One of the things I enjoy in Musubi no Hibiki, Hajimari no Ne is its use of "Yosakoi", a track based on a folk song from the region where Mutsunokami is from. In general, it is a "let's talk about our former masters" song, but in particular, it also expresses the bond between Mutsunokami and Nagasone.
The original "Yosakoi-bushi" is about a monk and a girl who eloped but were caught and exiled to never meet again. There is also a more modern song, "Nangoku Tosa wo Ato ni Shite" (Leaving the Southern Land of Tosa), where the homesick person sings the song from their home. Both the feeling of missing people and the knowledge that you won't meet someone again are heavily evoked in MusuHaji. It's also notable that, in modern times, this song (as a part of "Yosakoi Naruko Odori") is associated with the Yosakoi Matsuri, and with how SRS established that during festivals the boundaries between the realms of the living and the dead are thin, it is fitting how this track is used in the scenes that talk about the Touken Danshi's dead (or soon to be dead) masters.
Mutsunokami sings this song three times, and every time it shows how close he is with Nagasone (or at least that Nagasone wants to be close with him). In the first scene, he sings the part that translates to: "At the Harimaya Bridge at Tosa, Kochi, I saw a monk buy a hairpin. Come at night, come at night." Since monks aren't supposed to get cozy with women, there is usually a note of surprise when his profession is revealed. Shipping biases aside, it can be interpreted as "a meeting between two people that aren't supposed to meet because of their statuses." This fits the situation between Nagasone and Mustunokami, who belonged to people from enemy factions during Bakumatsu and were shown to not get along because of that in the past. Also, every time this song is sung, it's night.
During this scene, Nagasone surprises Mutsunokami by joining the song from his home. He shows that, unlike Izuminokami, he understands that Mutsunokami does grieve his former master, he just doesn't let it get in the way of the mission. Even though Mutsunokami dodges the subject of the former masters' deaths throughout the play, it is notable that Nagasone tries to talk about it at all. It's emphasized in Bakumatsu Tenrouden that Nagasone keeps things to himself, but here not only is he willing to talk about something so personal, it is implied that he regularly chats with Mutsunokami. There is also the fact that this sequence happens right after Nagasone talks about how their former masters must have been similar people, and during this scene Mutsunokami puts his hand on Nagasone's shoulder, which is a gesture Nagasone (and Hijikata) seems to associate with Kondo.
The second scene shows how comfortable the two are and how they have no problem with adapting to each other's habits. Mutsunokami compliments Nagasone's drinking, and it's probably worth noting that people from Tosa were stereotyped as never getting drunk (unless you are Chouson's master, who was known to be bad at holding his liquor). The song becomes a blend of "Kappore" (associated in the musicals with the Shinsengumi) and "Yosakoi-bushi" (a Tosa song). The two play hashiken, a game from Tosa, and it's Nagasone who initiates it. During the game they drink from the same cup. And despite Nagasone once again raising the uncomfortable topic, Mutsunokami stays with him for the rest of the night (according to theatergoers, they did different stuff after their scene ended but they were still on stage, and on the recorded last day it looks like they are stargazing while leaning on each other).
During the third scene, they don't interact, but Nagasone once again shows that he is willing to see things through Mutsunokami's perspective. He understands that Mutsunokami's experiences are different from the Shinsengumi swords', and tries to understand what Mutsunokami went through. He also encourages Izuminokami to talk things out, despite usually being the "actions over words" kind of guy (though, I guess, "talking" probably is the "action" here).
