Super Mario intro (easy):

^LP+MP ^LP+MP ^LP+MP ^HP ^LP+MP ^HK HK

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MLP intro (intermediate):

^LP+MP ^HK ^LP+MP ^LP+HP ^HK

^HP ^LP+HP ^HP  MK  HK

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Azumanga eye-catch (intermediate):

 HK  HK ^HP ^LP+HP ^LP+MP ^HP

 LK+MK  HK ^HP ^HP ^HP

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Metroid item fanfare (too hard):

vHK  HP  LP+HP  MP+HP  MP  LP+HP

 LK+MK  LP+HP  HK  LP+MP  HP

 LK+MK  MK

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tips:

-you can save yourself a lot of frustration when practicing by remembering
 to keep the button held down.

-none of these songs use more than two octaves, so you can transpose it
 if you think it would sound better or be easier to play