2016edo: Difference between revisions

Line 14:

2016 shares the mapping for 3 with [[224edo]], albeit with a 28 relative cent error.

2016edo has two reasonable mappings for 7. The 2016d val, {{Val|2016 3195 4681 5659}}, tempers out 5250987/5242880, 40353607/40310784 (tritrizo), and {{monzo|14, 11, -22, 7}}. As such, its circle of the interval 7/6 is the same as in [[9edo]]. The patent val, {{Val|2016 3195 4681 5658}} tempers out [[250047/250000]], along with {{monzo|7, 18, -2, -11}} and {{monzo|43, -1, -13, -4}}. This means that the symmetrical major third (400 cents, 1/3 of the octave) in 2016edo corresponds to [[63/50]].

2016edo has two reasonable mappings for 7. The 2016d val, {{Val|2016 3195 4681 5659}}, tempers out 5250987/5242880, 40353607/40310784 (tritrizo), and {{monzo|14, 11, -22, 7}}. As such, its circle of the interval 7/6 is the same as in [[9edo]].

The patent val, {{Val|2016 3195 4681 5658}} tempers out [[250047/250000]], along with {{monzo|7, 18, -2, -11}} and {{monzo|43, -1, -13, -4}}. This means that the symmetrical major third (400 cents, 1/3 of the octave) in 2016edo corresponds to [[63/50]].

In the 11-limit, 2016edo tempers out the {{monzo|0 0 -22 0 3 11}} comma, which equates a stack of 11 [[25/13]]<nowiki/>s with 3 [[11/1|hendecataves]].

=== Fractional octave temperaments ===

The patent val 7-limit in 2016edo gives rise to the to rank two temperaments of 72 & 624 with period 24 and 441 & 1407, period 21. The 2016d val gives rise to 171 & 306, period 9 and 270 & 936bd, period 18.

If we assume that 2016edo is a dual-seventh system, where 5659th and 5660th steps represent 7- and 7+, two distinct dimensions amounting to 49/1, then this allows for mixing of these two temperaments. 236 & 600 is the temperament which best represents that.

== Regular temperament properties ==

Line 40:

Line 47:

|2.3.5.7

|250047/250000, {{monzo|7, 18, -2, -11}}, {{monzo|43, -1, -13, -4}}

|[{{Val|2016 3195 4681 ~~5658~~}}]

|[{{Val|2016 3195 4681 5660}}]

|0.007

|0.066

Line 67:

Line 74:

|0.019

|3.2

|}

== Rank two temperaments by generator ==

{| class="wikitable center-all left-5"

!Periods

per octave

!Generator

(reduced)

!Cents

(reduced)

!Temperaments

|-

|21

|23\2016

|13.690

|441 & 1407

|-

|24

|29\2016

|17.262

|72 & 624

|}

@@ Line 14: / Line 14: @@
 shares the mapping for 3 with [[224edo]], albeit with a 28 relative cent error.
-edo has two reasonable mappings for 7. The 2016d val, {{Val|2016 3195 4681 5659}}, tempers out 5250987/5242880, 40353607/40310784 (tritrizo), and {{monzo|14, 11, -22, 7}}. As such, its circle of the interval 7/6 is the same as in [[9edo]]. The patent val, {{Val|2016 3195 4681 5658}} tempers out [[250047/250000]], along with {{monzo|7, 18, -2, -11}} and {{monzo|43, -1, -13, -4}}. This means that the symmetrical major third (400 cents, 1/3 of the octave) in 2016edo corresponds to [[63/50]].
+edo has two reasonable mappings for 7. The 2016d val, {{Val|2016 3195 4681 5659}}, tempers out 5250987/5242880, 40353607/40310784 (tritrizo), and {{monzo|14, 11, -22, 7}}. As such, its circle of the interval 7/6 is the same as in [[9edo]].
+The patent val, {{Val|2016 3195 4681 5658}} tempers out [[250047/250000]], along with {{monzo|7, 18, -2, -11}} and {{monzo|43, -1, -13, -4}}. This means that the symmetrical major third (400 cents, 1/3 of the octave) in 2016edo corresponds to [[63/50]].
 In the 11-limit, 2016edo tempers out the {{monzo|0 0 -22 0 3 11}} comma, which equates a stack of 11 [[25/13]]<nowiki/>s with 3 [[11/1|hendecataves]].
+=== Fractional octave temperaments ===
+The patent val 7-limit in 2016edo gives rise to the to rank two temperaments of 72 & 624 with period 24 and 441 & 1407, period 21. The 2016d val gives rise to 171 & 306, period 9 and 270 & 936bd, period 18.
+If we assume that 2016edo is a dual-seventh system, where 5659th and 5660th steps represent 7- and 7+, two distinct dimensions amounting to 49/1, then this allows for mixing of these two temperaments. 236 & 600 is the temperament which best represents that.
 == Regular temperament properties ==
@@ Line 40: / Line 47: @@
 |2.3.5.7
 |250047/250000, {{monzo|7, 18, -2, -11}}, {{monzo|43, -1, -13, -4}}
-|[{{Val|2016 3195 4681 5658}}]
+|[{{Val|2016 3195 4681 5660}}]
 |0.007
 |0.066
@@ Line 67: / Line 74: @@
 |0.019
 |3.2
+|}
+== Rank two temperaments by generator ==
+{| class="wikitable center-all left-5"
+!Periods
+per octave
+!Generator
+(reduced)
+!Cents
+(reduced)
+!Temperaments
+|-
+|21
+|23\2016
+|13.690
+|441 & 1407
+|-
+|24
+|29\2016
+|17.262
+|72 & 624
 |}