Ever been amazed by plant's geometry? Lindenmayer systems,in short , Lsystems have been inspired by the recurrence pattern observed in plants. We will learn more about Lsystems , its history , applications and learn how Lsystems were used to generate plants and trees models, snowflakes and so on. We later also make use of Lsystems to attempt to generate carnatic music snippets based on raagas.
Introduction
Lsystem was developed in 1968 by Hungarian botanist named Aristid Lindenmayer to model plants mathematically.It is a type of grammar , where we form words using set of rules.Lsystem is based on rewriting and recursive functions. Basically we generate complex models by recursively generating and replacing with versions defined by set of rules.
Lsystems consists of three components :
Alphabet , Axiom and Set of productions

(V,w,p) : Where Alphabet(V) is the starting letter/instruction

Axiom(w) means how Alphabet(V) can be changed.

Set of productions are instructions of certain subsequences or subsets that can be transformed to another.
 Lsystems  context free grammar , which means in case A>a , A can always be replaced by a and there no restrictions like context sensitive grammar.
Suppose we take an example where Alphabet  'a' , Axiom  'a'>'ab' and set of rules/ productions  'b''a'
so,
a
> ab (a>ab)
> aba (a> ab and b>a)
> abaab (a>ab , b>a and a>ab)
:
:
LSystems are similar to Chomsky grammar but replacement in Lsystem takes place simultaneously.We can follow same rules and generate recursive patterns ,how? Rather than characters we will be printing directions.
Turtle representation
Turtle can be shown at a position (x,y,Î´) where x,y are coordinates and alpha as angle , as per directions the turtle crawls thus leaving its impression.
 'F'  go forward (on constant length).
 'G'  go forward without leaving a trail.
 '+'  turn right on given rotation angle.
 ''  turn left on given rotation angle.
 'B'  go backwards
For 'F' we just move straight for certain specified length(s)
where new_x = x+s*cos(Î´) and new_y = y+s*sin(Î´)
For '+' we just turn Î´ angle if its positive then clockwise else anti clockwise and turn opposite for '' for respective cases.'G' case will be just moving forward and no line will be made. Backward is opposite side of forward.
Edge rewriting
We have an initial starting polygon and then each of its edge is replaced by the initial structure.
Here we can see that initially we have a polygon  triangle , whose edges are replaced by the rules and resultant figure's edges are again replaced by the rules to produce a snowflake .
Node rewriting
"The idea of node rewriting is to substitute new polygons for nodes of the predecessor curve."Here each polygon is substituted with nodes of previous curve .
Lsystems have been used to model plants and observe the patterns in plants , also used to draw fractals,snowflakes.
Snowflakes
import turtle
def lsystem(i,axiom):
s = axiom
e = ""
for i1 in range(i):
e = processe(s)
s = e
return e
def processe(old):
news = ""
for c in old:
news = news + applyRules(c)
return newstr
def apply(ch):
news = ""
if ch == 'F':
news = 'FF++FF' # Rule 1
else:
news = ch # no rules apply so keep the character
return newstr
def drawls(tturtle, instruction, angle, dist):
for c in instruction:
if c == 'F':
tturtle.forward(dist)
elif c == 'B':
tturtle.backward(distance)
elif c == '+':
tturtle.right(angle)
elif c == '':
tturtle.left(angle)
def main():
inst = lsystem(4, "F++F++F++F") # create the string
print(inst)
t = turtle.Turtle() # create the turtle
wn = turtle.Screen()
t.up()
t.back(200)
t.down()
t.speed(9)
drawls(t, inst, 60, 4) # draw the picture
# angle 120, segment length 4
wn.exitonclick()
main()
Output :
Plants modelling
Bracketed Lsystems
Now we introduce two new symbols '[' and ']' for modelling plants where
 '[' signifies starting of the branch.
 ']' signifies ending of the branch.
 Just like stack problem of parenthesis.
 A stack can be used to append the positions at starting of the bracket and ended after the bracket ended.
#inside the lsystem function
stack=[]
tturtle.left(angle)
elif command == "[":
stack.append((tturtle.position(), tturtle.heading()))
elif command == "]":
tturtle.pu() # pen up  not drawing
position, heading = stack.pop()
tturtle.goto(position)
tturtle.setheading(heading)
An axial tree with X>F[[X]+X]+F[+FX]X and F>FF axiom=X iterations=5 , angle(i)=22.5
X denotes X rule to be followed , wherever X is there , replace with X rule.
Carnatic Music generation using Lsystems?
Can we try to generate music Lsystems? We have now seen how using recursive techniques , Lsystem shows that plants can be modelled by replacing each node with structure similar to the total previous structure.Music also has elements of patterns associated with it .Carnatic music is a type of Indian Classical Music where we have songs composed by millions of arrangements of just 7 pitches!
