“I feel like Wriggle is a bit slow-headed. […] She's at the rank where it might be better for her to go play with Cirno instead.”
– ZUN, on Wriggle's “Unseasonal Butterfly Storm” spell card

The Baquartet (バカルテット, Bakarutetto), also called the Bakquartet or Bakartet, is an old grouping of plot-irrelevant, early stage bosses in Touhou Project which emerged in 2005 or 2006. In the English community, they may be nicknamed Team ⑨ (Team 9) or Team Nineball.¹⁾

During the 'early trilogy' era of Touhou Project, the early stage bosses were plot-irrelevant and had nothing to work with, aside from an off-hand comment about Wriggle and Cirno in Touhou Eiyashou. This would change in 2005 when Touhou Kaiedzuka gave birth to the '⑨' phenomenon and called Mystia 'bird-brained', then we sorta had a carefree Rumia in the first Touhou Bunkachou book released later that year.

It's been difficult to pinpoint when and where this idea originated. Both the full version of Touhou Kaeidzuka and the first Touhou Bunkachou book had released in August 2005, while the earliest known fan artwork carries a rough timestamp of August 2006,²⁾ long before the rise of Niconico and Pixiv. From what we have to work with, our best educated guess would be sometime in 2005 or 2006.

While it started a fan creation, the four had a cameo in Chapters 3 and 4 of Oriental Sacred Place in 2009, then in Chapter 12 of Oriental Sacred Place in 2010 with Tewi. After this, the four have mostly served as background characters in the latter volumes of the Touhou Sangetsusei series. The four would then appear at the Hall of Dreams' Great Mausoleum in Touhou Shinkirou in 2013, but were in random groups.

Historically, the group consists of four core members. There have been attempts to add new members to the lineup, forming a “Kyuuintet” (⑨インテット) or the “Baka Rangers” (バカレンジャー), but you then run into the issue of characters already belonging to a group, characters being too powerful, or characters simply being introduced far too late when the core had been cemented.

This grouping hasn't particularly aged well, especially after the 'character type' that this group was originally designed for had effectively stopped existing after the 'soft reboot' in 2007. Some might also argue that it helps feed a general preconceived notion about the characters that surface-level and bandwagon fans often regurgitate, ignoring recent developments like Mystia canonically having a job and a side hustle.

• The group's name would influence the 'Babaquartet' (ババルテット), which was the four-member iteration of what used to be the three-member 'Gensōkyō Sandaiba' (幻想郷三大婆) and what would later become the five-member 'Touhou Gotairō' (東方五大老). There are more than five milves.
• By the way, this group can be seen at the start of the famous promotional video for “Cirno's Perfect Math Class”, but it's easy to overlook since they clearly aren't the main focus.
• Personally, I'm not a huge fan of the group. Some of these one-on-one dynamics are fun to explore, but the four of them being brought together just gives this very strong 'odd one out' feeling.

• List of Touhou Group Tags

• Baquartet on the Niconico Encyclopedia
• Baquartet on the Pixiv Encyclopedia