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 tempered to log₂q octaves (which is exactly the prime qs just value). If M is a monzo, then <J|M> is M evaluated in terms of octaves. If we shift to weighted coordinates, so that 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.