Just intonation point: Difference between revisions

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'''JIP''' ('''just intonation point'''), or 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| log22 log23 log25 … log2''p'' }}, meaning that each prime ''q'' in the ''p''-prime limit is tuned to log2''q'' octaves (which is exactly the just value of the prime ''q'').

The '''JIP''' ('''just intonation point''') is a special tuning map that maps every monzo in some subgroup to its size in 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.

For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector or {{val | 1 1 1 ...}}.

= 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| log22 log23 log25 … log2''p'' }}, meaning that each prime ''q'' in the ''p''-prime limit is tuned to log2''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 <J|m> is the untempered JI value of m measured in octaves. In Tenney-weighted coordinates, where m = {{monzo|''m''2 ''m''3 ''m''5 … ''m''''p''}} is represented by the ket vector {{monzo|e2log22 e3log23 e5log25 … e''p''log2''p''}}, then J becomes correspondingly the bra vector {{val| 1 1 1 … 1 }}.

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-'''JIP''' ('''just intonation point'''), or 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''' ('''just intonation point''') is a special tuning map that maps every monzo in some subgroup to its size in 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.
+For prime limits, the JIP has a particularly simple definition in Tenney-weighted coordinates, where it is always the all-ones vector or {{val | 1 1 1 ...}}.
+= 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 }}.