Hello joachim192,
Welcome to Apple Support Communities.
We understand that your Mac isn’t going to sleep at the time interval you have set when you listen to your music via iTunes. You also mentioned that your Mac has slow performance updating. We are happy to help.
When you listening to your iTunes Music Library, are you using your Mac? If so, then this would be expected behavior for it not to go to sleep and show your screen saver.
We recommend these steps for screen saver preferences: Screen Saver preferences on Mac - Apple Support
- "Click the “Start after” pop-up menu, then choose how long your Mac needs to be inactive before the screen saver starts. If you don’t want a screen saver to start, choose Never."
Here’s more about setting sleep wake times: Set sleep and wake times for your Mac - Apple Support
"Specify sleep and wake settings for a portable Mac
Note: Some of the options may not be available, depending on your Mac.
- Choose Apple menu > System Preferences, then click Energy Saver.
- Do any of the following:
- Switch between graphics modes: Select “Automatic graphics switching” to switch between graphics modes automatically for better battery life.
- Set sleep times for when you’re using your portable computer’s battery: Click Battery, then drag the “Computer sleep” and “Display sleep” sliders, or the “Turn display off after” slider.
- Set sleep times for when you’re using your portable computer’s power adapter: Click Power Adapter, then drag the “Computer sleep” and “Display sleep” sliders, or the “Turn display off after” slider.
- Keep your Mac from going to sleep automatically: Select “Prevent computer from sleeping automatically when the display is off” in the Power Adapter pane.
- Put hard disks to sleep: Select the “Put hard disks to sleep when possible” option in the Battery or Power Adapter pane.
- Keep your display dimmed while using your battery: Click Battery, then select “Slightly dim the display while on battery power.”
- Update your system while it’s asleep: Select “Enable Power Nap while on battery power” in the Battery pane, and “Enable Power Nap while plugged in to a power adapter” in the Power Adapter pane.
- Allow your Mac to wake briefly so users can access shared services (if applicable): Select any of the available “Wake for…” options in the Power Adapter pane, for example, “Wake for network access.”
We recommend the steps here if your Mac is running slowly: If your Mac runs slowly - Apple Support
"Your computer’s startup disk may not have enough free disk space. To make disk space available, you can move files to another disk or an external storage device, then delete files you no longer need on the startup disk. macOS can also help you optimize storage by storing files in the cloud and helping you identify files you no longer need. Choose Apple menu 
> System Settings, then click General 
in the sidebar (you may need to scroll down). Click Storage on the right, then click the storage recommendations you want to use.
An app you’re using may require more memory than your Mac has easily available. You can use Activity Monitor to see how much memory your Mac and apps are using. See View memory usage in Activity Monitor. Also see the documentation that came with the app to find out its system requirements.
Quit any app that isn’t compatible with your Mac. For example, an app may require a different processor or graphics card. See the documentation that came with the app for information about its system requirements.
- To see what processor your computer has, choose Apple menu > System Settings, click General in the sidebar, then click About on the right.
- To see which graphics card your computer has, choose Apple menu > System Settings, then click General in the sidebar. Click About on the right, then click System Report. Click Graphics/Displays to see the name of the graphics card installed your computer.
- Quit any apps you’re not using.
- If you have problems with your disk, use Disk Utility to verify and try and fix any issues."
Let us know if you have further questions.
Take care.
