by their number of disjunctions.
Because of a perfectly balanced distribution of commas and disjunctions within the octave, for the same sum of commas + disjunctions, any interval of the Semantic-53 scale knows only one possible kleismic variation : the interval table of the Semantic-53 scale (see supporting documents) indicates the kleismic alternatives of each Semantic interval, and their 5-limit and 7-limit ratios.
Ultimately, 41 commas (of 3 kleismas) and 12 disjunctions (of 4 kleismas) separate the 53 notes of the Semantic scale, to generate 105 different intervals, among a global structure of 171 kleismas per octave. Approching these intervals by a whole number of kleismas is, therefore, the simplest way to express them, making the 171th of an octave a precise and convenient logarithmic unit to measure not only the Semantic’s, but all 5-limit or 7-limit intervals.
The notes of the Semantic scale being generated by a series of fifths (or inversely by a series of fourths), by multiplying the number of kleismas to a fourth, or inversely a fifth, by the proper number of generations, one will know the kleismas value of all the Semantic intervals. The interval table indicates these values for all the 105 intervals of the Semantic-53 scale.
One perfect fifth (3/2) is made up of 100 kleismas and its octave complement a perfect fourth (4/3), of 71.
So for example, two perfect fifths being a 9/8 more than an octave, the major tone (9/8) is made up of twice  100 k minus one octave (171 k) = 29 kleismas. Inversely, 16/9, the product of two fourths, is made up of twice 71 k = 142 kleismas.
The perfect major third (5/4), in the schismatic temperament that characterises the Semantic system, is equivalent to a series of 8 fourths : 8 times 71 minus 3 times 171 (3 octaves) = 55 kleismas.
A perfect major sixth (5/3) can be produced by the sum of a fourth and a major third, that is 71 + 55 = 126 kleismas, etc.
The values in kleismas of the three intervals between indian shrutis are finally :
– 3 kleismas for the pramana shruti or syntonic comma (81/80) ;
– 10 kleismas for the lagu (25/24) ;
– 13 kleismas for the limma (256/243 or 135/128).
The octave sums up 10 commas + 5 lagus + 7 limmas = 30 + 50 + 91 = 171 kleismas.

The 53 commas
Why 53 notes in the Semantic system ?
After the first cycle of 12 notes generated by a series of 12 fourths (or symmetrically 12 fifths), a series of 53 fourths (or 53 fifths) is the next cycle that produces an maximally even octave division, that sums up to only two step sizes of intervals, close to each other and balanced in an optimal way : 7 limmas and 5 apotomes with 12 notes, then 41 commas and 12 disjunctions with 53 notes.
As far as the dimensions of commas and disjunctions are close enough, as Alain Daniélou mentioned, these two types of commas cannot be mistaken, so the Semantic system can not be assimilated to a 53 commas-equal temperament, whose thirds and sixths, among others, are much more approximate.
In the hexagonal keys keyboard of the Semantic Daniélou-53 software, two yellow lines indicate the disjunction’s positions among the commas : crossing these yellow lines implies a move of one disjunction (of 4 kleismas) instead of a comma (of 3 kleismas).