2016edo: Difference between revisions

Line 14:

Prime harmonics (below 61) with less than 22% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47. With next error being 26% on the 37th harmonic, it is reasonable to make cutoff here.

2016edo is consistent in the no-7s 17-limit. 2016 shares the mapping for 3 with [[224edo]], albeit with a 28 relative cent error. Prime harmonics (below 61) with less than 22% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47. With next error being 26% on the 37th harmonic, it is reasonable to make cutoff here.

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]].

In the 11-limit, 2016edo tempers out the {{monzo| 0 0 -22 0 3 11 }} comma, which equates a stack of eleven [[25/13]]'s with three [[11/1]]'s. However, it does ''not'' temper out the [[jacobin comma]].

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 eleven [[25/13]]'s with three [[11/1]]'s.

2016 has a total of 576 numbers coprime to it, which means this is how many generators can reach any point in the octave by being stacked.

Line 69:

Line 65:

| 10.1

|-

| 2.5.11~~.13~~

| 2.3.5.11

| {{~~monzo~~| 5 -6 9 ~~6 }}~~, {{~~monzo~~| ~~-38 12 4 -1~~ }}~~, {{monzo~~| 0 -22 3 11 }}

| 14 8 -10 -1, -26 15 -5 4, -29 27 3 -6

| [{{Val| 2016 4681 6974 7460 }}]

| [{{Val| 2016 3195 4681 6974}}]

| 0.~~013~~

| 0.036

| 0.~~015~~

| 0.043

| 2.5

| 7.3

|-

|2.3.5.11.13

|196625/196608, 53144100/53094899, [14, 8, -10, -1, 0⟩,

[-13, 9, 5, -8, 4⟩

|[{{Val| 2016 3195 4681 6974 7460}}]

|0.032

|0.040

|6.7

|-

|2.3.5.11.13.17

|2601/2600, 120285/120224, 140625/140608, 161109/161051,

196625/196608

|[{{Val| 2016 3195 4681 6974 7460 8240}}]

|0.034

|0.036

|6.2

|-

| 2.5.11.13.19.41.47

@@ Line 14: / Line 14: @@
 {{Harmonics in equal|2016}}
-Prime harmonics (below 61) with less than 22% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47. With next error being 26% on the 37th harmonic, it is reasonable to make cutoff here.
+edo is consistent in the no-7s 17-limit. 2016 shares the mapping for 3 with [[224edo]], albeit with a 28 relative cent error. Prime harmonics (below 61) with less than 22% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47. With next error being 26% on the 37th harmonic, it is reasonable to make cutoff here.
-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]].
+In the 11-limit, 2016edo tempers out the {{monzo| 0 0 -22 0 3 11 }} comma, which equates a stack of eleven [[25/13]]'s with three [[11/1]]'s. However, it does ''not'' temper out the [[jacobin comma]].
-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 eleven [[25/13]]'s with three [[11/1]]'s.
 has a total of 576 numbers coprime to it, which means this is how many generators can reach any point in the octave by being stacked.
@@ Line 69: / Line 65: @@
 | 10.1
 |-
-| 2.5.11.13
+| 2.3.5.11
-| {{monzo| 5 -6 9 6 }}, {{monzo| -38 12 4 -1 }}, {{monzo| 0 -22 3 11 }}
+| 14  8 -10 -1, -26 15  -5  4, -29 27 3 -6
-| [{{Val| 2016 4681 6974 7460 }}]
+| [{{Val| 2016 3195 4681 6974}}]
-| 0.013
+| 0.036
-| 0.015
+| 0.043
-| 2.5
+| 7.3
+|-
+|2.3.5.11.13
+|196625/196608, 53144100/53094899, [14, 8, -10, -1, 0⟩,
+[-13, 9, 5, -8, 4⟩
+|[{{Val| 2016 3195 4681 6974 7460}}]
+|0.032
+|0.040
+|6.7
+|-
+|2.3.5.11.13.17
+|2601/2600, 120285/120224, 140625/140608, 161109/161051,
+/196608
+|[{{Val| 2016 3195 4681 6974 7460 8240}}]
+|0.034
+|0.036
+|6.2
 |-
 | 2.5.11.13.19.41.47