How can I query for all artists who were born after 1720 and died before 1900 on https://musicbrainz.org?
I need to retrieve their IDs and some information about them. Is it possible to get data in JSON format?
First, Musicbrainz only returns XML, as far as I know, so you'll have to convert the results to JSON.
To answer your question, it doesn't look like you'll be able to get the data you want in a single call. (The following is based off the XML Web Service Search documentation.)
This call will retrieve all artists who were born in a given year:
http://musicbrainz.org/ws/2/artist/?query=begin:1720
I believe you'd need to write 180 calls (one for each year between 1720 and 1900) to get the data you need. You'd also need to manually filter out artists who died after 1900, by looking at the <end> node within <life-span>. This is because the end field will only get you artists who died in a specific year.