Her keyboard is a 76-Key version of the Korg Triton Extreme, which is now discontinued. It is a very heavy instrument, as noted below.
Her surname is taken from former P-Model keyboardist Hikaru Kotobuki. Mugi's last name also has the same pronuciation as her voice actor Minako Kotobuki, but is written with different kanji.
The kanji of Tsumugi's given name (紬) refers to pongee - soft, thin cloth woven from raw silk.
Although Mugi does not like to flaunt the fact that her family is wealthy, it becomes rather obvious at certain points, and always to the shock of her friends. Her butler answers the phone when Ritsu calls her at home once and informs her that Mugi is on an overseas trip in Finland. Mugi's father owns the music shop that she and her friends frequently visit, so the workers there are always very courteous to her, and greet her using her proper title of "Lady Tsumugi". As a result, they offer her and her friends large discounts or free service. However, upon entering university, Mugi decided that she wouldn't depend on her wealth anymore and told her parents that the only thing she needed them to pay for would be her tuition.
At one point, Akira bluntly points out that the girls of HTT might be friends with Mugi because they're after her money, making Mugi cry. Yui and Mio later reassure her that they're friends regardless of money, and then immediately afterwards show excitement over some sweets she brought for them. Mio remarks that they're not after her money, but her sweets.
Mugi sometimes has to use the sweets and beverages she brings to manipulate the others (especially Yui) into doing club activities they would normally be lazy to do.
Mugi treasures her friends greatly: Upon seeing Yui, Mio and Ritsu again after not meeting for several days, Mugi cries that she hasn't seen them in precisely "57 1/2 hours".
Mugi is apparently very strong, as in the anime she is seen carrying loads of heavy equipment without breaking a sweat, including her Triton Extreme, which weighs 16.9kg / 37.26 pounds. She also knocked Akira over with a "light strike".
Mugi can speak in Kansai dialect perfectly. This may be why Minako Kotobuki was chosen to voice her, since she can speak Kansai dialect.
Yui states that Mugi's eyebrows resemble pickled radishes due to their thickness. She furthermore emphasizes it in the Ura-On! specials as well as dream sequences. In these exaggerated dreams, Mugi's eyebrows serve as an anchor that prevent her from dissolving. When Yui and Ritsu ate her eye brows (assumed to be radishes) in one occasion, she started to melt. Oddly, protractors served as makeshift anchors in the dream.
She seems to like yuri, since in Volume 1 of the manga, she had fantasies of both Yui/Mio and Ritsu/Mio. Additionally, Mugi initially showed signs of affection for Sawako-sensei, much to the surprise of Mio and Ritsu, indicated by her staring at Sawako through makeshift binoculars consisting of two rolled-up magazines.
In chapter 16 of volume 6, it is revealed that Tsumugi was home-schooled when she was young (as this maybe the reason why she is curious about the outside world). She has a childhood friend named Sumire Saito. Her interest in "yuri" came from when Sumire would buy manga, which Mugi was not allowed to see, for her.
In chapter 19, it is revealed that Tsumugi could be really scary when she is angry. This can be seen by the way she scolded a servant when he interrupted Tsumugi when she was talking with Mio on the phone.
In the first episode of the anime, Mugi makes a big deal out of mixing her fries with those of the other two girls. In Japanese culture, sharing your food with the group is a sign that you accept the group as your own, so Mugi was both being formal and very sweet in one go.
Mugi can be regarded as the female counterpart to George Harrison of The Beatles. Much like Harrison, who is famous for being the strange, sensitive "Quiet Beatle", Mugi is a gentle, eccentric and kind girl of few words, but makes what she says matter when she does speak. She also started as the single most talented member of her band, and is a prodigy famous for being successful in spite of refusing to confirm to restrictive social norms. Both are famous for their enthusiasm to try anything and everything new, and maintain equally close if sometimes distant friendships with their respective band-mates. Also much like Harrison, Mugi is the closest to the resident fellow eccentric Yui (much like how Harrison was close to Lennon).
Whether it be intentional or otherwise, Mugi's name sounds a little like the surname of Robert Moog, the man who pioneered the synthesizer and created the first publicly available, easy to use synthesizer: the Minimoog Model D.