OK, it's probably obvious that your spouse has unresolved jealousy issues and an inferiority complex. On top of that, she's uber-controlling. She wants to be the gatekeeper for all household issues, demoting you to inferior status. It sounds to me like she's infringing on your parental authority.
What person would dare tell a parent not to educate her own kids, especially in something as enriching as playing the piano? Well, someone who is insanely jealous of you and thinks you'll "steal" her daughter's attention away from her. It sounds to me like your spouse is making you as small as possible. You're probably not allowed to "show off" in front of her, either. She'll let you work though, because she wants your financial support.
And now for your daughter. Are you willing to quash her curiosity about music and desire to learn, plus ruin a special mother-daughter activity, just to appease your spouse? That sounds like putting your spouse's needs ahead of those of your child. While it's important to consider spousal needs in a marriage, this situation seems completely unjustified and over-the-top to me. It sort of sounds like your spouse is jealous that you might actually grow closer to your own child through a love of music.
And what will your daughter think in the long term? That you teach other kids about something you love, but you never have the time or inclination to teach her?
I think the next time your daughter asks, you go ahead and give her a lesson. If your spouse blows up about it, then you put your wife in a time out. Now maybe your daughter will feel horrible for "provoking" her mom, but I think she's old enough to understand that her mom is being completely unreasonable! I think a way to potentially resolve this would to give a joint lesson, so that your wife feels included. Maybe you ask your spouse for a song recommendation, so that she feels like she's consulted?
I confess I've had some issues with my spouse when he tries to keep me "small," out of jealousy. I tried reasoning with him (about the absurdity of the jealousy), as well as pointing out his many talents (to bolster his confidence), on top of reassuring him about my devotion to him. I would say thinks like, "I'm proud of you; shouldn't I want you to be proud of me too?" That didn't seem to sort out the problem, however. So I had to take a harder stance, something along the lines of, "I support you, and I deserve/demand the same from you." You might point out how you support your spouse when she teaches the kids or makes decisions on their behalf, and you insist on being accorded the same sort of support/autonomy/authority. This is a partnership, not a master/slave, diva/bit part relationship. Maybe that would work better? I guess what I'm saying is that if you let your spouse "get away" with infringing on your rights/values in important areas, such as parental authority or alienating you from your love of music, she'll take that as a license to take things further.
Now, I wouldn't advocate arguing over every little thing. I think you need to pick your battles. But this one seems like a worthy battle to me, because it involves your child's education, as well as your identity and authority. Your wife is allowed to feel envious--we probably all feel envious from time to time--but she can keep it to herself and not prevent your daughter from learning about music and bonding with you.


