What most fail to realize is that there are many more forms. The only problem is that most of them are “dead”. For instance, tanka was originally a form of waka which literally means “Japanese poem”. Waka had about five forms under its genre which were tanka, chōka, bussokusekika, sedōka, and katauta. As with most Japanese poetry, the common syllable count was 5-7 with these forms.
I partially understand why the other four forms died. They were all variations of 5-7 lines, but were longer. Why have five forms when you can have one? However, it’s their pattern variations and length that sets them apart and could make for good modern literature. There are modern attempts, but not a lot of awareness of the fact that they exist. In a regular poetry college course (a practicum, ideally), from my experience, you won’t hear about these forms. You probably wouldn’t even be asked to attempt one even if you did hear about them. I found about these forms via the internet.
Hopefully now that you have become aware (and have hopefully clicked the links within this entry to find out more), you might try out some of these on your own. I know I will. I might start with a chōka. I will leave you with a chōka by (I’m going to assume since it’s in article about chōka) Jack Huber:
Camouflaged
Suitably quiet,
he succeeds in blending in,
quite invisible,
while he waxes nonchalant.
Indeed, he'd rather be
anything but recognized,
just another sapling
camouflaged by the forest.
Every now and then
he is freed from his cocoon
and shows brilliantly
his talent for spectacle,
as crippling stage fright
releases its hold on him
ever so slightly.
Whispers become raucous songs,
telling the world, "Look at me!"