User:2^67-1/Sandbox: Difference between revisions

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[[User:Godtone]] notes the following JIP's, each one corresponding to a 1D continua contained therein, for which increasingly efficient approximations generally represents increasingly efficient [[7-limit]] temperaments:

* log2(256/243) / log2(25/24 * 49/48) = 0.8482245109 ; this is the JIP of p=1, q=1 (equiv. to p=-1, q=-1)

* log2(256/243) / log2(25/24) = 1.2766647429 ; this is the JIP of p=1, q=0 (equiv. to p=-1, q=0)

* log2(256/243) / log2(49/48) = 2.5275365063 ; this is the JIP of p=0, q=1 (equiv. to p=0, q=-1)

* log2(256/243) / log2(1225/1152) = 0.8482245109 ; this is the JIP of p=1, q=1 (equiv. to p=-1, q=-1)

* log2(256/243) / log2(50/49) = 2.5796543166 ; this is the JIP of p=1, q=-1 (equiv. to p=-1, q=1)

* log2(256/243) / log2(2401/2400) = 125.~~1...~~ ; this is the JIP of p=-1, q=2 (equiv. to p=1, q=-2)

* log2(256/243) / log2(60025/55296) = 0.6350918818 ; this is the JIP of p=1, q=2 (equiv. to p=-1, q=-2)

* log2(256/243) / log2(49/48) = 2.~~5275365063~~ ; this is the JIP of p=0, q=1 (equiv. to p=0, q=-1)

* log2(256/243) / log2(2401/2400) = 125.1044589 ; this is the JIP of p=1, q=-2 (equiv. to p=-1, q=2)

* log2(256/243) / log2(~~25/24 * 50~~/49) = 0.8540148427 ; this is the JIP of p=2, q=-1 (equiv. to p=-2, q=1)

* log2(256/243) / log2(30625/27648) = 0.5096257724 ; this is the JIP of p=2, q=1 (equiv. to p=-2, q=-1)

* log2(256/243) / log2(625/588) = 0.8540148427 ; this is the JIP of p=2, q=-1 (equiv. to p=-2, q=1)

Importantly, each JIP corresponds to a ''rational'', so that, for example, (''p'', ''q'') = (1, -2) is equivalent to (''p'', ''q'') = (2, -4) and to (''p'', ''q'') = (-1, 2) '''but not to (1, 2)'''. Note that all these JIPs lie on the JIL (just intonation line).

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 [[User:Godtone]] notes the following JIP's, each one corresponding to a 1D continua contained therein, for which increasingly efficient approximations generally represents increasingly efficient [[7-limit]] temperaments:
-* log<sub>2</sub>(256/243) / log<sub>2</sub>(25/24 * 49/48) = 0.8482245109 ; this is the JIP of p=1, q=1 (equiv. to p=-1, q=-1)
 * log<sub>2</sub>(256/243) / log<sub>2</sub>(25/24) = 1.2766647429 ; this is the JIP of p=1, q=0 (equiv. to p=-1, q=0)
+* log<sub>2</sub>(256/243) / log<sub>2</sub>(49/48) = 2.5275365063 ; this is the JIP of p=0, q=1 (equiv. to p=0, q=-1)
+* log<sub>2</sub>(256/243) / log<sub>2</sub>(1225/1152) = 0.8482245109 ; this is the JIP of p=1, q=1 (equiv. to p=-1, q=-1)
 * log<sub>2</sub>(256/243) / log<sub>2</sub>(50/49) = 2.5796543166 ; this is the JIP of p=1, q=-1 (equiv. to p=-1, q=1)
-* log<sub>2</sub>(256/243) / log<sub>2</sub>(2401/2400) = 125.1... ; this is the JIP of p=-1, q=2 (equiv. to p=1, q=-2)
+* log<sub>2</sub>(256/243) / log<sub>2</sub>(60025/55296) = 0.6350918818 ; this is the JIP of p=1, q=2 (equiv. to p=-1, q=-2)
-* log<sub>2</sub>(256/243) / log<sub>2</sub>(49/48) = 2.5275365063 ; this is the JIP of p=0, q=1 (equiv. to p=0, q=-1)
+* log<sub>2</sub>(256/243) / log<sub>2</sub>(2401/2400) = 125.1044589 ; this is the JIP of p=1, q=-2 (equiv. to p=-1, q=2)
-* log<sub>2</sub>(256/243) / log<sub>2</sub>(25/24 * 50/49) = 0.8540148427 ; this is the JIP of p=2, q=-1 (equiv. to p=-2, q=1)
+* log<sub>2</sub>(256/243) / log<sub>2</sub>(30625/27648) = 0.5096257724 ; this is the JIP of p=2, q=1 (equiv. to p=-2, q=-1)
+* log<sub>2</sub>(256/243) / log<sub>2</sub>(625/588) = 0.8540148427 ; this is the JIP of p=2, q=-1 (equiv. to p=-2, q=1)
 Importantly, each JIP corresponds to a ''rational'', so that, for example, (''p'', ''q'') = (1, -2) is equivalent to (''p'', ''q'') = (2, -4) and to (''p'', ''q'') = (-1, 2) '''but not to (1, 2)'''. Note that all these JIPs lie on the JIL (just intonation line).