810edo: Difference between revisions

(2 intermediate revisions by 2 users not shown)

Line 1:

== Theory ==

810 = 270 × 3, and 810edo has three copies of [[270edo]] in the 13-limit (and the 2.3.5.7.11.13.19 [[subgroup]]). It makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, only [[consistent]] to the [[9-odd-limit]]. [[11/9]], [[13/12]], [[13/9]], [[13/10]], and their [[octave complement]]s are all mapped inconsistently in this edo.

As an equal temperament, it [[tempering out|tempers out]] [[4914/4913]] in the 17-limit; and [[2024/2023]], [[2737/2736]], and [[3520/3519]] in the 23-limit. Although it does quite well in these limits, it is way less efficient as [[270edo]]'s or [[540edo]]'s mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied essentially tempered chords are worth the load of all the extra notes.

As an equal temperament, it [[tempering out|tempers out]] [[4914/4913]] in the 17-limit; and [[2024/2023]], [[2737/2736]], and [[3520/3519]] in the 23-limit. Although it does quite well in these limits, it is way less efficient than [[270edo]]'s or [[540edo]]'s mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied essentially tempered chords are worth the load of all the extra notes.

=== Prime harmonics ===

Line 11:

=== Subsets and supersets ===

Since 810 factors into {{factorization|810}}, 810edo has subset edos {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405 }}.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal<br />8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

Line 27:

Line 28:

| 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095, 4914/4913

| {{mapping| 810 1284 1881 2274 2802 2997 3311 }}

| -0.0281

| −0.0281

| 0.1025

| 6.92

Line 34:

Line 35:

| 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728, 4914/4913

| {{mapping| 810 1284 1881 2274 2802 2997 3311 3441 }}

| -0.0324

| −0.0324

| 0.0966

| 6.52

Line 41:

Line 42:

| 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728, 2024/2023, 2737/2736

| {{mapping| 810 1284 1881 2274 2802 2997 3311 3441 3664 }}

| -0.0257

| −0.0257

| 0.0930

| 6.28

|}

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro}}
+{{ED intro}}
 == Theory ==
 = 270 × 3, and 810edo has three copies of [[270edo]] in the 13-limit (and the 2.3.5.7.11.13.19 [[subgroup]]). It makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, only [[consistent]] to the [[9-odd-limit]]. [[11/9]], [[13/12]], [[13/9]], [[13/10]], and their [[octave complement]]s are all mapped inconsistently in this edo.
-As an equal temperament, it [[tempering out|tempers out]] [[4914/4913]] in the 17-limit; and [[2024/2023]], [[2737/2736]], and [[3520/3519]] in the 23-limit. Although it does quite well in these limits, it is way less efficient as [[270edo]]'s or [[540edo]]'s mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied essentially tempered chords are worth the load of all the extra notes.
+As an equal temperament, it [[tempering out|tempers out]] [[4914/4913]] in the 17-limit; and [[2024/2023]], [[2737/2736]], and [[3520/3519]] in the 23-limit. Although it does quite well in these limits, it is way less efficient than [[270edo]]'s or [[540edo]]'s mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied essentially tempered chords are worth the load of all the extra notes.
 === Prime harmonics ===
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
 Since 810 factors into {{factorization|810}}, 810edo has subset edos {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405 }}.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 27: / Line 28: @@
 | 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095, 4914/4913
 | {{mapping| 810 1284 1881 2274 2802 2997 3311 }}
-| -0.0281
+| −0.0281
 | 0.1025
 | 6.92
@@ Line 34: / Line 35: @@
 | 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728, 4914/4913
 | {{mapping| 810 1284 1881 2274 2802 2997 3311 3441 }}
-| -0.0324
+| −0.0324
 | 0.0966
 | 6.52
@@ Line 41: / Line 42: @@
 | 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728, 2024/2023, 2737/2736
 | {{mapping| 810 1284 1881 2274 2802 2997 3311 3441 3664 }}
-| -0.0257
+| −0.0257
 | 0.0930
 | 6.28
 |}