Firefox web browser:
To download tumblr songs, Firefox should be your most-used internet browser. You’ll also need the Greasemonkey addon. Greasemonkey allows you to install third-party scripts to customize the way you view webpages. The script you will be using for downloading tumblr mp3s is Tumtaster. It creates download links below music players on tumblr that allow you to easily download any song you may come across.
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Google Chrome web browser:
Since the ‘Greasemonkey addon’ isn’t for Google Chrome, you’ll need to download the ‘Blank Canvas Script Handler’, which is bascially the GreaseMonkey add-on for chrome. Once you download that, it should auto-install. Once it has installed, you want to download ‘Tumtaster’, which is the application used to download tumblr songs. [NOTE: If you don’t have the Blank Canvas Script Handler installed, you wont be able to download ANY songs.] Once you’ve sucessfully downloaded the two applications above, open up a new tab, and go to a music post [on tumblr]. You should see something like the picture below. If you don’t, something’s wrong, and you should start over if necessary. 
If you see the ‘Click to download’ box below the audio player, click it!
you should see a seperate player, in a new tab like this:
If you see that, just Right Click anywhere on the player, and click ‘Save video as’ [Yes, I know it’s not a video, but it says that. It will save as a .mp3 [music file]].
and just rename the music file to your liking and You’re Done! If you’ve found this post helpful, please reblog!
[Credit]
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I’m not sure if this works on other web browsers, but I know it works on Chrome, and Firefox for sure. ENJOY!
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UPDATE FOR CHROME:
https://chrome.google.com/extensions/detail/igkglbhdogeemdmnfdjgapgacfhgiikd <- This is better alternative tumblr song downloading PLUGIN FOR CHROME.
