I think another possibility is that the NASB and other translations of the Masoretic Text do not accurately represent what was in the original Hebrew.
The Jewish Publication Society Tanakh translates the Masoretic Text here slightly differently:
Disrobing on a chilly day, Like vinegar on natron,
Is one who sings songs to a sorrowful soul.
The Judaica Press translation, by Rabbi A. J. Rosenberg, reads:
A worn-out garment on a cold day is like vinegar on chalk,
and so is one who sings songs to a broken heart.
An annotation in the Oxford Jewish Study Bible (based on the JPS Tanakh) indicates that the Masoretic Text here might be corrupt:
The text of this verse is in doubt, and the first line [Disrobing on a chilly day] might be a mistake.
The comment continues:
As it stands, the proverb offers two images: undressing in the cold and pouring acid on a base. To these are compared the effect that cheerful music has on a melancholy person. The first action causes a chill, the second causes fizzing and hissing, which may suggest irritation and incompatibility. Without the first analogy (which many scholars excise), "vinegar on nitron" might imply neutralization of the sorrow.
Rashi, a medieval Jewish commentator who synopsized Talmudic commentary on Scripture, offers the following regarding this verse (Judaica Press translation):
A worn-out garment on a cold day: Heb. מעדה בגד, synonymous with (Isa. 64:5): “בגד עדים, a discarded garment,” a worn-out garment that is fit to be taken off since it is worn out, as we translate into Aramaic “ויסר” (Ex. 7:24) as ויעדי, and He shall remove. Now this is its interpretation: A worn-out garment on a cold day is like vinegar on chalk.
chalk: Heb. נתר, a kind of soft earth, like our earth that is called creide, which they would hew and make vessels therefrom, and if vinegar falls on it, [the earth] is dissolved and ruined. So…
is one who sings songs to a broken heart: [lit. on a bad heart. This one] resembles them both. Now what is one who sings songs? This is one who teaches the Torah to a wicked student, who has no intention in his heart to fulfill it.
The Alexandrian Jews who translated this verse into Greek in the 2nd century or so (BC) saw something rather different than what appears in the Masoretic Text (c. 7-10th c. CE):
ὥσπερ ὄξος ἕλκει ἀσύμφορον, οὕτως προσπεσὸν πάθος ἐν σώματι καρδίαν λυπεῖ. ὥσπερ σὴς ἱματίῳ καὶ σκώληξ ξύλῳ, οὕτως λύπη ἀνδρὸς βλάπτει καρδίαν.
As vinegar is bad for a sore, so trouble befalling the body afflicts the heart. As a moth in a garment, and a worm in wood, so the grief of a man hurts the heart.*
The other two answers you received so far indicate that "sad" may be the more appropriate reading, but given the above it seems there is uncertainty about what the "original" text actually said in the first place. It seems to me that the Septuagint is somewhat more coherent than the Masoretic Text here:
Masoretic Text (NASB)
Like a club and a sword and a sharp arrow
Is a man who bears false witness against his neighbor.
Like a bad tooth and an unsteady foot
Is confidence in a faithless man in time of trouble.
Like one who takes off a garment on a cold day, or like vinegar on soda,
Is he who sings songs to a troubled heart.
LXX (Orthodox Study Bible)
As a club, a sword, and a pointed arrow
So also is the man who testifies as a false witness against his friend
The tooth of an evil man and the foot of a lawless man
Shall perish in an evil day
As vinegar is bad for a festering wound
So calamity befalling the body grieves the heart
As a moth in a garment and a worm in wood,
So the grief of a man damages the heart
I'm not insisting that it is the case that the Masoretic Text is corrupt - just offering a third alternative.
* Brenton translation