^
A search of the Android market turns up quite a number; whether or not they're official is up to you to decide. The ones from CYBIRD are definitely real though.
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I'm gonna try and answer these one-by-one as best as I can; others please correct any wrong statements.
I obviously know that Aki-P writes all the lyrics to the songs for AKB48, but how is it for other groups based around idols. Do the members write songs or do they just perform them?
I'm not sure about the more underground idols, but in general, mainstream female idols don't write their own songs/lyrics. Taking the 'big names' (AKB, Morning Musume, etc.) as an example, they're usually taken under the wing of a producer and given songs to perform. Now, there are cases where a member within the group may decide to write his/her own songs (ex-AKB member
Hoshino Michiru, for example), but for the most part, it's written by someone else.
Do other musical artists that are not considered to be idols in JPop write their own songs? I don't mean groups like the B'z, since they are more rock oriented. I was thinking more of the solo JPop artists like Hamasaki Ayumi. I know she writes her own songs, but what about others that fit into her same musical genre as her? Is she just one of the rare performers in JPop that writes and performs or is it a common thing like in the United States?
This is a lot harder to say definitely, since nowadays there are more and more artists in the Japanese musical world today that write their own material (Kobukuro, Ayaka, Otsuka Ai, etc.). I think it's become more of a global trend that artists that write their own material tend to be valued more highly, so it's not necessarily just the US where this is more common.
Also, how do former idols that stay in the music industry fit into all of this? Do they end up writing their own songs or do they still only perform them?
I'm going to bring up Hoshino Michiru again for this, but she's still in the industry writing her own music. On the flip side, another ex-AKB member
Oshima Mai is currently signed under Avex and has 3 solo singles since leaving AKB; but none of those songs were written by her. I guess it just depends on whether that former idol has shown that potential of composing/writing his/her own songs.
One last question that is a little more specific. Itano Tomomi is in a unique position in that she is a solo artist as well as a member of AKB48. Dos she write her own songs for her solo album/s or just perform them?
She does not. All of her solo lyrics (as well as
Maeda Atsuko, who is also in the same position) is also written by Akimoto Yasushi.