By Jacques Dudon

The works of Alain Daniélou, Traité de musicologie comparée (1) and Sémantique musicale, were without doubt what inspired me to create my first microtonal fretting systems for string instruments, in the late 1960’s. In 1971, through a combination of circumstances, one of these instruments that I had taken to India, gave me the opportunity to meet the Director of the Vocal University of Banaras; Sri M.R. Gautam, who became my singing teacher and offered me an initiation into the basics of Indian music. I was thus able to reinforce my knowledge of shruti intonation; the 22 notes said to be the foundations of Indian music, as well as to discover the profound confusion present at that time in the North of India, concerning the theoretical demonstration of these shrutis (not their practical application, thankfully). Whilst reading a book on the musical theory of South India (2), I had the pleasure of discovering that without knowing it, the lute which had enabled me to meet my teacher was fretted exactly according to the “true” shrutis.

When I returned to France four years later, I built my first string instrument which enabled all 22 shrutis to be interpreted, which I named the dulcevina. I proudly presented the instrument to Alain Daniélou during a trip to Paris. He studied it carefully, played it a little, and then showed me all of his microtonal harmonium prototypes (of which certain details of the keys are shown on the photos on the page “Media Archives/Photos” of this website). I recall him telling me that he did not understand the use of one particular note, located between “GA” and “MA” (E and F), which was used quite commonly in popular Indian music, and he asked me if I had an idea about this. I was touched by the humility of a man who had such remarkable knowledge but was constantly researching his subject, constantly thirsty for greater knowledge. I replied that I too had often noticed this hypermajor third, and not only in Indian music, and that it intrigued me greatly too. Whilst we each had our own secret passage leading to this mysterious note, neither of us had the final answer, and this simple question has always stayed fondly in my memory, like a shared enigma. We also talked about projects that we had in common, such as the creation of instruments which would allow musicians to play all of the 53 commas per octave.

Later I began building all kinds of experimental instruments tuned to Indian shrutis or other systems, which is why Alain Daniélou’s scale – and more generally all just intonation systems – are quite familiar to me, having played microtonal instruments for many years, including string and keyboard instruments, as well as the “photosonic disk”; an optically generated sound instrument which is currently my main instrument. When the Alain Daniélou Centre invited me to write an introductory guide to the Semantic – the mythical instrument conceived by Alain Daniélou before his death, and the only instrument in existence at this time which enabled a precise exploration of his system, I accepted wholeheartedly, just as I have accepted to write these lines on his founding book, Sémantique Musicale. Fourty years after it was first published, my interpretation of this book is of course different to the one I had at the time, when I discovered just intonation as a teenager. Today I retain three main ideas from this book, which carry the same powerful conviction about just intonation in its different aspects: tonal creation, perceptions and language. These are:

– Just intonation allows our musical systems to evolve,
– Just intonation allows us to explore our own mechanisms of perception,
– Factors 2 3 5 are the key to musical language.

His previous book, “Traité de musicologie comparée” (Treatise on comparative musicology) (1) presents the fundamental idea that just intonation is the basis of all music. Drawing on Indo-European traditions, Sémantique musicale reaffirms this observation by offering a just intonation reference system with the aim of inspiring evolution amongst all different musical approaches, as well as in the conception of our instruments.

The difference between a collection of intervals in just intonation, as interesting as they may be from a historical or cultural point of view, or a temperament which offers more specific arrangements, and a system like the one Alain Daniélou developed, lies in the quality of the relationship between the different sounds and in the ability of the system to respond, at least partly, to the practical issues involved in playing an instrument. By using clearly defined, specific generational laws, each system has its own sound environment and expresses its own harmonic landscape. Just as we can recognise a tree by its fruit, it is through practical musical applications that we can discover the identity and value of a system, and that it is able to reveal its own particular palette of sound colours. This is of course only possible providing the adapted instruments are developed, and played.

