Just intonation point: Difference between revisions

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If m is a monzo, then <J|m> is m evaluated in terms of 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.

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|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>.

[[Category:Regular temperament theory]]

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 If m is a monzo, then &lt;J|m&gt; is m evaluated in terms of 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.
+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>.
 [[Category:Regular temperament theory]]