390edo: Difference between revisions

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== Theory ==

390et is ~~only consistent to the~~ [[~~3-odd~~-limit]]. ~~It can be used~~ in the 2.3.7.11.13.17.23.31.41 [[subgroup]]. Using the patent val, it tempers out [[~~32805~~/~~32768~~]] ~~in the 5-limit; 283115520/282475249, 184528125/184473632, [[589824/588245]], 2460375/2458624, 67108864/66976875, [[6144/6125]], 102760448/102515625,~~ [[3136/3125]]~~, [[2401/2400]] and 5250987/5242880~~ in the 7-limit~~. It~~ [[support]]s [[~~trilobite~~]].

390et is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[65edo]]. But its approximation to higher [[harmonic]]s are improved, so that it is suitable for use in the 2.3.7.11.13.17.23.31.41 [[subgroup]].

Using the [[patent val]] nonetheless, it tempers out [[2401/2400]] and [[3136/3125]] in the 7-limit, [[support]]ing [[hemiwürschmidt]].

=== Prime harmonics ===

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=== Subsets and supersets ===

390 factors into 2 × 3 × 5 × 13 ~~with~~ subset edos {{EDOs|2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, and 195}}. [[780edo]], which doubles it, gives a good correction to the harmonic 5.

Since 390 factors into 2 × 3 × 5 × 13, 390edo has subset edos {{EDOs| 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, and 195 }}. [[780edo]], which doubles it, gives a good correction to the harmonic 5.

== Regular temperament properties ==

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

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

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

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

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

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

! 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]] (¢)

! [[TE error|Absolute]] (¢)

![[TE simple badness|Relative]] (%)

! [[TE simple badness|Relative]] (%)

|-

~~|2.3~~

~~|{{monzo|-103 65}}~~

~~|{{mapping|390 618}}~~

~~| 0.1314~~

~~| 4.27~~

|-

|2.3.7

| 2.3.7

|118098/117649, 34451725707/34359738368

| 118098/117649, 34451725707/34359738368

|{{mapping|390 618 1095}}

| {{mapping| 390 618 1095 }}

| 0.0395

| 0.1685

| 5.48

|-

|2.3.7.11

| 2.3.7.11

|118098/117649, 1362944/1361367, 235782657/234881024

| 118098/117649, 1362944/1361367, 235782657/234881024

|{{mapping|390 618 1095 1349}}

| {{mapping| 390 618 1095 1349 }}

| 0.0693

| 0.1548

| 5.03

|-

|2.3.7.11.13

| 2.3.7.11.13

|729/728, 16848/16807~~, 10648/10647~~, 1574573/1572864

| 729/728, 10648/10647, 16848/16807, 1574573/1572864

|{{mapping|390 618 1095 1349 1443}}

| {{mapping| 390 618 1095 1349 1443 }}

| 0.0839

| 0.1415

| 4.60

|-

|2.3.7.11.13.17

| 2.3.7.11.13.17

|729/728, 1089/1088, 16848/16807, 95823/95744~~, 65637/65536~~

| 729/728, 1089/1088, 16848/16807, 65637/65536, 95823/95744

|{{mapping|390 618 1095 1349 1443 1594}}

| {{mapping| 390 618 1095 1349 1443 1594 }}

| 0.0838

| 0.1292

| 4.20

|}

@@ Line 3: / Line 3: @@
 == Theory ==
-et is only consistent to the [[3-odd-limit]]. It can be used in the 2.3.7.11.13.17.23.31.41 [[subgroup]]. Using the patent val, it tempers out [[32805/32768]] in the 5-limit; 283115520/282475249, 184528125/184473632, [[589824/588245]], 2460375/2458624, 67108864/66976875, [[6144/6125]], 102760448/102515625, [[3136/3125]], [[2401/2400]] and 5250987/5242880 in the 7-limit. It [[support]]s [[trilobite]].
+et is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[65edo]]. But its approximation to higher [[harmonic]]s are improved, so that it is suitable for use in the 2.3.7.11.13.17.23.31.41 [[subgroup]].
+Using the [[patent val]] nonetheless, it tempers out [[2401/2400]] and [[3136/3125]] in the 7-limit, [[support]]ing [[hemiwürschmidt]].
 === Prime harmonics ===
@@ Line 9: / Line 11: @@
 === Subsets and supersets ===
-factors into 2 × 3 × 5 × 13 with subset edos {{EDOs|2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, and 195}}. [[780edo]], which doubles it, gives a good correction to the harmonic 5.
+Since 390 factors into 2 × 3 × 5 × 13, 390edo has subset edos {{EDOs| 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, and 195 }}. [[780edo]], which doubles it, gives a good correction to the harmonic 5.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" |[[Comma list|Comma List]]
+! rowspan="2" | [[Comma list|Comma List]]
-! rowspan="2" |[[Mapping]]
+! 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]] (¢)
+! [[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
+! [[TE simple badness|Relative]] (%)
-|-
-|2.3
-|{{monzo|-103 65}}
-|{{mapping|390 618}}
-| 0.1314
-| 0.1314
-| 4.27
 |-
-|2.3.7
+| 2.3.7
-|118098/117649, 34451725707/34359738368
+| 118098/117649, 34451725707/34359738368
-|{{mapping|390 618 1095}}
+| {{mapping| 390 618 1095 }}
 | 0.0395
 | 0.1685
 | 5.48
 |-
-|2.3.7.11
+| 2.3.7.11
-|118098/117649, 1362944/1361367, 235782657/234881024
+| 118098/117649, 1362944/1361367, 235782657/234881024
-|{{mapping|390 618 1095 1349}}
+| {{mapping| 390 618 1095 1349 }}
 | 0.0693
 | 0.1548
 | 5.03
 |-
-|2.3.7.11.13
+| 2.3.7.11.13
-|729/728, 16848/16807, 10648/10647, 1574573/1572864
+| 729/728, 10648/10647, 16848/16807, 1574573/1572864
-|{{mapping|390 618 1095 1349 1443}}
+| {{mapping| 390 618 1095 1349 1443 }}
 | 0.0839
 | 0.1415
 | 4.60
 |-
-|2.3.7.11.13.17
+| 2.3.7.11.13.17
-|729/728, 1089/1088, 16848/16807, 95823/95744, 65637/65536
+| 729/728, 1089/1088, 16848/16807, 65637/65536, 95823/95744
-|{{mapping|390 618 1095 1349 1443 1594}}
+| {{mapping| 390 618 1095 1349 1443 1594 }}
 | 0.0838
 | 0.1292
 | 4.20
 |}