I don't think being greek has anything to do with chapters having houses or participating in social events with other orgs (although it is great fun).
Being greek has to do with the traditions and rituals laid down as the foundation by the founders. Sisterhood is not about the word, but the intention behind the ritual that creates those bonds. Sorority sisters share secrets that are only privelege to it's members.
Sigma Alpha Iota began as music sorority because the founders didn't want just any club, they wanted a sisterly bond to exist among it's members.
Mystic Cat is correct when he states Phi Mu Alpha's exemption from title IX (although for a very short time, the did admit women). The exemption is made for social fraternities and sororities. SAI requested exemption from IX and recieved it because the requirements for membership were broadened to one music class. Our founders did not intend for men to share in this sisterhood with us (hehehe

). It is a broad misconception, even among sisters, that we are a professional organization. We are not. We exist to foster relationships and social connections among women interested in music.
No, WildCat, we are not affiliated with NPC, but that does not take away from the careful planning, love, and dedication put forth by our founders to supply us with this beautiful sisterhood. When someone tells me that SAI is not greek, I imagine myself in the middle of an initiation ceremony and recall all the beautiful words said. They do not know our ritual and they do not know our secrets. They are speeking from an uninformed mouth, and that is okay because only the best are SAI.
