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Intro, More musical examples , The Koch Snowflake , One way of going inwards to smaller pitch intervals , Automated counterpoint
Let's start with the seed 0 1 2 0,
Let's play that tune a little faster, and then add, the seed as an ornament to each of the notes of the seed 0, 1, 2 and then 0, like this:
0 1 2 0 (add 0) 1 2 3 1 (add 1) 2 3 4 2 (add 2) 0 1 2 0 (add 0)
which is 0 1 2 0 1 2 3 1 2 3 4 2 0 1 2 0 .
Now lets add the seed again to each of these notes. It will start the same way as before, but our melody is longer now: 0 1 2 0 1 2 3 1 ... So add the seed to all those new notes like this:
0 1 2 0 (add 0) 1 2 3 1 (add 1) 2 3 4 2 (add 2) 0 1 2 0 (add 0) 1 2 3 1 (add 1) 2 3 4 2 (add 2) 3 4 5 3 (add 3) 1 2 3 1 ( (add 1)
and so on. So far we have added the seed to the notes 0 1 2 0, 1 2 3 1 of the previous melody.
Another way of looking at it is that we add the entire pattern so far to each of the original seed notes, to get
0 1 2 0, 1 2 3 1, 2 3 4 2, 0 1 2 0 (add 0) 1 2 3 1, 2 3 4 2, 3 4 5 3, 1 2 3 1 (add 1) 2 3 4 2, 3 4 5 3, 4 5 6 4, 2 3 4 2 (add 2) 0 1 2 0, 1 2 3 1, 2 3 4 2, [0 1 2 0 (add 0)
Both ways give you the same result, which is 0 1 2 0 1 2 3 1 2 3 4 2 0 1 2 0 1 2 3 1 2 3 4 2 3 4 5 3 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6 4 2 3 4 2 0 1 2 0 1 2 3 1 2 3 4 2 0 1 2 0 , (faster again)
We can keep on doing this as many times as you like. Here are the next two steps, each one faster than the one before.
These clips are for the recorder voice, Scale = equal temparament , Arpeggio = pentatonic . The seed numbers are degrees in the arpeggio, so 0 = first note of the arpeggio, 1 is the second note, and so on.
To experiment further with it, you can modify the Tune Smithy file recorder_0120_1.ts. It's saved with the number of layers set to 1. To get the other examples, increase the number of Seed Layers by 1 each time, and reduce Note (secs) value.
If you are musical, you may get a better idea of this by listening to some more examples to hear what it means.
Some of the fractal tunes bring it out particularly clearly.
Try the string quintet. Here the first violin plays every note of the melody line, while, the second violin plays the note that begins each pattern at the second layer, the viola plays the notes of the third layer, and so on down to the contrabass. The pattern is 0 1 0.
Here is the 'score'
Key: 1 = first violin, 2 = second violin, 3 = viola, 4 = cello, 5 = double bass.
Vertical lines show seconds. Notice that all the parts are playing exactly the same tune, at different speeds. Also here, they are all in different octaves. So, it is a canon by augmentation, one that works automatically because of the fractal structure. You give FTS a musical phrase to use as a seed, and it turns it into a canon by augmentation for you.
marimba with string quartet is the same tune as string quintet, with a marimba for the top line, and changes to the volumes of the notes in the tune, and the durations of the notes in the top line.
The counterpoint is based on the idea of passing notes. All the instruments start at octaves, then the marimba plays its seed as a kind of passing tone on the sustained octaves of the other instruments. Then the violin plays its second note, and the marimba joins it to start a new seed - at this point they are both playing a passing tone in octaves above the cello and double bass who are still on the first note. Finally they all join together in octaves on the second note for the double bass, however this is a passing note over an implied fifth part which would play the first note of each of the double bass's seeds, but here isn't played on a separate instrument. So it goes on.
Even wtih one part only playing, one hears some kind of pattern superimposed from the first notes of each seed, which gives a kind of counterpoint to the tune.
So basically it is a very simple idea. It's not an attempt to make music following traditional rules of harmony at all, but making something new that will work with a fractal tune.
