Although you can simply upload MP3 files directly to Tumblr when creating an audio post, it's not always the best thing to do. As of December 2013, Tumblr still places a strict size limit of 10MB on uploaded audio files, meaning any piece of music you post in this manner is necessarily limited in duration, quality or both. To get around this limitation, you can stream the audio file from elsewhere. There are several websites you can use as a source, each with their upsides and downsides.
If the music you want to post on Tumblr is on Spotify, you can simply stream it from there. Type the name of the piece of music in the search field that appears when you click the "Audio" button on your dashboard to see matching results from Spotify and then click the song you want to select it. The finished post shows the music in a Spotify-branded player; clicking the "Play" button launches Spotify and automatically plays the song.
SoundCloud also integrates with Tumblr in a manner similar to Spotify, with matching results from the service also appearing when you type something into the search field in the audio post editor. You can also post a piece of music from SoundCloud by copying the URL to it, selecting the "External URL" tab in the audio post editor and then pasting the URL in the text field. The finished post shows the music in a SoundCloud-branded player. Clicking the "Play" button plays the song directly from within the post, without launching an external application or opening a new Web page.
If you upload the file to a hosting service that allows you to link directly to it, you can stream it through Tumblr's generic audio player. Copy the direct link to the file -- it has to end in ".mp3" or the process won't work -- and paste it into the text field on the "External URL" tab in the audio post editor. What service you use doesn't matter -- you can use your own website, a dedicated audio hosting service such as Kiwi6 or a cloud storage service with the option to create direct links to shared files such as Dropbox.
Choosing a Method
Spotify has a very large library of music, and is most likely to have the song you want to stream. The problem is that listening to songs from the Spotify library requires having a Spotify account, which is only available in specific countries, so some of your readers may not be able to listen to the song. SoundCloud's library is less likely to have popular music, since it's primarily an audio distribution platform for unsigned artists. It does, however, have no playback restrictions, and you can also upload your own music to it, though you're limited to a maximum of two hours of uploads for free. If you're using a service other than Spotify or SoundCloud to host the MP3 files, what limitations you have or, indeed, whether you have any at all depends on the service you're using.