Python/Musical intervals (numpy matplotlib)
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This code creates three graphs of musical intervals that deviate from just intonation by a user-selected frequency. The graphs exhibit something analogous to an acoustical beat, except that these beats involve only phase shifts because each wave is a pure sinusoidal. To ensure that all three timescales are identical, it is important that the first graph plotted have the lowest beat frequency (here it is the perfect fifth.)
The code requires numpy and matplotlib. The modules scipy and sound file are called, but may be removed since they are not used in this version of the code. A sample output is shown below:
Plots
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Code
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import numpy as np #------------------------numerical operations on large arrays
import matplotlib.pyplot as plt #-----------creates and displays plots
import scipy, sys,os, time, math #----------scipy not used here
from scipy import signal #------------------signal not used here
import soundfile as sf #--------------------soundfile not used here
start_time = time.time()#-------------------lets me see run time for code
#FORMATTING CONSTANTS ----------------------these determine plot's appearance
figHigh,figWide=3,50#-----------------------figure Height and Width
Amp=1; samplerate=44100#--------------------standard CD sampling (not used)
dt=1/samplerate#----------------------------dt to samplerate conversion
slw=1#-------------------------------------slw signal's line width
barx=1#------------------------------------barx quasi-period marker width
barb=1.4#-----------------------------------barb beat marker width
fs=10#--------------------------------------fontsize, txp (latter not used?)
show_images=False#--------------------------show_images: option to show images
graph_dots=False#---------------------------graph_dots plots dots (not lines)
beat_dots=False#----------------------------beat_dots marks beats with dots
blu,pur="#0072B2","#CC79A7"#----------------colorblind friendly colors
ver,sky,yel,bgr="#D55E00","#56b4e9","#F0E442","#009E73"
MAR=2#--------------------------------------MAR-normalized margin
#ESSENTIAL CONSTANTS------------------------these determine what gets plotted
f_00=100#-----------------------------------Longest quasiperiod
tmar=MAR/f_00#------------------------------margin at start and stop
Df_q=.5#------------------------------------Df_p change p-tone pitch
Df_p=0#-------------------------------------THIS SHOULD ALWAYS EQUAL ZERO
BTS=1#--------------------------------------BTS=beats_to_show
for count in range(3):#---------------------counts the three intervals plotted
if count==0:#-------------------------- 3/2 fifth
interval="fifth 3-2"#--------------- p, q, f_0 defined
p , q , f_0 = 3 , 2 , f_00#-------- 1/f_0 = quasiperiod
q_0=q
if count==1:#------------------------------ 7/5 tritone
interval="tritone 7-5"
p , q = 7 , 5
f_0=f_00*q_0/q
if count==2:#------------------------------ 8/5 minor sixth
interval="min_6th 8-5"
p , q = 8, 5
f_0=f_00*q_0/q
print("count =%i: %s"%(count,interval))
T_0=1/f_0#---------------------------------- T_0 quasiperiod
f_p= p*f_0#--------------------------------- Defines p-wave pitch
f_q= q*f_0
f_b=abs(p*Df_q - q*Df_p)#----------------- f_b=2*beat freq
T_b=1/f_b#---------------------------------- Defines beat period
#Next, I verify that the first graph requires the longest time period
if count==0: T_M=T_b#----------------------- Check for margin error
if T_b>T_M:sys.exit("Margin error: i=%i"%i)#-
cents=abs( 1200*( np.log2(1+Df_p/f_p)
-np.log2(1+Df_q/f_q) ) )#------ deviation from just (cents)
tmax=BTS*T_b+2*tmar#------------------------ tmax=time plotted
twopi=2*np.pi# (twopi=2*pi)----------------- define twopi=2*pi
om_p,om_q=(f_p+Df_p)*twopi,(f_q+Df_q)*twopi#-define omega_p omega_q
datapoints=int(round(tmax/dt))#------------- datapoints
print("%i datapoints"%(datapoints))#-------- not standard for CD players
t=np.linspace(-tmar,#------------------------create numpy array for time
BTS*T_b+tmar,#-----------------Define timespan with margins
num=datapoints)
yp=Amp*np.cos( om_p*(t) )#------------------ yp (numpy array)
yq=Amp*np.cos( om_q*(t))#------------------- y = yp + yq
y=yp+yq#------------------------------------ (all are numpy arrays)
plt.figure(figsize=(figWide,figHigh))#------ plt=variable name of plot
i=0# --------------------------------------- initialize beat line location
while i*T_0 < tmax:#------------------------ Plot quasiperiod markers
plt.axvline(x=1/f_0+i/f_0, color=pur,linewidth=barx)
i+=1
if graph_dots:#------------------------------ Plot signal (dots or line)
plt.scatter(t,y,s=dotsize, color="k")#--- I used dots to verify lines
else:
plt.plot(t,y, color='k', linewidth=slw)#--Lines easier to see
plt.xlim(-tmar, BTS*T_M+tmar)#----------------Sets limits on axis
plt.axhline(y=0, color='k', linewidth=slw/4)# Plot axis (y=0)
a0=Amp #------------------------------------- Plot y_p and y_q signals
a1=Amp; offset=a0+2*a1 #-------------------- Offset signals downward
arr=np.empty(datapoints); arr.fill(offset)#-- Numpy array speeds things up
plt.plot(t,a1*yp-arr, color=ver, linewidth=slw)#Plot p-wave
plt.plot(t,a1*yq-arr, color=bgr, linewidth=slw)#Plot q-wave
i=0#----------------------------------------- Plot beats T_b=1/f_b
while i*T_b<tmax:# Plots vertical markers for the beats
plt.axvline(x=i*T_b, color=sky, linewidth=barb)
i+=1
i=0#----------------------------------------- Label quasiperiods T_0
while i*T_0<tmax:
s=int(i)#---------------------------------- s=string representing i
plt.annotate(s, xy=(i/f_0,2.25*Amp),#------ Places s on t axis at i*T_0
xycoords='data', fontsize=fs, horizontalalignment='center',
color=blu, verticalalignment='bottom')
i+=1
titlestring="%s %5.1f to %5.1f"%(interval,f_p+Df_p,f_q)
titlestring+=" (%5.2f cents off)"%(cents)#-------Title for graph
path2image=titlestring+".svg"#-------------------Names svg file
path2image=os.path.join("images", path2image)#---Location for image
plt.xlabel(titlestring,fontsize=fs,loc="left")#---TITLE
plt.savefig(path2image,dpi=1600,bbox_inches="tight")
if show_images: plt.show()#----------------------Show image
print("T_b=%f for %s"%(T_b,interval))
print("---------------- %s seconds" % (time.time() - start_time))
soundfile and playsound (pip installed packages)
editI discoverd that soundfile cannot create large OGG files, probably due their compression. So instead I made WAV files. And I installed playsound so I could hear the interval each time I ran the code. The latest version of the code currently resides at: