Si vous n'êtes pas redirigé vers la page d'accueil après quelques secondes, cliquez sur ce lien
Click this link if you are not automatically redirected to the home page after a few seconds
Little story On an ordinary autumn morning, as I went to my work, driving maybe a little too fast on the highway (about 150 km/h, I know it's bad), I glanced absent-mindedly at the tachometer, witch central display told me that it was forty-eight past eight and that my car covered exactly 84,8484 kilometres. After the first turmoil - No, I'm not so late, I just forgot to set my clock to wintertime - the symmetrymaniac musician who lies dormant inside of me (but sleeps with one open eye), resolved to use some other day those so aesthetically arranged two numbers in a future musical composition... Thus was born this modest topical canon. Principle Two themes will be used. The first one will take the place of the digit '8' (let's name it T8), and the second one (T4), half the size of T8, will take the place of the digit '4'. If we multiply their values by 1.5 in the highest voice, we get the following layout : Ideally, all voices should end together in a final chord. Furthermore, I would have liked to take also into consideration the speed of 150 km/h. So I decided to add a second episode, with reduced values : unchanged values for the highest voice and values reduced by a half for the lowest voice. Adding a last T4 theme at the end of the lowest voice makes it possible to have a total of 150 quarter-notes for each voice (we omit the rests preceding the entrance of a voice and the final pause sign). Choice of the themes Theme T8 : Theme T4 : These two themes will go through numerous transpositions and inversions witch make them evolve from D minor to other keys and then return to initial key at the end of the piece. Final result The canon itself is rather poor and will serve as a basis for the harmony. It will be played either on the organ (for a richer sound) or on the piano or the harpsichord (for too sensitive people). I added two accoustic guitar parts, unified by some imitations and weak symmetries. That finally gives : (it begins with the same first 9 notes than the theme from the Two swapping monozygotic canons Bidon Canon for 3 keyboards It's a perpetual 3 in 1 canon on the following theme : The rest of the melody is a development in the classical sense, including variations on this theme and its rythmic cells. Voices enter at regular time intervals of 9 quarter-notes, twice at the descending fourth : The first voice starts in G major, the second one in D major, the third one in A major. Then the piece evolves, regularly modulating to the fifth every 9 quarter-notes (In the strict sense, those are not modulations, but rather 'shaded tones'). It travels all over the whole cycle of fifths, until the initial theme is played again in G major. So, if we use the following notation, we obtain a canon at the descending fourth with two responses that are both real and tonal : Summary graph Chinese jazz-rock in the form of a canon for 6 voices The first part of the piece is a 6 in 1 canon, as free as it is complex, on this theme : Some voices are inverted (mirror) and/or in double values (by augmentation). They enter with irregular and sometimes very short time intervals : In the middle of the piece, while both voices in double values go on playing their own part, the other four voices (in simple values) have finished. They are replaced by a free voice and two canons 'with holes', forming a weak "(6 in 3) + 1" canon. I confess I'm not plainly satisfied by the structure of this second part. Despite its interresting mood, it reveals the limits of the run for complexity. At the outset I wished to add to the voices in double values two retrograde and proportional canons (with 2/3 and 3/4 as temporal ratios, witch is pretty thin on the ground), the first one being inverted and starting with the theme from the Canon hexaphone spectroïde . So I would have obtained a triple canon that don't pull any punches. But along the way I finally thought that it would be a pity if the other three voices were drown out and that it would even be better if they could play alone during some bars. So I let the snatches of those two additional canons in the state they were. quater-note between the 1st and the 2nd voice quarter-notes between the 2nd and the 3rd voice between the 3rd and 4th and 5th voices between the 5th and the 6th voice Canon C78 Three themes were build on the basis of the R78 rythm . Let's name them respectively T78a, T78b and T78c : Notes pitches were specifically selected so that each of the three themes can fit into the following construction : A first 5 in 1 canon, with a time-lag of one whole-note for each voice entrance ( T = 8 eighth-notes), progressively moving away from C-3, alternately to high et low registers, first by inversion, then by simple transposition, each time with an interval of 9 semitones for the first note : A second 5 in 1 canon with the same inversions and transpositions, but this time with a time-lag of one double-dotted half-note for each voice entrance T = 7 eighth-notes) : The six resulting canons are arranged in such a way that the last two notes of one canon are synchronized with the two first notes of the following one. This graph shows how the six canons are ordered and sums up each voice characteristics. Academic 4 in 1 canon by inversion So, a rather classic canon : Voices enter at regular time intervals of five bars. The second one is inverted towards G3. The third one is transposed to the descending octave. The fourth one is inverted towards C#3. (So, it is transposed to the descending octave towards the second voice). The whole is truncated at bar 35 and each voice is extended with a pause sign. Melody Summary graph Canon on Fibonacci series Principle All bars have the same time signature (4/4) and the number of notes begining in each bar follows Fibonacci series As this piece is a 2 in 1 canon, where the follower enters one bar after the leader, this number of notes must be distributed between the voices. Furthermore, as Fibonacci series grows rather quickly for this usage, we decide not to exceed 21 new notes in a bar. When the melody reaches this number, its retrograde is