1. Even numbered lines have the same rhyming sound.
2. Couplets.
3. The end words of the odd numbered lines with the title tell a brief story.
4. The number of couplets is unlimited.
5. The odd numbered lines do not rhyme but tell a story.
6. The even numbered lines are mono-rhymed.
In this example the even numbered mono-rhymes are coast, toast, most, host, boast.
The odd numbered lines tell the story. This Fourth of July July Fourth bursts with independence. The story is in bold.

This Fourth of July

Some grandchildren are away this July,
Two are in New Jersey on the east coast.
We won’t see grandchildren on this Fourth
so to them I salute a happy toast.
As the fireworks display bursts
less abundant than most
I will remember the times with
the grandchildren at the parties we host.
This year we celebrate adolescent independence
and the love of the grandchildren we boast.

My Thanks to Linda Varsell Smith for her contributions above.

Raccontino (Italian meaning narrator or story teller) is a poetic narrative written in any number of couplets linked by a single rhyme. Found at Writer’s Café. I’ve been unable to find a history or original example of this form. One source on line suggests this is an English form but no time frame is indicated nor other reasons given for this assumption. Because the name has Italian roots, is syllabic rather than metric (which is more typical of Italian poetry than English poetry) and carries a single rhyme (which is much easier in Italian than English), my assumption is that the frame itself also has Italian roots.

The Raccontino is:
• narrative, tells a story.
• written in any number of couplets.
• syllabic. The number of syllable is set by the first line. Whatever number of syllables occurs in the first line should continue throughout the poem.
• rhymed xa xa xa xa xa xa etc. x being unrhymed.

Pasted from http://www.poetrymagnumopus.com/index.php?/topic/1162-raccontino/
My thanks to Judi Van Gorder for years of work on this fine PMO resource.

Specifications restated
Raccontino is most likely of Italian origin.
It is:
A single stanza consisting of any even number of lines.
Rhymed: the rhyme pattern is xaxaxaxa…
Formulaic: The end words in the un-rhymed lines tell a brief story.
Isosyllabic: All lines having the same number of syllables, at the poet’s discretion.