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 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 ~~[[Mathematical guide/Matrix operations#Dot product~~|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 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 ~~...~~}}.

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 ==

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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 covector {{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 covector {{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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 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 [[Mathematical guide/Matrix operations#Dot product|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 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 ...}}.
+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 ==
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 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 covector {{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 covector {{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>.