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 is the target of optimization in optimized tunings including [[TOP tuning|TOP]] and [[TE tuning~~|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 }}.

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 }}.

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| log22 log23 log25 … }} ≈ {{val| 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>.

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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 is the target of optimization in optimized tunings including [[TOP tuning|TOP]] and [[TE tuning|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 }}.
+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>.