juxtaposed to it. The complete melody that we obtain is thus symmetrical. So this canon is both straight and retrograde. Let's note that the second half of the piece follows Fibonacci series reluctanctly, except for the last three bars. This lastest anomaly stems from the double whole-note witch must start the melody (a new note in the first bar, none in the second one). To obtain the real decreasing Fibonacci series in the second half of the piece, all we would have to do is to replace that double whole-note by one whole-note followed by a one-bar rest. Realisation Here is a solution that respects this pattern, a strict canon at the fifth in D minor for two brass instruments : Here are two canons on the same theme, almost with the same melody. They are meant to be played one after the other, although the first one is rather written for keyboards and the second one for wind instruments (but well, if you dwell on this... ) The First One The first one is a crab canon, accompanied by two free voices witch constitute the bass and remain silent after two thirds of the piece, except for the final chord : Summary diagram The Second One The second one is a perpetual 3 in 1 canon. The 3 voices enter at regular time interval at the ascending fourth (less 2 octaves for the third voice, so as to obtain a bass). The 'shaded tones' technique applied here is the same as used in the Canon for 3 keyboards , but we travel all over the circle of fifths in the opposite direction. The cadence witch closes the piece in D minor is not part of the canon. Final result If we juxtapose them, we obtain a strange form, both bipartite and tripartite, homogeneous and contrasting. Two swapping monozygotic canons Both of them begin with these two melodic parts (with 4 whole-notes between them in the first canon; directly juxtaposed in the second one) : These notes are differently distributed between the higher two voices of each canon. And this is what we get when we juxtapose them : The counterpart of this method in painting could be the one used by Salvadir Dali in the Metamorphosis of Narcissus (1937), where two identical profiles come from very different objects and contexts. Canon in tribute to the city of Mons In this 3 in 1 canon (the second voice is in mirror) you can hear the same theme 11 times in each voice, alternately straight and inverted , with different pitches : This theme is always present, and the time interval between two of its successive entries in the same voice is shorter and shorter : From 39 eighth-notes between first and second entry, to 37 sixteenth-notes between the last two ones. The three voices enter with a time interval of 13 eighth-notes, so the piece is similar to a closer and closer stretto, particularly between the lowest and the highest voices. And here is another arrangement for the last 18 bars : The canon is unchanged, but played by other instruments and accompanied by dissonant chords. It's a little more oppressing : Credits Sorry, this page is not available yet... The softwares I used I composed the music with (score screenshots were taken from this software), and with a personal software named FabZik witch I would had offered to you with a great pleasure if I had written it in a distributable release. The website was developed with and The administration section (but normaly you have no access to it) relies on The morphing from Bill Bruford to Bruce Lee, witch illustrates the Chinese jazz-rock was created with the freeware Resources taken from other websites The characters of the Lumecon who appear in the page witch present the Canon in tribute to the city of Mons were cut-out from a stamp photography, found on the website of Some of the sounds produced by the were involuntarily offered by the following singers : in by or The other ones were recorded from a or were generated by a The is a animation, just like the The punk duck witch I cut into a puzzle comes from an images collection by cool_colonia4711, appearing at The 'Camels crossing' pannels were found more or less all over the web, notably at 's web site. Hexaphone spectroid canon Principle The structure of this canon refers to the breaking down of a complex sound wave into a fondamental sound and his harmonics . This principle is applied here to a melody instead of a sinusoid sound wave : On voice plays a 'fundamental' melody and we add its 'harmonic' melodies : The 1st voice plays the melody as it is. The 2nd voice plays it twice as quickly, with a doubled frequency (at the octave). The 3rd voice plays it 3 times as quickly, with a trebled frequency (atthe redoubled fifth). The 4th voice plays it 4 times as quickly, with a quadrupled frequency (at the redoubled octave). The 5th voice plays it 5 times as quickly, with a quintupled frequency (at the twice redoubled major third). The 6th voice plays it 6 times as quickly, with a sextupled frequency (at the twice redoubled fifth). The following graph shows how each voice will play the theme in different forms : In theory, there's no reason why we can't add an infinity of 'harmonics' to a 'fundamental' melody. However I had to limit myself to the first five 'harmonics' because the tools I use for composing and playing my tunes don't allow to make hear intervals smaller than a semitone. And yet, if we want to obtain the sixth harmonic of a note (to multiply its frequency by 7), we inevitably get a frequency out of our European chromatic scale. Example : The sixth harmonic of A-3 has for frequency 440 Hz x 7 = 3080 Hz, a note between F#-6 et G-6. Construction of the theme My first idea was that each note played by the bass (the 'fundamental' melody) should correspond to the entrance of another voice. So the rythm of this melody would have been : But I was not very inspired by it. I rather choosed a more regular rythm, superimposing three pulses witch slice the theme in 6 periods of 5 times : 10 periods of 3 times : 15 periods of 2 times : I choosed the following pitches for the notes : (In this sound excerpt, I increased the tempo to spare your nerves) Final result In essence, this canon is perpetual and starts with a major chord (at least if we limit ourselves to the first five 'harmonics'; over that, we get a 'spectral' chord). I choosed to play two consecutive cycles. In