You might also like to listen to major and minor scales fractal, for a more complex pattern. The recorder is playing it exceedingly fast at the top, then glockenspiel, wood block, and church organ follow at increasingly slower speeds. It takes some time for the church organ to change a note - more than two and a half minutes. The pattern is modulating back and forth through major and minor keys all the time and is meant to suggest some natural sounds, like the wind, as it can be on occasion.
When you listen to some of the other sound clips, the structure might not be so apparent. You can use the program to distribute the notes of the fractal tune amongst the instruments according to various methods. Also you can choose 'scales' which instead of ascending in a regular way, have some steps up, and some steps down. Effects of timings and dynamics, changing the instrumentation (such as mixing percussive and lyrical instruments), and shifting instruments for the parts up and down in pitch also add to the complexity of the resulting composition.
In fact, several of the pieces also use identical note height patterns, but with variations of timing, choice of scale, instrumentation and dynamics. For instance, flute , flute2, music box, and piano improvisatio all use the pattern 0 1 2 6 0.
The mathematical connection is with fractals of the type of the Koch snowflake:
There is a good introduction to the koch snowflake at this site: Fractals (introduction for kids - also suited for interested non mathematical adults
A fractal tune with pattern 0 1 0:
(treble clef, pentatonic scale, scale of A, sharps in dark blue)
Notice that the highest note in the melody increases with each step.
Two steps after the last frame in the animation, as a graph. This time in the whole tone scale, to give equal spacings in pitch between the notes.
This is part of it in the whole tone scale played very fast on pizzicato strings.
As you play the melody over larger and larger spans of time, you find that the melody almost repeats, but never repeats completely until you reach the end of the cycle, which could be many hours or even days, depending on the number of layers you choose to iterate.
The fractal melodies are like the Koch snowflake, but you build them up outwards rather than inwards, because it works more easily that way in music.
You can't go inwards to smaller and smaller details so easily in music because there are minimum pitch intervals between notes in the usual types of scale. However it can be done with unusual scales..
One can make scales with indefinitely small intervals, in an attempt at a closer parallel to the geometric case, and they can be interesting.
Try a scale with steps of an octave, then a third of an octave, then a ninth of an octave, and so on, or in cents:
0 cents 1200 cents 1600 cents 1733.333333 cents 1777.777777 cents 1792.592592 cents, ... (approaching 1800 cents and never quite reaching it).
The seed is 0 1 0 as before.
As you add extra notes between each one shown, the space beneath the curve remains clear, apart from some notes very close to the ones shown. The space above the curve shown eventually fills up with lines, as between any pair of notes you can find another one as close as you like to 1800 cents - one and a half octaves above the first note of the scale. It's not a continuous curve, but it has a type of exact self similarity. Can you see that the whole pattern is echoed in it's centre third? Also each third is echoed in its centre third? Can you see how the echoes continue to smaller and smaller copies?
The self similarity is of the same general type of pattern as the Koch snowflake, fractals with exact self similarity. The method of construction is similar too, adding identical smaller copies of a pattern to each of it's components. One could perhaps more exactly call this fractal the musical equivalent of Cantor's dust (Maths Encyclopedia entry).
Cantor's dust is what you are left with if you start with a line, remove the middle third of that, the middle third of each one left, and so on. The lowest notes of the fractal play out Cantor's dust. You have to suppose that each note that you hear is divided yet further into smaller notes. Cantor's dust has the paradoxical property of having no total length, yet having as many points in it as the complete line (see the Maths Encyclopedia article for details).
The higher notes show the result of doing another Cantor's dust construction on each of the middle thirds that was removed at every stage, then another one on each one of those, and so on. Eventually, every point in the line is reached by this method, so it's a way of filling the line by repeating the Cantor's dust construction infinitely often.
This is what it sounds like played on a marimba, with the notes quite fast. There would need to be many more notes between each of the ones played, indeed, infinitely many.
Notice the self similarity of the rhythmic patterns. Try listening to one of the pitches of notes only. Can you hear that each of each pair of low notes is in fact double. They are too close together to see as separate notes in the picture.
These fractal tunes are based on the idea of passing notes. In the original form, all the parts start in unison or at octaves (or more generally non octaves, whatever is the interval of equivalence).
Then the first, fastest part plays its seed above that unison / octave, as a passing note. The second part then plays its second note, with the first part joining it to start a new seed.
That unison of second and first part is a point of rest, but also a passing note with respect to the third part.
