Golden Meantone is defined to be the linear tuning, with octave period, such that the ratio between the cent measures of a diatonic semitone and a chromatic semitone is the Golden Ratio: \begin{equation} \label{golden} \varphi = \frac {\sqrt{5}+1}{2}; \end{equation}
Please note the abuse of language when using the word «semitone»: in this framework, it doesn't mean half-a-tone as in customary 12-equal division of the octave, however, a diatonic semitone (-5 fifths) + a chromatic one (7 fifths) does equal a tone (2 fifths).
Golden Meantone turns out to be the meantone tuning generated by a 696.214-cent fifth, not too far from 50-equal division of the octave.
Let's prove this fact. Name F the unknown fifth and Ω the gauge of the octave (normally 1200 cents).
Now, a diatonic semitone corresponds to -5 fifths plus 3 octaves, whereas a chromatic semitone corresponds to 7 fifths minus 4 octaves, hence, keeping into account definition \eqref{golden}, we have: \begin{equation} \label{ratio} \varphi = \frac{-5F+3Ω}{7F+4Ω}. \end{equation} Solving for F, multiplying both numerator and denominator for 12-7φ, and keeping into account that φ^2 = φ + 1, we get: \begin{equation} \label{solve} F = \frac{4\varphi+3}{7\varphi+5}\Omega = \frac {48\varphi-28\varphi-21\varphi+36-28} {84\varphi-49\varphi-35\varphi+60-49} \Omega = \frac {8-\varphi} {11} \Omega ; \end{equation} putting Ω = 1200 in \eqref{solve}, we eventually get F = 696.214 cents up to three decimal digits. Please note that \eqref{solve} also shows that F is linear in Ω, which is relevant in case ot tempered octave (slightly different from 1200 cents). F is not too far from 50-edo's fifth, which is equal to 696 cents, so that 50-edo approximates Golden Meantone, as far as not too many fifths are used.