The particularity of a just intonation system is that is only uses natural musical accords, and thus shares, with many other traditional scales, a common core of ratios which are part of a universal language. Language, however, implies a selection of sounds, and this is where the use of certain rules is fundamental in order to define its limits, since without them neither the instruments nor the music could be created. The limit that Alain Daniélou chose was that of harmonic 5; a very wise limit, and historically acknowledged as being that of Indo-European music. It includes, as it name implies, all the possible frequency ratios between prime and composite numbers up to 5 (i.e. 2, 3, 5) and therefore a large part of the most harmonious musical intervals. It was therefore logical that this system would include the 22 Indian shrutis (the word “shruti” meaning “that which is heard”, which is particularly significant given the subject of the book), which Alain Daniélou had the good sense to extend to a total of 53 sounds per octave.  This number did not come about by chance, given the relative dimension of the comma (81/80), which is the main interval between the shrutis, compared to the other two, the limma (256/243) and the lagu (25/24).  The limma can be divided into approximately 4 commas, and the lagu into 3 commas. Since the 22 shrutis divide one octave into a total of 7 limmas, 5 lagus and 10 commas, this brings us to (7 x 4) + (5 x 3) + 10 = 53 commas.

Before going any further, I would like to look back to the origin of the syntonic comma, or pramana shruti, the simplest harmonic coincidence of the 5-prime limit, and the cornerstone of Alain Daniélou’s system: a major third can be generated in 5-limit tuning in two different ways, either by a series of fifths (powers of 3) in order to reach 81, or directly via the 5th harmonic, and with these two notes brought to the same octave, 81/64 and 5/4, they have a ratio of a comma 81/80.

It is also possible to reach the number of 53 through the cycle of fifths, since there are also approximately 4 pythagorian commas (appearing with each cycle of 12 fifths and very close to the previous commas) this time in the descending limma, obtained through a cycle of fifths C-G-D-A-E-B of which the total is again (4 x 12) + 5 = 53 fifths, thus dividing the octave into just as many commas.

What is presented as a cycle (p. 70) is in fact a spiral, obtained by a succession of six series of symmetric fifths around the notes 36/25, 6/5, 1/1, 5/3, 25/18, 125/108, since Alain Daniélou’s trade “secret” was to choose notes separated not by major thirds but by major sixths (or inversely, minor thirds). Had he chosen major thirds, he would have had to use seven series of eight fifths instead of six series of nine fifths, in order to cover the 53 notes. This would have complicated the system and further accentuated the “wolf fifth” (the interval between the furthest notes). In this solution the series uses factor 53 (=125), either “+++” or “- – -”, which encompasses the circle of 53 commas, and although this choice is not explicit in all his tables, Alain Daniélou seems to opt for this method which places 53 to the numerator. It may be noted that the two series cannot be superimposed, since their notes are separated by 8,11 cents and not schismas of 1,95 cents as is the case between all other adjacent series.

A few years after writing this book, Alain Daniélou initiated the creation of a keyboard instrument based on these 53 sounds, the S52, and later a second one using a subset of 36 of the same sounds, which later became the Semantic Daniélou-36 (presented in detail in a separate section ).

Developed for teaching purposes, Alain Daniélou’s choice of names for the notes was very simple and practical. For western readers, it is important to note that although there is no reason not to apply the intervals to western tuning, C D E F G A B should be considered as a translation of the Indian notes SA RE GA MA PA DHA NI, but not in terms of absolute pitches with regards to a given tuning, rather as a modal concept of relative pitches with regards to a reference note (C = SA = 1/1). Unlike most microtonal notations, Daniélou did not use the two common alterations, sharp and flat, to describe the sharp of a note and the flat of the following note (C# and Db for example). Hence each note, altered or not, has the same chromatic status within a same given set. We can see another adaptation of Indian notation with “komal”, translated here by “flat”, and “tivra” by “sharp”. The fact that the latter is, like in Indian notation, only employed for the interval of an augmented fourth (F#) is no coincidence, and highlights the fact that F# is associated with the sounds of the major harmonies of A, E and B, whilst D, Eb, Ab and Bb are the result of a cycle of descending fifths, following another route which associates them with minor harmonies.

