I will try to explain little bit why I think Red and Blue LED's are optimum for growing chaeto. (Note: In my research sometimes I had to visit interesting sites and forums. Thanks to the folks that grow cannabis. It appears that they put a lot of work into light optimization).
In my short note I will try to comment only about the chaeto and cyanobacteria.
Chlorophyll is a green pigment found both in algae and cyanobacteria. The chlorophyll has various states (pigments): Chlorophyll a, Chlorophyll b, Phycoerythrin, Phycocyanin, and few more others.
Chaeto has mainly Chlorophyll a and b. I think there are two more other components (c1 and c2), but it looks that they are similar to Chlorophyll a in terms of light absorption.
The absorption spectrum for Chlorophyll a has two peaks at 430 nm (violet-blue) and 662 nm (deep red).
The absorption spectrum for Chlorophyll b has two peaks at 453 nm (blue) and 642 nm (red).
Cyanobacteria has the other two pigments listed above (Phycoerythrin and Phycocyanin).
The absorption spectrum for Phycoerythrin has a maximum at 550nm (green).
The absorption spectrum for Phycocyanin has a maximum at 620 nm (orange).
The Violet UV LED's have a spectrum range from 410 to 420 nm.
The Royal Blue LED's have the spectrum distributed with a maximum at ~455 nm.
The Philips Deep Red LED's have the maximum output at 660 nm.
White LED's would have one peak at ~450 nm and the other one between 550 and 600 nm.
Combining all of the above, I think that for growing chaeto, the best is to have a combination of Royal Blue, Deep Red and some Violet UV LED's. I attached a picture for clarification.
White LED's, fluorescent lamps and incandescent lights will grow chaeto (no doubt), but a lot of energy will be wasted, consumed in a zone where chaeto does not absorb. Instead, cyanobacteria will use this portion of the light.