Just intonation point: Difference between revisions
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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 [[interval size 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 {{w|dot 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 {{vmprod|''J''|'''m'''}} 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 covector {{val| 1 1 1 … 1 }}. | |||
As seen in the 5-limit [[projective tuning space]] diagram, it is the red hexagram in the center. Equal-temperament 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. 84/53 ≈ 1.585/1.000 and 123/53 ≈ 2.322/1.000. | |||
[[Category:Regular temperament theory]] | |||
[[Category:Math]] | |||
[[Category:Terms]] | |||