Just intonation point

The just intonation point (JIP) is a special tuning map that maps every monzo in some subgroup to its span in cents (or any other logarithmic unit), relative to the point 1/1 (which maps to 0 cents).

For instance, in 5-limit JI, the JIP is ⟨1200.000 1901.955 2786.314]; if we take the bracket product of this tuning map with any monzo, we get its size in cents. Of course, one can always build the JIP using different units than cents.

For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector, ⟨1 1 1 ...].

Units

It may be helpful to think of the units of each entry of the JIP — as with a normal (temperament) tuning map — as [math]\mathsf{¢}\small /𝗽[/math] (read "cents per prime"), [math]\small \mathsf{oct}/𝗽[/math] (read "octaves per prime"), or any other logarithmic pitch unit per prime. For more information, see Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis.

Mathematical definition

The JIP, 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]\frac{84}{53} ≈ \frac{1.585}{1.000}[/math] and [math]\frac{123}{53} ≈ \frac{2.322}{1.000}[/math].