Update:
First of all, use the Help / Solution documentation for an up-to-date solution.
http://api.soundcloud.com/resolve?url=http://soundcloud.com/USER_NAME&client_id=YOUR_CLIENT_ID
Replace USER_NAME with the username from the regular SoundCloud link, for example https://soundcloud.com/USER_NAME , and your application client ID. Note: there is no longer .json in /resolve .
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