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).

If m is a monzo, then <J|m> is the untempered JI value of m 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, 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].