Kenmark, assuming that you have the necessary permissions, here's a step-by-step guide to help you create an MP3 album of your favorite songs from online radio:
Step 1: Download Audacity from Sourceforge.net here. Available for Mac, Windows and Linux platforms.
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Step 2: Open Audacity and choose View -> Float Mixer Toolbar. In the drop-down menu on Audacity's mixer toolbar, choose "Wave Out" or "Stereo Mix" as the input source. Open the Audacity Preferences dialog and choose Mono channel in the Audio I/O tab. Select your sound card name in Recording Device [Sigma Tel C-Major Audio in my case]
Step 3: Open the radio station of choice using the web browser or Real player or Yahoo! Messenger. Hit the play button and switch to Audacity.
Step 4: As the song of your choice is about to play, hit the keyboard shortcut - R to begin recording of the song or press the Record button on the Audacity interface. When the song end, hit the Space Bar to stop recording.
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Now when your next favorite song is playing on the radio station, hit the R button again. Now Audacity will record the song in a separate track. This is helpful as you will realize in a minute. Repeat the R-space bar cycle until you record all the songs to your PC or Mac.
Step 5: The last step - Choose File -> Export Multiple - this will save the individual Audacity track as separate MP3 files or WAV files. Choose the export folder and select "Split files based on Tracks". Click export.
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Step 6: Or export a single MP3 or WAV music track. Advanced users may consider
Audacity effects to fine-tune their internet recordings
Step 7: That's it. Burn this folder as an Audio CD or transfer them your iPod or mobile phone and enjoy the web music offline.