Before exploring the microtonalities developed by Alain Daniélou, it is important to fully grasp the chromatic “skeleton” of his system, composed of the diatonic scale of the Bilaval raga, identical to Ptolémée’s Diatonon Syntonon, (also called the “Zarlino scale”), which is completed by an F# 45/32 as the third of D 9/8, via an extension of descending fifths from F to Db. This chromatic structure can also be summarised as a series of 8 fifths (3/2) from Db to D, completed by 4 fifths A, E, B, F#, a major third above F, C, G and D.  The + or – signs which follow the names of these 12 notes indicate that they are raised are lowered by the number of corresponding commas. The 22 shrutis, as they are generally known today, are named C, Db, Db+, D-, D, Eb, E, E+, F, F#, F#+, G, Ab, Ab+, A, A+, Bb, Bb+, B, B+.

Alain Daniélou’s choice to use the first three prime numbers to build his tonal language was entirely justified by the harmonic nature of sounds produced by most instruments, namely the voice. He reached a global set of 53 pitches, unequally divided by 4 types of commas, which, depending on the harmonies, could give slight variations of the value of a schisma (ratio of 32 805/32768, i.e. 1,95 cents), which, according to Alain Daniélou’s notation and that of Indian musicians, did not change the name of the note.

To recall briefly the origin of this schisma, in practical terms it is the coincidence of the culmination (brought to the same octave) of two distinct paths, one minor and the other major, with a total of 8 fifths (3/2) and a major third (5/4),  in one equivalent note.

It is important to note that even if Alain Daniélou seems exclusively interested in modal music in this book, such a system is not at all limited to modal music alone. Just intonation, in general terms, accommodates all transpositions and polyphonic music and is not at all inconsistent with the principle of changing the tonic, which has been used in Indian and other musical cultures for a long time. Furthermore, a shift in the tonic to a different note through the introduction of variations considerably multiplies the number of intervals and therefore the expressive possibilities of an unequal system. Given the possible inversions and schismic micro-variations, which are indispensable in restoring the exact value of each of its consonances, practically speaking, the global system is composed of many more than 53 intervals, which should simply be considered as a further source of richness and a display of its natural diversity.

Although he chose not to elaborate on this subject, it should be acknowledged that Alain Daniélou did not compromise his system by simplifying it to a mechanical division of the octave into 53 tempered commas, which would have been restrictive in both the quality of the thirds and all of its possible harmonic subtleties. By fully respecting the natural consonances of factors 3 and 5 while at the same time applying a certain selective logic, he preserved the harmonic qualities and essential polarities. His obstinate determination to defend and rationalise the phenomenon of consonances rightfully brought him the recognition as being one of the pioneers in the field of just intonation.

Alain Daniélou’s vision was that of a universal language. His research into an archetypal system, even as far as the emotional content of intervals, is coherent with the hierarchy he proposes for these intervals, which follows a fairly explicit family-based logic throughout his book. His philosophy of a language of intervals, not only in terms of their microtonalities but also their semantic content, reveals a systemic vision which remains today just as remarkable to witness.

I will not provide here all the details of the families of intervals as defined by Alain Daniélou, but will instead recommend that readers consult the “Cycle des intervalles”, on p. 70 of Sémantique Musicale, which summarises them quite well. I call the reader’s attention to the fact that there is a risk of confusion between the signs + and – associated with the names of the notes, which follow the logic of a chromatic keyboard (ie. in series of 12), and the signs + and – associated with the families of intervals, which contain on average 9 notes each, and which Alain Daniélou named “initial series, series -, series +, series – -, series + +, series – – -, series + + +”. Indeed we may note that a series with a given name contains notes which have another suffix. For example, a “series – -” contains more “-” notes than “- -” notes, etc. This is because the signs given by Alain Daniélou to the different series actually symbolise the exponents of the powers of 5 contained within the note ratios they include. Once this is understood, we can simply translate “series -, – -, – – -” by “series 5, 52, 53” and “series + , + + , and + + +” by “series 1/5, 1/52, 1/53”, whilst the initial series, which is not concerned by this problem, is the series 3n.