Let us see an example of Abhogi Raagam Varnam(a type of composition) named "Evvari Bodhana"
 Raagam : Descending and ascending order of Arrangement of pitches of Carnatic music S R G M P D N analogous to Do Re Mi Fa So La Ti.
 Varnam : A type of Carnatic Music Composition.For simplicity let us call it a song.
 Abhogi Raagam : incr  S R G1 M D S decre  S D M G1 R S D
 Arohana means increasing order of pitches , Avarohana means decreasing order of pitches
G1 here denotes the lower G,and here DD2 which means higher pitch D.
Let us see an excerpt of the song's pitches 
(Here ',' denotes a pause .)
R , G , G R S , S R S S D M D , 
M D S D , S D S R G , M G G R S 
If you observe , all the pitches are in line with the Abhogi Raagam.
Let us divide it into parts such that all parts are the substrings of either increasing or descending order of the raagam ,considered as the main string.
R,G, 1
G R S ,2
S R 3
S 4
S D M 5
D , 6
M D S 7
D , 8
S D 9
S R G ,10
M G11
G R S12
We have seen that, the music is being generated keeping in mind the source of pitches are according to the rules of raagam. The raagam'subtrings are either being inverted or combined with others to generate musical composition.If we represent graphically , the structure of graph seems that it can be mimicked usign Lsystems. But here we can see that , the raagam's substrings have been placed and its not necessary that all the pitches are used at once. Suppose if we want to use the same formula to generate music that was used to generate axial tree then where F signifies set of pitches/pitch then , what is F going to denote as F need not be complete raagam set as it can be any of the substring ?
Without Lsystems probably we could pick out random substrings from the raagam string and compose a musical piece , but will Lsystem be of any use in here?
Here we try to slowly replicate the logic of Lsystem to produce carnatic music on lines of raagam , here we experiment on Mohana raagam whose swaras go like : s r2 g2 p d2 s
We make r2,g2,d2 as notes through which sub parts of raaga emanate as s and p are natural notes(prakruthi swaras) and are not allowed to have variations.
import numpy as np
import pandas as pd
import random as rd
# 3 pitches
octave = ['s','r1','r2','g1','g2','m1','m2','p','d1','d2','n1','n2']
high = ['S','R1','R2','G1','G2','M1','M2','P','D1','D2','N1','N2']
freq=[1, 16/15, 9/8, 6/5 ,5/4, 4/3 ,45/32, 3/2, 8/5, 5/3, 16/9, 15/8]
base_freq = 261.63 #
freqs = {octave[i]: base_freq * freq[i] for i in range(len(octave))}
hfreqs = {high[i]:2* base_freq * freq[i] for i in range(len(octave))}
freqs.update(hfreqs)
freqs[''] = 0.0
print(type(freqs))
from scipy.io.wavfile import write
import numpy as np
samplerate = 44100
def get_notes():
'''
Returns a dict object for all the piano
note's frequencies
'''
octave = ['s','r1','r2','g1','g2','m1','m2','p','d1','d2','n1','n2']
high = ['S','R1','R2','G1','G2','M1','M2','P','D1','D2','N1','N2']
low=['sa','ri1','ri2','ga1','ga2','ma1','ma2','pa','da1','da2','ni1','ni2']
freq=[1, 16/15, 9/8, 6/5 ,5/4, 4/3 ,45/32, 3/2, 8/5, 5/3, 16/9, 15/8]
base_freq = 261.63
note_freqs = {octave[i]: base_freq * freq[i] for i in range(len(octave))}
hfreqs = {high[i]:2* base_freq * freq[i] for i in range(len(octave))}
lfreqs = {low[i]:(1/2)* base_freq * freq[i] for i in range(len(octave))}
note_freqs.update(hfreqs)
note_freqs.update(lfreqs)
note_freqs[''] = 0.0
return note_freqs
def get_wave(freq, duration=0.5):
amplitude = 262
t = np.linspace(0, duration, int(samplerate * duration))
wave = amplitude * np.sin(2 * np.pi * freq * t)
return wave
def l_systems(note):
if note=='r2':
choices = ['r2','r2s','r2g2','r2g2p','r2g2pd2','r2g2pd2S']
ind = rd.choice(choices)
return ind
if note=='g2':
choices = ['g2','g2r2s','g2p','g2pd2','g2pd2S']
ind = rd.choice(choices)
return ind
if note=='d2':
choices = ['d2','d2S','d2p','d2pg2','d2pg2r2','d2pg2r2s']
ind = rd.choice(choices)
return ind
return note
def get_song_data(music_notes):
note_freqs = get_notes()
notes=""
for note in music_notes.split(''):
if(notes!=""):
notes=notes+""+l_systems(note)
else:
notes=l_systems(note)
print(notes)
#for i in range(0,len(music_notes)1):
#notes=notes+(l_systems(music_notes[i]+music_notes[i+1]))
#print(notes)
song = [get_wave(note_freqs[note]) for note in notes.split('')]
song = np.concatenate(song)
return song.astype(np.int16)
def main():
music_notes = 'sr2g2pd2Sr2g2Sr2pg2'
data = get_song_data(music_notes)
data = data * (16300/np.max(data))
write('song.wav', samplerate, data.astype(np.int16))
if __name__=='__main__':
main()
We used basic modification to change 'sr2g2pd2Sr2g2Sr2pg2' to this: 'sr2g2r2spd2pg2Sr2g2pg2r2sSr2spg2pd2'.
