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|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''). | '''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''). | ||
If ''m'' is a monzo, then <''J''|''m''> is the untempered JI value of ''m'' 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}}. | 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''<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, 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>. | 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, 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>. | ||
Revision as of 19:07, 7 May 2021
JIP (just intonation point), or commonly denoted "J", is a point in p-limit tuning space which represents untempered p-limit JI. Specifically, it is equal to ⟨log22 log23 log25 … log2p], meaning that each prime q in the p-prime limit is tuned to log2q octaves (which is exactly the just value of the prime q).
If m is a monzo, then <J|m> is the untempered JI value of m measured in octaves. In Tenney-weighted coordinates, where m = [m2 m3 m5 … mp⟩ is represented by the ket vector [e2log22 e3log23 e5log25 … eplog2p⟩, then J becomes correspondingly the bra vector ⟨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, such as ⟨53 84 123 …], have integer elements which are in proportions relatively similar to the proportions of the corresponding elements in J = ⟨log22 log23 log25 …] ≈ ⟨1.000 1.585 2.322 …], e.g. [math]\displaystyle{ \frac{84}{53} ≈ \frac{1.585}{1.000} }[/math] and [math]\displaystyle{ \frac{123}{53} ≈ \frac{2.322}{1.000} }[/math].