Music intervals graphic with TIKZ - trigonometry

Question

I am drawing a picture showing how to position segments for the natural music scale. My trigonometry lessons are long ago forgotten, so I apologize for this is going to be a very dummy question.
Consider the followintìg tikz picture. I need to place the segment re-D so that its measure will be 8/9 the measure of the segment ut-C. Now I just placed it at a random distance of 8/9 between ut and ut'.
What mathematical function should I use (sen, cos...???) and how do I write it down for tikz?
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}[scale=2]
\coordinate (o) at (0,0);
\coordinate [label=below:ut] (ut) at (2,0);
\coordinate [label=below:ut'] (ut') at (4,0);
\coordinate [label=above:C] (C) at (2, 2.4);
\coordinate [label=above:c] (c) at (4, 1.2);
\coordinate (O) at ($ (c)!2!(C) $);
\draw (ut) -- (ut');
\draw (ut) -- (C);
\draw (C) -- (c);
\draw (ut') -- (c);
\coordinate [label=below:re] (re) at ($ (ut)!8.0/9!(ut') $);
\coordinate [label=above:D] (D) at ($ (C)!8.0/9!(c) $);
\draw (re) --(D);
\node [fill=black, inner sep=1pt] (c') at ($ (ut)!1.0/6!(C) $) {};
\node [fill=black, inner sep=1pt] (c') at ($ (ut)!2.0/6!(C) $) {};
\node [fill=black, inner sep=1pt] (c') at ($ (ut)!3.0/6!(C) $) {};
\node [fill=black, inner sep=1pt] (c') at ($ (ut)!4.0/6!(C) $) {};
\node [fill=black, inner sep=1pt] (c') at ($ (ut)!5.0/6!(C) $) {};
\node [fill=black, inner sep=1pt] (c') at ($ (ut')!1.0/3!(c) $) {};
\node [fill=black, inner sep=1pt] (c') at ($ (ut')!2.0/3!(c) $) {};
\draw (ut') -- (c);
\end{tikzpicture}
\end{document}
Thank you,
A

Answer 1

If you are still interested, here is an answer to your question.
In the following code, I assumed that c is always half as high as C on the diagram. You may change the fraction for the position of D. Right now it is set at 8/9. No trigonometry was required. I removed the code defining the origin (o). It was not used in any way.
\begin{tikzpicture}[scale=1]
\newlength{\height}\setlength{\height}{2.4cm}
\newlength{\width}\setlength{\width}{4cm}
\def\fraction{0.889} %8/9
\coordinate [label=below:ut] (ut) at (0,0);
\coordinate [label=below:ut'] (ut') at (\width,0);
\coordinate [label=above:C] (C) at ($(ut) + (0,\height)$);
\coordinate [label=above:c] (c) at ($(ut') + (0,0.5\height)$);
\draw (ut) -- (ut') -- (c) -- (C) -- cycle;
\draw let
\p{C} = (C),
\p{c} = (c),
\n{y_D} = {\fraction*\y{C}},
\n{x_D} = {\x{c}*2*(1-\fraction)}
in
(\n{x_D},\n{y_D}) coordinate[label=above:D] (D) -- (\n{x_D},0) coordinate[label=below:re] (re);
%you could replace the previous line by
%(\n{x_D},\n{y_D}) -- (\n{x_D},0);
%because (D) and (re) aren't used later
\foreach \x in {1,...,5}{
\fill[black] ($ (ut)!\x/6!(C) - (1pt,1pt)$) rectangle ++(2pt,2pt);
}
\foreach \x in {1,2}{
\fill[black] ($ (ut')!\x/3!(c) - (1pt,1pt)$) rectangle ++(2pt,2pt);
}
\end{tikzpicture}
\end{document}