The Pure Grimoire (無垢なる魔道書 Mukunaru Madōsho), also known as No Name (ノーネーム Nō Nēmu), is a grimoire that turned Mai Natsume into a girl as its vessel has to be female. Mai is the Pure Grimoire herself, since she is a Remix Heart.
Simply speaking, No Name is a Blank Grimoire (白紙の魔道書 Hakushi no Madōsho, Grimoire of the White Paper), that gains power of any Code (
In BlazBlue: Remix Heart, No Name was also known as the Nameless Grimoire (名もなき魔道書 Namonaki Madōsho). Before it fused with Mai, the grimoire was sentient and was capable of communicating with its vessel when she was unconscious.