The location and order of the non-commensurate mean motion resonances up to seventh order, illustrating how the pattern repeats itself relative to each first-order MMR. Note that each differently colored cluster can be enumerated up to any k using the “Farey sequence” F_k, which is the sequence of reduced fractions between 0 and 1 that have denominators less than or equal to k, e.g., ﹩{F}_{2}=\{0,1/2,1\}﹩ and ﹩{F}_{3}=\{0,1/3,1/2,2/3,1\}﹩. The resonant period ratios within a cluster that contains the first order ﹩J\,:J-1﹩ MMR occur at ﹩P/P^{\prime} =\tfrac{J-1+r}{J+r}﹩ for ﹩r\in {F}_{k}﹩. When plotting ﹩P/P^{\prime} ﹩ elsewhere in this paper we stretch the horizontal scale to assign equal measure to each group of resonances associated with a single J. This is done by setting plots’ horizontal coordinates uniformly in ﹩J={(1-P/P^{\prime} )}^{-1}﹩.