= Octave tuning in a two-mass complete spring graph
{title2=$m_2/m_1=(43\pm5\sqrt{33})/32$}
For two masses, connect each to a fixed anchor and to the other mass with equal stiffness $k$. Set $m_1=1,k=1$ to choose units. The squared <angular frequencies> have characteristic equation $m_2z^2-2(m_2+1)z+3=0$. Requiring a <frequency ratio> of two means that these roots are $z$ and $4z$. Their sum and product give $16m_2^2-43m_2+16=0$, yielding the two reciprocal ratios in the header. Restoring units multiplies all squared frequencies by $k/m_1$, leaving the permitted mass ratios unchanged. The term octave refers here only to this two-to-one ratio.
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