Now we iterate for 5 times where each time the generated snippet goes more changes and different orientation notes emerge.
def get_song_data(music_notes,iteration):
note_freqs = get_notes()
notes=""
# iterate for 5 times
# each iteration , updated string gets even more updated .
for i in range(0,iteration):
for note in music_notes.split(''):
if(notes!=""):
notes=notes+""+l_systems(note)
else:
notes=l_systems(note)
print(notes)
song = [get_wave(note_freqs[note]) for note in notes.split('')]
song = np.concatenate(song)
return song.astype(np.int16)
def main():
music_notes = 'sr2g2pd2Sr2g2Sr2pg2'
data = get_song_data(music_notes,5)
data = data * (16300/np.max(data))
write('song.wav', samplerate, data.astype(np.int16))
Generated snippet : link
Now we automate it even further . We generate substrings on our own for given raagam and then generate for the following swaras with variations  r,g,d,n
def substrings(pos,aro):
res=[]
res1=[]
stri=''
#generate substrings : eg 'r2','r2s'..
#based on arohana and avarohana(ascending and descending order of swaras)
for i in range(pos,len(aro)):
if(aro[i]=='2' or aro[i]=='1'):
strin=aro[pos:i+1]
res.append(strin)
else:
if(aro[i]=='S' or aro[i]=='s' or aro[i]=='p'):
stri=stri+aro[i]
strin = aro[pos:i+1]
res.append(strin)
return res
# We generate randomised systematic substrings for all vikruthi swaras
def l_systems(note,aro,ava):
if note=='r2' or note=='r1':
#choices = ['r2','r2s','r2g2','r2g2p','r2g2pd2','r2g2pd2S']
pos = aro.find(note)
pos1=ava.find(note)
res=substrings(pos,aro)
res1=substrings(pos1,ava)
res1=res1[1:]
res=res1+res
res.append('')
res=np.asarray(res)
#print(res)
#choices=res
ind = rd.choice(res)
return ind
if note=='g2' or note=='g1':
pos = aro.find(note)
pos1=ava.find(note)
res=substrings(pos,aro)
res1=substrings(pos1,ava)
res1=res1[1:]
res=res1+res
res.append('')
res=np.asarray(res)
#print(res)
#choices=res
ind = rd.choice(res)
return ind
if note=='d2' or note=='d1':
pos = aro.find(note)
pos1=ava.find(note)
res=substrings(pos,aro)
res1=substrings(pos1,ava)
res1=res1[1:]
res=res1+res
res.append('')
res=np.asarray(res)
#print(res)
#choices=res
ind = rd.choice(res)
return ind
if note=='n2' or note=='n1':
pos = aro.find(note)
pos1=ava.find(note)
res=substrings(pos,aro)
res1=substrings(pos1,ava)
res1=res1[1:]
res=res1+res
res.append('')
res=np.asarray(res)
#print(res)
#choices=res
ind = rd.choice(res)
return ind
return note
def get_song_data(music_notes,iteration,ragam):
note_freqs = get_notes()
aro = ragam[0]
ava = ragam[1]
notes=""
for i in range(0,iteration):
for note in music_notes.split(''):
if(notes!=""):
notes=notes+""+l_systems(note,aro,ava)
else:
notes=l_systems(note,aro,ava)
print(notes)
song = [get_wave(note_freqs[note])[0] for note in notes.split('')]
t=[get_wave(note_freqs[note])[1] for note in notes.split('')]
song = np.concatenate(song)
t=np.concatenate(t)
return song.astype(np.int16),t
This time we experiment with Abhogi raagam : Abhogi snippet
Conclusion
With this article at OpenGenus, you must have a strong foundation on L System.
As the knowledge regarding carnatic music evolves , more variations can be accomodated in the code.Hence Lsystems are simple algorithms used for change and regeneration and using Lsystems we have tried to produce carnatic music given the raagam and their arohana and avarohana.We can further improve the algorithm by tuning the sound so that it appears more realistic or like some Indian Instruments.We can also improve the algorithm in such a manner that it produces systematic compostions(Varnam,Krithi,Kirthi etc)
This is the colab link to musical lsystems