How to change numbers on axis of colour bar

I have produced a contour plot and have a colour bar but it's labels are such random numbers which I assume just comes from the data and I've been trying to find a way to relabel it so the numbers go from 1-12 instead and go up in integers. Is there any way I can do this?

I tried changing the limits but it didn't change anything at all.

When the colour bar gradient is smooth, I've found this works well:

#Create the colour bar
cbar = plt.colorbar() #alternatively: plt.colorbar(ticks=range(1, 13))

#Adjust ticks and trim the limits
cbar.set_ticks(range(1, 13))
cbar.ax.set_ylim(1, 12) #optional extra, to clip the overhang

If the gradient wasn't as smooth (i.e. fewer levels in the contour plot), this method wouldn't work as well.

Reproducible example

import matplotlib.pyplot as plt

#Test data
import numpy as np
xx, yy = np.meshgrid(*[np.linspace(0, 1)]*2)
z = np.cos(xx**2) * 9 + np.sin(yy**2) * 9 - 4

#Plot and adjust colour bar ticks
plt.contourf(xx, yy, z, cmap='PiYG', levels=100)
plt.gcf().set_size_inches(8, 3)
plt.axis('off')


cbar = plt.colorbar(label='colour bar label')
#alternatively: plt.colorbar(ticks=range(1, 13))

#Adjust ticks and trim limits
cbar.set_ticks(range(1, 13))
cbar.ax.set_ylim(1, 12) #optional extra, to clip the overhang