From Farey sequences to resonance diagrams

Farey sequences were discovered in the beginning of the nineteenth century proving very useful in number theory. These series are found to be tightly related to tune resonance diagrams in accelerators. So far, no connection has been made between these two fields. This paper aims to serve as a bridge between these two areas of knowledge introducing the concept of resonance sequences for the most efficient exploration of the tune space.

Figure 1

Tune diagram displaying all resonances up to order 5 in the unitary square.

Figure 2

Graphical representation of the resonance sequence of order 5 attached to $1/1$. The resonance lines are represented by $(a,b)$ and the corresponding Farey fraction by $p/q$. Note that $(a,b)=(p,q-p)$. The free term $c$ to complete the resonance equation is $c=a$.

Figure 3

Graphical representation of the resonance sequence of order 5 attached to $1/2$. The resonance lines are represented by $(a,b)$ and the corresponding Farey fraction by $p/q$. Note that $(a,b)=(2p,q-2p)$. The free term $c$ to complete the resonance equation is $c=a/2$.