Each of these families, also called “interval categories” (p. 64), which are also in fact a “series” of fifths, is said to possess certain specific properties, which trigger a global type of emotion, which Alain Daniélou resumes very clearly, and which in my opinion seems to conform with our current perception of these interval families. These basic emotions are then broken down, according to their intervals, into several variations whose differences seem, in comparison, to lack contrast. They demonstrate that for the author the main semantic criteria concerning the intervals was that they belonged to these different families, and I would largely share this observation were it not for another criteria – the order in the cycle of fifths – and hence the position of the notes in the octave, strikes me as being more subtle but just as important. One question therefore remains unanswered, this second criteria being precisely the only one we have left in order to distinguish the intervals of an equal temperament, by definition ambiguous: which is the mechanism by which we can perceive them? Moreover, is the perception of pitch in tempered music so radically different from the perception of pitch in just intonation music? For most listeners, I am not certain it is. What can however make a difference, in just intonation, is another quality of harmonic fusion, of differential resonance, of pulse, etc. which could be made available to listeners in order to broaden our scope of sensations. Between a piece of music played on a piano tuned in equal temperament and the same piece played on a piano tuned (appropriately) in just intonation, or even just in an improved temperament, the differences are noticed mainly in terms of timbre.

Yet the perception of timbre is far more complex, global, intuitive and personal – than analytic and selective. We have all had the experience of being able to fully appreciate any kind of music in a general manner, for example the harmony of its sounds and melodies, not to mention the rhythm, without having the slightest analytic notion of what we were hearing. Musicians can if they wish add the recognition of pitch to their perception, in order to name what they are hearing within one or another system, but the emotion is present first. This means that between microtonalities, harmonies, harmonics and timbre, there are multitude of possible uses for our memorised standard figures which determine and even personalise what we hear within sounds.

As far as the intervals involving the power of 5 are concerned, the family classifications described by Alain Daniélou is in my view particularly pertinent. Personally, I would be inclined to make a clear distinction between shruti-type intervals (series 3n, 5, 1/5), which I would qualify as chromatic, and all the others, products of the power of 5 and not used in Indian music, which I would describe as “extra-chromatic”. Concerning the latter, I have very often noticed that an interval based on factor 25 (5²) has a very different feel to it, depending on if it is included in a mode which also contains factor 5, or not. In one case it is linked to other notes by factor 5, and in the other case, where factor 25 is not divided (which seems to be the case in the experiments of Alain Daniélou), I perceive it as being very foreign to the family of 5 or of 1/5. I have noted the same phenomenon with factor 125 (53), whose nuances are very different to those of factor 25. On the subject of 53, it is regretful that Alain Daniélou did not dedicate a specific family of emotions to this factor alone. The reserves he showed about it were most likely due to the modal analysis his method employed (p. 63), which dismisses 125/64 or 128/125, because of an evident dissonance with 1/1. I believe that factor 5^3 avoids 1/1 of its own accord in order to combine with other reference notes, and the way in which for example it divides a fourth such as the one between 27/16 and 9/8, into two almost equal semifourths (125/108 and 144/125), offers some very beautiful musical applications.

The context, in terms of mode, sound environment and even our thoughts about the sound when we hear these intervals, is thus essential. In a number of group experiments carried out during psychoacoustics workshops I host regularly, it would appear that an interval placed in a context, within a given mode, always carries more meaning than if it is played on its own. Even if it is perceived in a less “neutral” manner, it awakens our senses into a more “global” way of listening, which is after all the natural musical context of an interval. Alain Daniélou suggests something similar when he points out that after hearing the notes of a raga, they become much more precise, since they are animated by a just sensation: “Each note then takes on an expressive colour. The musician plays with feeling, with emotion. At that moment the intervals become extremely precise” (p. 27). In the case of a raga, with its hierarchy between the notes, its key-phrases, its ornaments, etc., whose interpretation – and therefore context – are highly coded, microtonalities as well as the semantic content of each interval, are even more precise.

My work on the microtonal modelling of ragas and of all traditional scales has allowed me to bring to light the importance of double notes, which Alain Daniélou’s research also verified. We have become used to thinking that a hexatonic scale contains six notes, a heptatonic scale contains seven notes, etc., yet this is in fact only the tempered, thus incomplete aspect of their intonations. Due to what I name harmonic coincidences, each scale contains at least one or two double notes, if not more, separated by “commas” or “schismas”. They are widely used in Indian music, in which they represent the keys to different gamaks or ornaments. These double notes are so essential that they alone often encompass the entire scale within their respective harmonic genesis. The “commal” definition of intervals which allows us for example to distinguish between two shrutis of the same degree, is quite audible and fits perfectly with Alain Daniélou’s way of deciphering intervals by factors 3 and 5. The distinction between the “schismic” polarities, reputed for being difficult to differentiate, is a problem in this type of interpretation, as Alain Daniélou acknowledged himself. He deduced that as a result of the “overlapping of cycles”, instead of a note which is undetermined in terms of its affiliation, it was preferable to have a less ambiguous note, a comma higher or lower (p. 68). However, the perception of these ambiguous intervals, in the musical context of a mode, or even just with one or two other neighbouring notes, in the majority of cases completely resolves the problem.

