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The '''just intonation point''' ('''JIP''') is a special [[tuning map]] that maps every [[monzo]] in some [[subgroup]] to its [[span]] in [[cent]]s (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 {{val | 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, {{val | 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 [[Vals and tuning space|tuning space]] which represents untempered ''p''-limit JI. Specifically, it is equal to {{val| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'' }}, meaning that each prime ''q'' in the ''p''-prime limit is tuned to log<sub>2</sub>''q'' octaves (which is exactly the just value of the prime ''q''). 

The JIP is the target of optimization in optimized tunings including [[TOP tuning|TOP]] and [[TE tuning]]. If m is a monzo, then &lt;J|m&gt; is the untempered JI value of m measured in octaves. In Tenney-weighted coordinates, where m = {{monzo|''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>''p''</sub>}} is represented by the ket vector {{monzo|e<sub>2</sub>log<sub>2</sub>2 e<sub>3</sub>log<sub>2</sub>3 e<sub>5</sub>log<sub>2</sub>5 … e<sub>''p''</sub>log<sub>2</sub>''p''}}, then J becomes correspondingly the bra vector {{val| 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 {{val| 53 84 123 … }}, have integer elements which are in proportions relatively similar to the proportions of the corresponding elements in J = {{val| log<sub>2</sub>2 log<sub>2</sub>3 log<sub>2</sub>5 … }} ≈ {{val| 1.000 1.585 2.322 … }}, e.g. <span><math>\frac{84}{53} ≈ \frac{1.585}{1.000}</math></span> and <span><math>\frac{123}{53} ≈ \frac{2.322}{1.000}</math></span>.

[[Category:Regular temperament theory]]
[[Category:Math]]
[[Category:Terms]]