Recording streamed music off the internet onto C.D.?

TunesKit Screen Recorder will be a good choice for you to record audio/video on the computert. You can edit the recorded music/video to any compatible format as you want. 

I found this link helpful：Top 15 free screen recorder for windows and Mac

I don't think it's illegal to record from a radio station for non commercial personal use . 

Sound Tap is what many people use and it is a superb product and gives great results. It records from your desktop and can record up to 32 / 96.Recording from radio is illegal but who is to know if you don't tell anyone.

Why not use an audio recording program to record system or microphone voice lossless. Windows 10 also has a built-in feature to record voice

Take VideoSolo Screen Recorder into consideration if you really want to pay for a screen recorder. It is cheap but useful in recording music and video. You can use it to record any music that is unable to download.

Thanks for the feedback!  I'm not wanting to do anything illegal so I'll send a message to the radio station that I want to record and ask them what there take on it is...  I could see how recording tracks off of Tidal might be wrong (how much does the actual artist make from Tidal?  And Jay Z has plenty of $ already...) but I'm not sure if the radio station that I am trying to record cares; I'll keep you posted.  Also, my take on it is, there is no way to purchase these dj sets but if I recorded them and found new artists that I like I could then go and buy any music that they do have for sale, as well as promote their music to friends who have similar tastes... seems that that would be beneficial to the artists, no?  

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I stand corrected then I didn't realize you could record streams.

I record orchestral concert streams from my Cambridge CXN. I use the analog output feeding a portable Edirol recorder, then take the memory card to my computer for a little processing. The Edirol has the ability to record 24/96, but I don't bother going that high because most of the streams are 128-192 bit mp3 streams. I can stream Tidal to my CXN, so I guess I could record their MQA at 24/96, but yes that would be illegal. Your recordings to a CD player would top out at 16/44, as that is the resolution spec for CDs. That's why they wouldn't be high rez.

I don't think this would be possible pretty sure you can only rip files that are actually in your possession. Not to mention it's certainly illegal as well.