Q:

Seamless animation of Wave procedural texture

Following advice from this question on Stack Exchange I’m now able to seamlessly animate generated textures. I’m now using these methods to animate stripes via the Wave texture.
Something that I’ve noticed is that the stripes only will appear seamless if the scale is set 1.88889 or multiples thereof (see gif below). Is there significance to this number? What does it relate to?

A:

You wrong. Perfect number is 1,884955781 ( or Pi / 5 * value in modulo )
For answer the question, first you need to know, that exactly Wave Texture node do.
So I build in simpler nodes exact copy of Wave Texture node:

This node setup produce similar result as Wave Texture node, and you can check this:

So now we can follow the nodes, to know where 1.8888 is hidden.
We begin in modulo coordinates here, from reference question. As i see, you have 3 repeats, so I think you set 0.3333 in modulo. You didn't give any information about it, but it's looks like that I'm right.
 
x,y,z now is from 0 to 0.3333.
Next stop is my node setup, in first nodes texture coordinates are multiplying with the result of this two:

First, multiply 10: 0.3333*10=3.3333
Next it is our X-number, that you think is 1.8888. I call him X, so result is 3.3333*X.

Last stop - sine. You need to know, that Sine repeats in 2*Pi period. So now we can get eaquation:
3.3333*X=2*Pi

So X will be:
X=(2*Pi)/3.3333

X=1,884955781

So, you wrong. perfect number is 1,884955781

