I want to automatically update cell comments with time stamp last time cell changed

I am trying to automatically add a comment to each cell, to show when the cell was last changed. I have the following VBA,

Private Sub Worksheet_Change(ByVal Target As Range)
    With Target
        .ClearComments
        .AddComment ("Last Modified " & Date)
        .Comment.Visible = False
    End With
End Sub

which updates an individual cell's comments with "Last Modified (date)" whenever the cell content is changed. It works just fine if I enter a value into a cell, or copy and paste values of one cell to another cell. But it does not work if I change multiple cells, such as copying and pasting a range.

How do you get this to work on a range of cells?