Hearing for example the note Db (ati komal D) in the context of the raga Marva leaves no doubt about the major nature of the shruti, as a major third of DHA (here A+), and also the vadi note (= reference note) of the raga and not a note produced by a cycle of descending fifths of which none are present in the scale. Therefore this Db is indeed 135/128 and not 256/43. In less common modes, or in the context of misra ragas (mixtures of ragas) etc., even if there was a certain ambiguity as to the path by which the shruti arises, its schismic definition would then be double, without it causing a problem. They also express shared emotions, which we know to be possible. The “blue” note sought by all musicians is a good example of this type of ambiguity. With both a questioning feel to it, while expressing the blues, or even pain, the note’s precise intonation and its place in the melodious context can also express consolation, overcoming, and liberation. Ask three or four different microtonalists what this interval is, and you will have a fine example of complexity.

It is therefore conceivable that modes, or even simple tetrachords or chords, could, if the intervals were placed in an already harmonised context, make them easier to understand, through the memory of more or less cultural archetypal figures or at least by approaching what I would call a “perception of belonging”, a wholeness, whose usefulness we have just observed, in completing further and finer analysis based on consonances. If we take the question a little further, being able to directly associate the value of an interval with the fundamental degrees of a simplified reference system (which does not have to be tempered) should perhaps also be an operation which requires verification. The harmonic series of vocal harmonics which constantly accompany our spoken voice, seem in this respect to be a fundamental model which is deeply rooted in our perceptions. Not only does it contain some of the most consonant intervals – by definition all the epimoric ratios of the form (n+1) / n, but also offers a whole selection of perfectly consonant tetrachords. Other scales which are more cyclic, pentatonic, heptatonic or even dodecaphonic, also seem to be engraved in our memory, to a different degree according to our culture and education, but are certainly part of the most shared patterns of perception throughout the world.

The Indians themselves, with their prodigious memory of modes and the precision of their intonations, use a system of chromatic notation which does not make any reference to the shrutis. Inversely, the ambiguity of the sounds placed between the notes of a system can also be something that musicians seek, even more so if the intermediate intervals between these sounds offer harmonic ratios, as is the case of the neutral seconds or thirds in Arab and Persian music, or semifourths, frequently found in African pentatonic scales, etc. Is this vague sensation of slightly “foreign” notes compared to our usual systems not in fact one of the attractions of the different kinds of music across our planet?…

A language is necessarily composed of a limited number of signs, and from there it was only a small step to setting the limits of the tonal language of harmonics 2, 3 and 5, perfectly in keeping with the Indian theory; a step which Alain Daniélou took without hesitating. The almost coincidental number of notes per octave (53) generated by the full scope of this system, and the number of phonemes used in the Sanskrit alphabet which he mastered perfectly, the observation of geometrical archetypes and the visual recognition of numbers, the structure of atoms, cristallisations, thought mechanisms, everything pushed Alain Daniélou, spurred by a wonderful creative and intellectual impulse, to make this hypothesis the omnipresent basis of his arguments and convictions, all throughout his book. He could have included in his elementary figures Pythagoras’ remarkable triangle or the Egyptians’ rope with 12 knots, who only needed those same numbers (3/2²/5) in order to create a right angle.

How relevant is this hypothesis fifty years later, in our practical approach to music? As far as the evolution of our western intonation system is concerned, regretfully we can observe that it is far from following the path opened by Alain Daniélou. The equal temperament with 12 sounds per octave not only still reigns supreme but its domination is spreading to all music throughout the world; the so called “Indian” harmoniums, alas inappropriately tempered, have never been so popular, and the lack of transmission of traditional models of intonation and other knowledge is very alarming. Twelve equally spaced sounds within an octave will only ever give 12 intervals which are harmonically incoherent, with – amongst other defects – a denatured major third, and the same one everywhere, since we have lost the intelligence of our historic temperaments. When we know that an infinite choice of intervals exists in nature and in traditional music, it is clear that 12 poor intervals will never offer the lasting fulfilment of a satisfying tonal language.