the first one, only the last instance of the theme is played by each voice. So they start one by one, from the lowest to the highest voice, on a more and more hasty way. The second cycle is conventional. The first chord is repeated at the end, to close the piece. Gérard Grisey is generally recognized as the first composer who refered to the spectral analysis of a sound in the form of one of his works : the last episod of Périodes (1974), later included in Les Espaces Acoustiques, witch I heartly recommend you. However, the principle behind Périodes, the 'instrumental synthesis', strongly differs from the one used in my little canon, prooving that the starting idea is rich and may lead to much different results. Dromadaire Blues I was about 17 when I wrote this piece, on a Yamaha DX100 synth, with a second hand Rolland MSQ-100 sequencer, a pencil and a rubber. It is some kind of opera overture, in pompous style. At this time I listen much to the first albums by Queen, Mike Oldfield and ELP. The instrumentation is more recent. As for the title, this is what inspired me the bars 44 to 57 (from 1:26 to 1:54). Fibonacci series Definition Mathematician Leonardo Fibonacci (Pisa ca 1175 - Pisa p 1240) published in 1202 his Liber abbaci, witch spreaded in Europe the mathematical knowledge he brought back from his journeys in the Indo-Arab world. The series that takes its name from him is an infinite series of natural numbers in witch each term is given by the sum of the preceeding two terms. First and second terms of the series are both fixed to 1 : Some interesting properties Extended in the negative whole numbers, it is symmetrical to 0, with alternated sign + and - : The ratio of a term to the preceeding one tend to the golden number, i.e. the positive root of the equation Therefore, a curve circumscribed in Fibonacci squares tend to a logarythmic spiral, witch coefficient is Golden number to the power of 4 Length of segment B length of segment A ... and logo creaters seem to like this spiral : In the decimal system, the same last digit comes each 60 values. The last 2 (3, 4, 5, ...) digits are subjected to cycles too, but unfortunately neither PHP nor Javascript can handle such large numbers with the needed precision. In binary, the last digit is subjected to a cycle of 3 values. the last 2 digits are subjected to a cycle of 6 values. the last 3 digits are subjected to a cycle of 12 values. the last 4 digits are subjected to a cycle of 24 values. and so on. In this table, we can observe that each Fibonacci number of rank 3k is even, the others are odd. F(nk), i.e. Fibonacci number of rank nk (n and k are natural numbers), is always a multiple of F(n). A consequence is that if F(k) is a prime number, then k is a prime number too, at least from rank 5. And this list is far from being exhaustive... Exercise If F(n) is Fibonacci number of rank n, Proof that with And and Personnal use Faced to so much richness, I'm almost embarrassed of using none of those amazing properties in my Canon on Fibonacci series Maybe for another time... The ratio of a term to the preceeding one tend to the golden ratio To know more about it Some links about Fibonacci Series and the Golden ratio : In French Wikipedia in French In English Wikipedia in English !!! Activate on your navigator to obtain the first 78 Fibonacci numbers !!! on your navigator to obtain the tables showing the periodicity of the last digit !!! on your navigator to obtain the tables showing the cycles in Fibonacci series in binary !!! The Golden ratio The Golden ratio ( ) is the positive root of the equation From an aesthetic point of view it represents an ideal ratio for the sides of a rectangle : If from a rectangle witch has two sides in the golden ratio we remove the biggest square as possible, what's left is another rectangle with two sides in the golden ratio Among other properties, the golden ratio is strongly bound to the Fibonacci Series If and then Harmonics A pure sound wave comes in the form of a sinusoid. But a natural sound is rarely pure : it generally goes with several harmonics witch enrich its timbre. A harmonic is itself a sinusoid wave, witch frequency is an integer multiple of the fundamental frequency. Depending of the sound source, the amplitude of the different harmonics differs (timbre). And it also differs in the course of time between the striking up an the end of the sound (envelope) This is the principle of the breaking down of a sound into its different harmonics. It is used as a basis for my Hexaphone spectroid canon Harmonics A pure sound wave comes in the form of a sinusoid. But a natural sound is rarely pure : it generally goes with several harmonics witch enrich its timbre. A harmonic is itself a sinusoid wave, witch frequency is an integer multiple of the fundamental frequency. Depending of the sound source, the amplitude of the different harmonics differs (timbre). And it also differs in the course of time between the striking up an the end of the sound (envelope) This is the principle of the breaking down of a sound into its different harmonics. It is used as a basis for my Hexaphone spectroid canon How to activate JavaScript on your navigator ... Mozilla Firefox In the menu Tools, select the item Options : Select the Content tag and check the 3rd box : Activate JavaScript : ... Internet Explorer In the menu Tools, select Options Internet : Select the Security tag, zone Internet and click on Customize level : Dans the section Script, choose Activate : Welcome to my musical compositions notebook. If you decide to linger, you will discover and hear I see that you deactivated JavaScript from you navigator. It's a great pity because you won't see all the wonderful animations I spent hours and hours to build only for your entertainment. I don't want to influence you, but if I were in your position, a would activate it. If you want, I can even remind you how it can be done. So please, activate it !!! Contents (oh well, that means "Fabrice's music")
Caution : If you are a real born English speaking person, you could be horrified by my language. So, if you want to help me, you can send me better translations, then I shall be eternally grateful to you and your name will appear forever in the Credits page of this website.
