Just intonation point: Difference between revisions
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Clarify interpretation of the JI point |
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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 | '''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''}}. If M is a monzo, then <J|M> is M evaluated in terms of octaves. If we shift to weighted coordinates, so that 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. | As seen in the 5-limit [[projective tuning space]] diagram, it is the red hexagram in the center. | ||
Revision as of 22:24, 6 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]. If M is a monzo, then <J|M> is M evaluated in terms of octaves. If we shift to weighted coordinates, so that 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.