In spite of the lack of instruments and of musicians to fully express them, there would nevertheless appear to be a flutter of curiosity around microtonal music. Advancing audio techniques have enabled a limited sphere of initiated musicians to explore different scales, via computers and digital keyboards, with growing precision. A small number of adventurous guitarists have their electric guitars refretted in order to experiment different systems, mostly of just intonation, the only one which allows them to play chords through the natural distortion and saturation effects of amplifiers. The work of a number of dedicated researchers has enabled the lie concerning the so-called “equal” intonation system of J.S. Bach to be rectified after all these years: it has now been proved that the expression “WELL tempered clavier” means circular, without forbidden tones, and never implied that it was an equal temperament. However this does not mean that his works are often played in their original temperament, except by a few rare musicians whose number will perhaps increase in the future.

At the beginning of the 3rd millennium, the microtonal trend is following the opposite path to that of Alain Daniélou: there are a number of different schools of thought, primarily those of high temperaments, namely systems with 19, 24, 31, 34, 41, 72, 144 etc. tones per octave, and the just intonation system which, following Harry Partch, continues to be attracted to 7-limit, 11-limit, 13-limit systems, or higher.

A few years before Alain Daniélou, in America, Harry Partch (1901 – 1974), another creator of several microtonal harmoniums (3), developed a “monophonic” system which also drew from positive and negative polarities of harmonics combined together, but extended to factor 13, which gave a framework of 43 sounds per octave, which revolutionised the history of overseas microtonal music. What a great shame that the two seasoned travellers, with their shared passion for musical traditions outside of Europe, never met each other!

After Harry Partch, the last few decades have witnessed the emergence of a number of just intonation musicians, including Lou Harrison, La Monte Young, Terry Riley, Jon Catler, Robert Rich, and many more, as well as the theorist Erv Wilson,whilst David B. Doty was editing the only review dedicated to just intonation 1/1, in Saint Francisco at the Just Intonation Network. All of these composers using just intonation explored systems reaching at least as far as 13-limit ratios. In Europe “historical” microtonal musicians limited their work to the exploration of systems such as fourths, fifths, sixths and even twelfths of tempered tones (24, 31, 36, 72 degrees by octave).

Just as painters who would be unwilling to criticise their respective colour palettes, however different their approaches, microtonalists are respectful, even curious, regarding the different systems adopted by fellow experts, and the only quarrel that may arise would be over questions of logic or concerning historical or musicological interpretations. Having always defended the right to be different, it is only normal that they would approve of a diversity of systems. There is no harm in being more interested in one system than another, provided the aim is to share and develop the fruit of this research. It is important to listen to a broad number of systems, composers and of musical styles, in order to grasp a notion of the different theories of microtonal music. Traditional music, with its oral transmission, also contains a wide variety of intonations, chosen and reproduced simply because they sound better than others and because they are familiar. What can they teach us about the work of Alain Daniélou? Here I would like to give a number of examples drawn from traditional music, some of which tend toward his theory and others which do not.

The Tibetan monks of the famous sects Gyütö and Gyüme intone their songs by amplifying harmonics 5 and 6 respectively (or 10 and 12 from their low subharmonic). As far as I am aware – yet I would happily be informed otherwise – there is no Tibetan monk song in which harmonics 7 or 11 are intentionally made to resonate. Tuvan and Mongolian singers who use overtone singing, sing melodies which explore the harmonics of their root note, generally ranging from 4 to 12, amongst which they knowingly avoid harmonics 7, 11, 13, and 14, thus keeping only the 5-limit intervals. Their altered scale is therefore composed of harmonics 6 – 8 – 9 – 10 – 12, and when they come back to the root notes they add to these four notes the sixth 5/3 (and not 13 or 14) in order to obtain a pentatonic scale similar to that of the raga Bhupali (1/1 9/8 5/4 3/2 5/3). We can also note that this scale is no doubt one of the most frequently used pentatonic scales throughout the world, being found in practically identical form in almost all Amerindian as well as North and South American ethnic cultures.