It's up o you... Use your mouse or the keyboard of your computer. Pianoafafa E-mail If you want to send me... Congratulations Encouragements Better English translations Questions Correctives Constructive criticisms Insults Spam Viruses and Trojan horses ... you can fill this form : Tour name Your email address Subject Send Construction of the rythm R78 We wish a rythm that can be heared, depending of the context, as having seven times (7/8) or eight times (8/8). Let's superimpose two pulses, one given by whole-notes, one by double dotted half-notes : You can find another example of rythm construction by superimposing different pulses in the description of the Hexaphone Spectroid Canon The final result fits to our requirements : Incidentally, we obtain a 'unretrogradable' rythm as would have said, i.e. a rythm witch is its own retrograde. So, when we will retrograde it (cf. infra), it will retain the properties we expect from it. As it is, we can't get a very singsong melody from it, we must enhance it by the insertion of some extra notes. I arbitrarily decided that the number of notes in each 7/8 bar will be growing in each bar : 1-2-3-4-5-6-7 (I leave the last bar). This can give, for example : Let's retrograde this last rythm : Not too bad. We can proudly call it R78 and use it to create the three themes of the canon C78 Saturnin's Tetralogy Canon and double canon in memory of Saturnin This piece begins with a 4 in 1 canon. One voice (the bass, played on tuba) has doubled note values. When it has played half of its part, the other voices have finished their own. At this moment, a second episod begins : While the tuba continues playing its melody in double values, the clarinet plays the same part again, and the oboe and the trombone play a canon by inversion on a second melody. Melody 1 Melody 2 Here is the final result (only half of each note value is played) Saturnin's Tetralogy Like the first , the second movement of the Teralogy begins with a 4 in 1 canon, where one voice (the bass, played on tuba) has doubled note values. The second episod starts as soon as the highest three voices are turned off : While the tuba goes on playing his part in doubled note values, the clarinet and the trombone play again their own part unchanged, but they are now preceded - and not followed, as they were in the first episod - by the oboe, which melody is now a simple transposition at the fifth of the leader, instead of being an inversion. So this melody is heared 7 times, in 5 different forms which gives Saturnin's Tetralogy 3. Terrine sarladaise (Summary : the story so far and beyond) ... in the form of a double canon by inversion and augmentation, where the first follower precedes its leader. The second movement started in D and ended in E, witch doesn't go down well in certain circles. So, in order to show a minimum of respect for the tonal system, this one will start in E and end in D. With a view to constrast, I wanted it to be anarchic and nervous (punk ?). In the opposite, I opted for a collage so as to keep some unity in the whole work. Out of laziness too. Melody 1 Melody 2 and altogether that comes to this Saturnin's Tetralogy Saturnin returns Unlike the first movement , the fourth and last one begins with a double canon and ends with a 4 in 1 canon. Like in the three other movements, one of the four voices (clarinet) plays the same melody twice without a change, and this melody is played by the tuba in double values. In the second half of the piece, this melody is also played by the trombone and the oboe, witch during the first half played a canon at the descendent fifth on another melody. Melody 1 Melody 2 This is the final result Implemented technicals Sorry, this page is not available yet... Saturnin's Tetralogy This is a suite of canons with some common features. They are all written for 4 voices. One voice plays twice the same unchanged part while another one - the bass - plays it in doubled note values (inverted or not). Each piece includes a 4 in 1 canon (except the third one) and a double canon (except the second one). You can listen to them independently - just click on their graphic representation below, where each melody is assigned to a different colour ... or listen to the whole suite in one go Pieces list Title Part. Genre Minim. Perpet. Inv. Retrog. Proport. Accomp. Listen.