Tetrachordal otonal scale: Difference between revisions

Line 17:

* Category 0 and 1 are excluded, as they get nonsensical results like category 1 having 0.5 scales.

* The amount of notes a scale of a given category x has <math>S(x)</math> falls between the ranges of these functions:<math>2x+1+\operatorname{floor}\left(\frac{x-1}{3}\right) ≤ S(x) ≤ 2x+1+\operatorname{floor}\left(\frac{x-2}{2}\right) </math> And the amount of notes are closely related to the scale tree of diatonic scales.

The data can be summarized in this table:

{| class="wikitable"

|-

! <math>x</math> !! <math>J(x)</math> !! <math>K(x)</math> !! <math>J(x) - K(x)</math> !! <math>2^{J(x)-1}</math> !! <math>2^{J(x)-1} - J(x)</math> !! <math>S_1(x)</math> !! <math>S_2(x)</math>

|-

| 2 || 1 || 1 || 0 || 1 || 0 || 5 || 5

|-

| 3 || 1 || 1 || 0 || 1 || 0 || 7 || 7

|-

| 4 || 1 || 1 || 0 || 1 || 0 || 10 || 10

|-

| 5 || 1 || 0 || 1 || 1 || 0 || 12 || 12

|-

| 6 || 2 || 2 || 0 || 2 || 0 || 14 || 15

|-

| 7 || 1 || 0 || 1 || 1 || 0 || 17 || 17

|-

| 8 || 2 || 1 || 1 || 2 || 0 || 19 || 20

|-

| 9 || 2 || 1 || 1 || 2 || 0 || 21 || 22

|-

| 10 || 2 || 1 || 1 || 2 || 0 || 24 || 25

|-

| 11 || 2 || 0 || 2 || 2 || 0 || 26 || 27

|-

| 12 || 3 || 2 || 1 || 4 || 1 || 28 || 30

|-

| 13 || 2 || 0 || 2 || 2 || 0 || 31 || 32

|-

| 14 || 3 || 1 || 2 || 4 || 1 || 33 || 35

|-

| 15 || 3 || 1 || 2 || 4 || 1 || 35 || 37

|-

| 16 || 3 || 1 || 2 || 4 || 1 || 38 || 40

|}

Simple TOs can be visualized as edo or MOS detempers, which makes them particularly useful either for NEJI targets or essential tempering.

@@ Line 17: / Line 17: @@
 * Category 0 and 1 are excluded, as they get nonsensical results like category 1 having 0.5 scales.
 * The amount of notes a scale of a given category x has <math>S(x)</math> falls between the ranges of these functions:<math>2x+1+\operatorname{floor}\left(\frac{x-1}{3}\right) ≤ S(x) ≤ 2x+1+\operatorname{floor}\left(\frac{x-2}{2}\right) </math> And the amount of notes are closely related to the scale tree of diatonic scales.
+The data can be summarized in this table:
+{| class="wikitable"
+|-
+! <math>x</math> !! <math>J(x)</math> !! <math>K(x)</math> !! <math>J(x) - K(x)</math> !! <math>2^{J(x)-1}</math> !! <math>2^{J(x)-1} - J(x)</math> !! <math>S_1(x)</math> !! <math>S_2(x)</math>
+|-
+| 2 || 1 || 1 || 0 || 1 || 0 || 5 || 5
+|-
+| 3 || 1 || 1 || 0 || 1 || 0 || 7 || 7
+|-
+| 4 || 1 || 1 || 0 || 1 || 0 || 10 || 10
+|-
+| 5 || 1 || 0 || 1 || 1 || 0 || 12 || 12
+|-
+| 6 || 2 || 2 || 0 || 2 || 0 || 14 || 15
+|-
+| 7 || 1 || 0 || 1 || 1 || 0 || 17 || 17
+|-
+| 8 || 2 || 1 || 1 || 2 || 0 || 19 || 20
+|-
+| 9 || 2 || 1 || 1 || 2 || 0 || 21 || 22
+|-
+| 10 || 2 || 1 || 1 || 2 || 0 || 24 || 25
+|-
+| 11 || 2 || 0 || 2 || 2 || 0 || 26 || 27
+|-
+| 12 || 3 || 2 || 1 || 4 || 1 || 28 || 30
+|-
+| 13 || 2 || 0 || 2 || 2 || 0 || 31 || 32
+|-
+| 14 || 3 || 1 || 2 || 4 || 1 || 33 || 35
+|-
+| 15 || 3 || 1 || 2 || 4 || 1 || 35 || 37
+|-
+| 16 || 3 || 1 || 2 || 4 || 1 || 38 || 40
+|}
 Simple TOs can be visualized as edo or MOS detempers, which makes them particularly useful either for NEJI targets or essential tempering.

Tetrachordal otonal scale: Difference between revisions

Eufalesio (talk | contribs)