However, there are some important non-European musical cultures (the Slendro tuning of the Indonesian gamelan, Aka and Baka Pygmy songs – which often use the septimal tone 8/7 -, Thai and Cambodian classical music, etc.) in which factor 5 is totally absent. Lou Harrison had 7-limit Slendro scales authenticated by masters of the Indonesian gamelan. Factor 5 is also often absent in many african, arabian and persian modes. It is unquestionable that systems other than those of the 5-limit exist, and moreover their music has provided the basis for many prodigious repertoires.

Must we then exclude or maintain prime numbers above 5? There is nothing to criticise in Alain Daniélou’s system in itself, which possesses all the necessary qualities in order to develop the most beautiful music, in all forms, and holds the promise of being extendible to different universes of sound which are yet to be discovered. In my mind it would be a mistake to want all just intonation systems to be exclusive, or even separate from one another. There is no universal system, just as there are no impenetrable barriers, and we can observe many complementary qualities and permeable limits which demonstrate new theories of consonance.

Differential coherence (4), for example, explores the harmonies between different sounds through an acoustic phenomenon which is perfectly audible and reproducible, involving the use of chords between different sounds to generate in our ears new, lower notes whose frequencies correspond with the difference between the frequencies they are composed of. All of us can witness, or already have, that for example the minor third 6/5 generates in its undertones a minor sixth whose frequency is 6/5 – 5/5 = 1/5. If this differential sound is part of the tonal context (scale, harmony, etc.) of the interval, the latter is deemed as consonant. Indian shrutis in the context of ragas and traditional scales generally possess rich qualities in terms of differential coherence.

Harmonic coincidences (5) make up another field of consonance, which is particularly instructive concerning the origins of musical systems and directly addresses the question of harmonic fusion inherent to the chords between different sounds. While the comma 81/80 and the schisma 32 805 / 32 768 are the two main harmonic coincidences between factors 3 and 5, there are some very subtle coincidences between factors 3, 5 and 19 for example, such as the schisma 1 216 / 1 215, (approximately 1,42 cent), which oblige us to acknowledge that factor 19 is “transparent” when it arises in 5-limit ratios. We can find this coincidence between a limma 135/128 and another limma which is much simpler, 19/18, which then generates, through the successive fifths 19/12 then 19/16, a wonderfully deep, coherent minor third which results directly from harmonic 19. In order to offer an answer to a problem often raised by Alain Daniélou, certain prime numbers can therefore prove to be complementary to a system, in bridging the complexities generated by the extention of its combinations of factors and offering more consonant intervals.

To conclude, microtonal music is far from having delivered all its secrets. Harmonic 5, amongst others, is one of them and still contains immense resources which a system such as that of Alain Daniélou enables us to explore in a rational manner, inside its traditional models, beyond modal music, genres and forms, and pointing in the direction of new microtonal explorations which are yet totally unknown. Our work is now to build the instruments, be it by traditional or new manufacturing techniques, which will provide its means of expression and at the same time provide the essential tools to study the foundations of different types of music from India, Europe, and all over the world, from the past, the present – or the future. This would also be of considerable benefit to vocal practices, with which the spectrum of the 5-limit harmonies is one of the most coherent. The very question of the evolution of our listening experience is at stake, in order that we may achieve, through just intonation, the kind of harmony that the world is awaiting.

(1) Alain Daniélou, Traité de musicologie comparée, 1959, Hermann, Paris.
(2) P. Sambamoorthy, South indian music, Book IV, 1954, The indian music publishing house, Madras.
(3) Harry Partch, Genesis of a music, 1949, Da Capo, New York.
(4) Jacques Dudon, “Cohérence différencielle : une nouvelle approche de la consonance”, Actes des Journées de l’informatique musicale 1998, éditions du CNRS-LMA, Marseille — or in its English version, “Differential coherence : experimenting with new areas of consonance”, 2003, 1/1 – The Journal of the Just Intonation Network, Vol 11 #2, San Francisco.
(5) Jacques Dudon, “La confusion des genres : tentatives de résolution”, Actes du colloque “Autour de l’harmonie #1″, 2004, Thésaurus Coloris, Carcès, Mondes harmoniques, AEH, Le Thoronet.