Polyhymnia teaches songs of praise and the history of the deities. She specializes in Arcanist because Arcane is the land she has spent most of her life on. But she is learning more and more about Shadowbinder too, now that clan Pergamon has settled in a bit better and gotten the progress of healing started up. It has been so interesting that she now plans to improve her knowledge on all the other deities too. Who knows where her students end up so it would be useful if she could teach them more than general knowledge about them.
Also, embarking on this endeavor would mean she could delegate her singing lessons to other teachers without feeling too guilty about it. She loves the songs and is a decent composer herself, but she hates her own voice and teaching the songs has always made her uncomfortable. Calliope already helps her out from time to time so all she would need to do is persuade Erato to do the same. Tiore and Fjeln also use her help and resources in their plays and performances, so she might be able to convince them to take on a few of her students as a return service. Besides, those two probably already know how to perform the songs better than she ever did.