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

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The blackwood comma is the characteristic [[3-limit]] comma tempered out in 10edo.

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

* 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)

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)'''.

Also note that continua separated by 2401/2400 are meaningfully different, but due to the efficiency of 2401/2400, one may want to examine the continuum of all 7-limit temperaments supported by [[10edo]] for which [[2401/2400]] is tempered.

{| class="wikitable center-1 center-2"

|+ Selected temperaments with integer ''p'' and ''q''

|-

! rowspan="2" | ''p''

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| 0 || 0 || [[Blackwood]] || [[256/243]] || {{monzo| 8 -5 0 0 }}

|-

| 0 || 1 || [[Archytas]] (squared) || [[4096/3969]] || {{monzo| 12 -4 0 -2 }}

| 0 || 1 || [[Archytas]] (squared) || [[64/63|4096/3969]] || {{monzo| 12 -4 0 -2 }}

|-

| 0 || 2 || [[Buzzard]] || [[65536/64827]] || {{monzo| 16 -3 0 -4 }}

|-

| 0 || 3 || [[Slendric]] (squared) || [[1058841/1048576]] || {{monzo| -20 2 0 6 }}

| 0 || 3 || [[Slendric]] (squared) || [[1029/1024|1058841/1048576]] || {{monzo| -20 2 0 6 }}

|-

| 0 || ∞ || [[Semaphore]] || [[49/48]] || {{monzo| -4 -1 0 2 }}

@@ Line 6: / Line 6: @@
 The blackwood comma is the characteristic [[3-limit]] comma tempered out in 10edo.
+[[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>(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>(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>(25/24 * 50/49) = 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)'''.
+Also note that continua separated by 2401/2400 are meaningfully different, but due to the efficiency of 2401/2400, one may want to examine the continuum of all 7-limit temperaments supported by [[10edo]] for which [[2401/2400]] is tempered.
 {| class="wikitable center-1 center-2"
 |+ Selected temperaments with integer ''p'' and ''q''
 |-
 ! rowspan="2" | ''p''
@@ Line 20: / Line 33: @@
 | 0 || 0 || [[Blackwood]] || [[256/243]] || {{monzo| 8 -5 0 0 }}
 |-
-| 0 || 1 || [[Archytas]] (squared) || [[4096/3969]] || {{monzo| 12 -4 0 -2 }}
+| 0 || 1 || [[Archytas]] (squared) || [[64/63|4096/3969]] || {{monzo| 12 -4 0 -2 }}
 |-
 | 0 || 2 || [[Buzzard]] || [[65536/64827]] || {{monzo| 16 -3 0 -4 }}
 |-
-| 0 || 3 || [[Slendric]] (squared) || [[1058841/1048576]] || {{monzo| -20 2 0 6 }}
+| 0 || 3 || [[Slendric]] (squared) || [[1029/1024|1058841/1048576]] || {{monzo| -20 2 0 6 }}
 |-
 | 0 || ∞ || [[Semaphore]] || [[49/48]] || {{monzo| -4 -1 0 2 }}