frames on note page, correct frame numbering

Question

TLDR: Using \addtocounter{framenumber}{*value*} decreases counter by 3xvalue.

Hello,

for an upcoming presentation I'd like to create a dual screen presentation. I want my noteslides to show the actual frame + my notes. So I'm using frames inside my notes. This works pretty well, but of course these note frames increase the framenumber counter and the second slide my listeners will see will have the frame number 3 (instead of 2).

I thought this could easily be fixed with

\addtobeamertemplate{note page}{\addtocounter{framenumber}{value}}. 

In theory now every frame inside a note (unwantedly) increases framenumber and every note decreases the framenumber which leads to no change of framenumber inside a note after all. But that's where things get strange. Using the above code results in decreasing framenumbers. Using different values instead of -1 shows that framenumber is always increases by three times the value.

\documentclass{beamer}

%may have to be modified to fit your system
\usepackage[ngerman]{babel}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[latin1]{inputenc}

\usetheme{Frankfurt}

\setbeamertemplate{footline}[frame number]

\usepackage{pgfpages}
\setbeamertemplate{note page}[plain]
\addtobeamertemplate{note page}{%
    \addtocounter{framenumber}{0}} %no effect while 0

\setbeameroption{show notes on second screen}

\begin{document}

\begin{frame}
    \titlepage
\end{frame}
            \note{\begin{frame}{title}{}
                    \begin{itemize}
                        \item notes for introduction 
                    \end{itemize}
                  \end{frame}
                }

\begin{frame}{First Slide}{Subtitle}
    \begin{itemize}
        \item information
        \item more information
    \end{itemize}
\end{frame}
            \note{\begin{frame}{First Slide}{Subtitle}
                    \begin{itemize}
                        \item information + notes
                        \item more information +notes
                        \begin{itemize}
                            \item even more notes
                        \end{itemize}
                    \end{itemize}
                  \end{frame}
                }
\end{document}

result with value=0 result with value=-1

Why is this happening? What do I have to change to get the wanted effect?

Workaround I already know: Using \addtocounter{framenumber}{-1} in each note leads to correct numbering. Shouldn't \addtobeamertemplate{note page}{...} have exactly the same effect as writing it in every note?

Greetings an thanks if you read till here. :-P

Edit: Basically samcarter had the solution. I added a few lines, to modify the textsizes an enumeration items.

\AtBeginNote{
    \usebeamertemplate{frametitle}

    \setbeamertemplate{itemize/enumerate body begin}{\footnotesize}
    \setbeamertemplate{itemize/enumerate subbody begin}{\footnotesize}
    \setbeamertemplate{itemize/enumerate subsubbody begin}{\footnotesize}

    \setbeamercolor{itemize item}{fg=blue}
    \setbeamercolor{itemize subitem}{fg=blue}
    \setbeamercolor{itemize subsubitem}{fg=blue}

    \setbeamertemplate{itemize item}[circle]
    \setbeamertemplate{itemize subitem}[endash]
    \setbeamertemplate{itemize subsubitem}[triangle]
}

Answer 1

I wouldn't use notes outside a frame environment. The content belongs to the frame and this just creates additional frames.

\documentclass{beamer}

%may have to be modified to fit your system
\usepackage[ngerman]{babel}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[latin1]{inputenc}

\usetheme{Frankfurt}

\setbeamertemplate{footline}[frame number]

\usepackage{pgfpages}
\setbeamertemplate{note page}[plain]
\setbeameroption{show notes on second screen}

\AtBeginNote{\usebeamertemplate{frametitle}}

\begin{document}

    \begin{frame}
        \titlepage
        \note{
            \begin{itemize}
                \item notes for introduction 
            \end{itemize}
        }
    \end{frame}

    \begin{frame}{First Slide}{Subtitle}
        \begin{itemize}
            \item information
            \item more information
        \end{itemize}
        \note{
            \begin{itemize}
                \item information + notes
                \item more information +notes
                \begin{itemize}
                    \item even more notes
                \end{itemize}
            \end{itemize}
    }
    \end{frame}

\end{document}