When the second part finishes its seed, the third part moves on to its next note, where it is joined by the first two parts, so now the three parts are all playig a unison / octave passing note with respect to the fourth part.
So it goes on. When the last part goes onto its second note, then it is still felt as an intermediate note because the seed has just started, and there is lots still to go.
Even with one part only playing, one hears some kind of pattern superimposed from the first notes of each seed, which gives a kind of counterpoint to the tune.
So basically it is a very simple idea. It's not an attempt to make music following traditional rules of harmony at all, but making something new that will work with a fractal tune.
What makes it fractal is that each part plays the same tune as all the others, with varied speed.
The musical seed gets transformed as it moves up and down in the scale, but the same thing is happening in all the parts, so they still all play the same tune - the same notes, at varied speeds.
This corresponds to visual fractals that look the same at all scales - the fractal tunes sound the same at all speeds (if one had parts to play for each of the layers of ornamentation).
So they are canons by augmentation - i.e. a canon in which each part plays the same tune, with some parts playing it slower than other parts.
The best way to hear this is to try speeding up the fractal tune so that you hear each part in turn playing the tune at its original speed. You will find it helpful to select Bs | Tempo, Note time & volume for tune| more | Time by | start of tune . The usual setting here is previous note , as that means that none of the notes need to be skipped, and the tune slows down if necessary to play them all. However here the idea is to increase the speed of the slower parts to match the original speed of the fastest part - so you want to skip the faster notes when they get exceedingly fast.
Listen for instance to ascending above the clouds.ts. First play the seed. Now try playing the tune, and you'll hear the fractal tune based on this seed played on the flute. Now play it eight times faster (click the down button for the Note time (secs) box six times). You'll hear the same tune at the same speed, now on the Cor Anglais. Play it eight times faster again, and you'll hear it on the Bass flute. Now play it eight times faster yet again and you hear it on marimba. Now go back to the original and listen to the fractal tune played on all those instruments, and see how many of the layers of the canon by augmentation you can follow.
Many of the ones in the original drop lists are strict canons by augmentation - try them out and see. Some are prob. not so easy to hear as others. For instance, experimental.ts is a canon by augmentation but with the guitar and flute at the same pitch, perhaps one mightn't hear it so easily.
If you find that you can't hear that this one is a canon so easily, try playing it at four times the speed and mute the guitar by selecting Voice | Rests into that part.
Others are like the eskimo_hexatonic.ts - it is a canon in the first three parts. The fourth part plays the same notes as the others in a canon, but with notes regularly spaced instead of following the rhythm of the seed.
Then, others like african_jungle.ts are still based on an augmented canon though one would be hard put to follow it - that's because the notes of the canon are re-distributed to all the parts in a complex fashion - and some of the parts may be transposed up / down by octaves - and then sustain ( Bs | Tempo, Note time & volume | Sustain ) is used to link together the notes in each part. To hear the original canon this fractal tune is based on, select Parts | Choose Parts By | By Layer with simultaneous notes .
Sometimes some melodic notes get dropped from the canon altogether, if they are played on a non melodic percussion part - see for example percussion_medley_with_15_tET_brass.ts . To hear all the notes of the canon here, one would need to play these parts on melodic instruments as well.
In this way, all the fractal tunes in these first two drop lists are based on canons by augmentation.
There are various notions of fractal, including ones that are exactly the sane at every levels (layer), like FTS, or ones that have similar structures at every level, but are not identical - the word fractal is a general name for a field of study rather than a precise mathematical term as such. It is indeed a field of maths, but its boundaries are somewhat vaguely delimited, if you ask what exactly counts as a fractal and what doesn't. Terms within the field like fractal dimension etc. and particular types of fractal have been given precise mathematical definitions. Examples of fractals that aren't identical at all levels include the mandelbrot set, and randomised fractals, also the term gets applied to natural fractal like features such as coastlines, mountains, and trees..
There are lots of fractal music programs, however I wonder if FTS is unique so far in using _exactly_ similar fractals to make music?
That's for the basic form of the fractal tune idea. The later transformations and remappings and fractal rhythms and polyrhythms make fractal tunes that are no longer exactly self similar, however, they are still based on the same passing notes idea for harmony / counterpoint.