Just intonation point: Difference between revisions
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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 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 | 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 ...}}. | ||
= Mathematical Definition = | == 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, 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''). | ||