Just intonation point

JIP (just intonation point), or commonly denoted J, is a point in p-limit tuning space which represents untempered p-limit JI. Specifically, it is equal to ⟨log₂2 log₂3 log₂5 … log₂p], meaning that each prime q in the p-prime limit is tuned to log₂q octaves (which is exactly the just value of the prime q).

The JIP is the target of optimization in optimized tunings including TOP and TE tuning. If m is a monzo, then <J|m> is the untempered JI value of m measured in octaves. In Tenney-weighted coordinates, where m = [m₂ m₃ m₅ … m_p⟩ is represented by the ket vector [e₂log₂2 e₃log₂3 e₅log₂5 … e_plog₂p⟩, then J becomes correspondingly the bra vector ⟨1 1 1 … 1].

As seen in the 5-limit projective tuning space diagram, it is the red hexagram in the center. ET maps which are relatively close to this hexagram, which means low tuning error, such as ⟨53 84 123 …], have integer elements which are in proportions relatively similar to the proportions of the corresponding elements in J = ⟨log₂2 log₂3 log₂5 …] ≈ ⟨1.000 1.585 2.322 …], e.g. [math]\displaystyle{ \frac{84}{53} ≈ \frac{1.585}{1.000} }[/math] and [math]\displaystyle{ \frac{123}{53} ≈ \frac{2.322}{1.000} }[/math].