Try this.

While on your home screen, push the volume up/down button.

The volume should appear.
Touch the little arrow pointing down on the side of the volume toast message.

This will bring down the entire volume menu.

Set each one as desired.

Now, open Settings.

Go to:

Sound
Here you should see the individual settings for the ringer, notifications, alarms, and anything else.

This is where you decide what tone you want for each.

As the beep-beep-beep-beep-beep seems to be so annoying, make that your alarm sound, hahaha (sarc.)

Now back out of there and stay in the first screen of settings.

Select Notifications.

Here you can disable or adjust the notifications for each app.

There is a way to get more adjustment settings for notifications, by activating your System UI Tuner.
But this is only applicable on certain devices, as the options that this opens varies greatly from device to device.

On your homescreen, pull down your quick settings menu from the top.

Locate the settings pinwheel, then tap and hold it until it animates (spins)- about ten seconds- then let go.

Then go into Settings, and scroll down to near the bottom until you see System UI Tuner.

Select that, and look for Power Notification Controls.

This will give you 5 steps of notification intensity for each app separately.

Like I said, this is not applicable to all devices, but if you do have it it is quite useful.

 

I think I'm in same sitch as OP. Basically, the phone is making the default notification sound every few seconds without showing visual evidence of a new notification. The sound itself is not annoying; it's having to hear it repeatedly. My only solution so far is to mute the volume for all notifications which is not ideal cuz I want to hear sounds for some notifications when they arrive.

 

Engage the Do Not Disturb mode, and then go through your notifications for each app that you DO want to get notifications for and enable Override Do Not Disturb.

Each device is different, but most should have these options.

On my 7.1.1 I had to activate the System UI Tuner to get the option to be available.

You can look up how to enable the System UI Tuner for your individual device.