374edo: Difference between revisions

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== Theory ==

374et is ~~only~~ consistent to the [[3-odd-limit]]. Omitting the harmonic 5, it is consistent to the [[31-odd-limit]]. Using the patent val, it tempers out ~~40500000/40353607, 184528125/184473632,~~ [[5120/5103]] ~~and~~ [[2100875/2097152]] in the 7-limit; ~~1073741824~~/~~1071794405, 161280/161051~~, ~~820125~~/~~819896, 2097152/2096325~~, 12005/11979, ~~2621440/2614689, 496125/495616, 1296000/1294139, 1265625/1261568, [[200704/200475]], 5767168/5764801, 1375/1372, 184549376/184528125, 1479016/1476225, 275653125/275365888,~~ [[41503/41472]]~~, 1362944/1361367, 166375/165888, 3294225/3294172 and 322102/321489~~ in the 11-limit. It [[support]]s [[quintakwai]] and [[quartemka]].

374et is in[[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[5/1|5]] is about halfway between its steps. Omitting the harmonic 5, it is consistent to the [[31-odd-limit]].

Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[5120/5103]], 1071875/1062882, 1500625/1492992, [[2100875/2097152]], and 9765625/9680832 in the 7-limit; 1375/1372, 4375/4356, 12005/11979, and [[41503/41472]] in the 11-limit. It [[support]]s [[quintakwai]] and [[quartemka]].

=== Prime harmonics ===

Line 9:

Line 11:

=== Subsets and supersets ===

374 factors into 2 × 11 × 17, ~~with~~ subset edos {{EDOs|2, 11, 17, 22, 34, and 187}}. [[748edo]], which doubles it, gives a good correction to the harmonic 5.

Since 374 factors into 2 × 11 × 17, 374edo has subset edos {{EDOs| 2, 11, 17, 22, 34, and 187 }}. [[748edo]], which doubles it, gives a good correction to the harmonic 5, but its approximation of harmonic 3 has drifted too far to render it inconsistent in the [[9-odd-limit]].

== Regular temperament properties ==

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

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

|-

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

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

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

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

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

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

! colspan="2" |Tuning Error

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

! colspan="2" | Tuning Error

|-

![[TE error|Absolute]] (¢)

! [[TE error|Absolute]] (¢)

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

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

|-

|2.3

| 2.3

|{{monzo|593 -374}}

| {{monzo| 593 -374 }}

|{{mapping|374 593}}

| {{mapping| 374 593 }}

| -0.2268

| −0.2268

| 0.2267

| 7.07

|-

|2.3.7

| 2.3.7

|{{monzo|4 -22 11}}, {{monzo|51 -18 -8}}

| {{monzo| 4 -22 11 }}, {{monzo| 51 -18 -8 }}

|{{mapping|374 593 1050}}

| {{mapping| 374 593 1050 }}

| -0.1699

| −0.1699

| 0.2018

| 6.29

|-

|2.3.7.11

| 2.3.7.11

|41503/41472, 1362944/1361367, 70493667328/70027449129

| 41503/41472, 1362944/1361367, 70493667328/70027449129

|{{mapping|374 593 1050 1294}}

| {{mapping| 374 593 1050 1294 }}

| -0.1675

| −0.1675

| 0.1748

| 5.45

|-

|2.3.7.11.13

| 2.3.7.11.13

|~~41503~~/~~41472~~, 20449/20412, 652288/649539~~, 10648/10647~~

| 10648/10647, 20449/20412, 41503/41472, 652288/649539

|{{mapping|374 593 1050 1294 ~~1384~~}}

| {{mapping| 374 593 1050 1294 138 4}}

| -0.1401

| −0.1401

| 0.1656

| 5.16

|-

|2.3.7.11.13.17

| 2.3.7.11.13.17

|~~22528/22491,~~ 2058/2057, ~~34816~~/~~34749~~, 8624/8619, ~~8281~~/~~8262~~

| 2058/2057, 8281/8262, 8624/8619, 22528/22491, 34816/34749

|{{mapping|374 593 1050 1294 1384 1529}}

| {{mapping| 374 593 1050 1294 1384 1529 }}

| -0.1546

| −0.1546

| 0.1546

| 4.82

|-

|2.3.7.11.13.17.19

| 2.3.7.11.13.17.19

|1729/1728, 2912/2907, 22528/22491~~, 2058/2057, 5929/5928~~, 34816/34749

| 1729/1728, 2058/2057, 2912/2907, 5929/5928, 22528/22491, 34816/34749

|{{mapping|374 593 1050 1294 1384 1529 1589}}

| {{mapping| 374 593 1050 1294 1384 1529 1589 }}

| -0.1622

| −0.1622

| 0.1444

| 4.50

|}

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|374}}
+{{ED intro}}
 == Theory ==
-et is only consistent to the [[3-odd-limit]]. Omitting the harmonic 5, it is consistent to the [[31-odd-limit]]. Using the patent val, it tempers out 40500000/40353607, 184528125/184473632, [[5120/5103]] and [[2100875/2097152]] in the 7-limit; 1073741824/1071794405, 161280/161051, 820125/819896, 2097152/2096325, 12005/11979, 2621440/2614689, 496125/495616, 1296000/1294139, 1265625/1261568, [[200704/200475]], 5767168/5764801, 1375/1372, 184549376/184528125, 1479016/1476225, 275653125/275365888, [[41503/41472]], 1362944/1361367, 166375/165888, 3294225/3294172 and 322102/321489 in the 11-limit. It [[support]]s [[quintakwai]] and [[quartemka]].
+et is in[[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[5/1|5]] is about halfway between its steps. Omitting the harmonic 5, it is consistent to the [[31-odd-limit]].
+Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[5120/5103]], 1071875/1062882, 1500625/1492992, [[2100875/2097152]], and 9765625/9680832 in the 7-limit; 1375/1372, 4375/4356, 12005/11979, and [[41503/41472]] in the 11-limit. It [[support]]s [[quintakwai]] and [[quartemka]].
 === Prime harmonics ===
@@ Line 9: / Line 11: @@
 === Subsets and supersets ===
-factors into 2 × 11 × 17, with subset edos {{EDOs|2, 11, 17, 22, 34, and 187}}. [[748edo]], which doubles it, gives a good correction to the harmonic 5.
+Since 374 factors into 2 × 11 × 17, 374edo has subset edos {{EDOs| 2, 11, 17, 22, 34, and 187 }}. [[748edo]], which doubles it, gives a good correction to the harmonic 5, but its approximation of harmonic 3 has drifted too far to render it inconsistent in the [[9-odd-limit]].
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
+|-
-! rowspan="2" |[[Comma list|Comma List]]
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" |[[Mapping]]
+! rowspan="2" | [[Comma list|Comma List]]
-! rowspan="2" |Optimal<br>8ve Stretch (¢)
+! rowspan="2" | [[Mapping]]
-! colspan="2" |Tuning Error
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! colspan="2" | Tuning Error
 |-
-![[TE error|Absolute]] (¢)
+! [[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
+! [[TE simple badness|Relative]] (%)
 |-
-|2.3
+| 2.3
-|{{monzo|593 -374}}
+| {{monzo| 593 -374 }}
-|{{mapping|374 593}}
+| {{mapping| 374 593 }}
-| -0.2268
+| −0.2268
 | 0.2267
 | 7.07
 |-
-|2.3.7
+| 2.3.7
-|{{monzo|4 -22 11}}, {{monzo|51 -18 -8}}
+| {{monzo| 4 -22 11 }}, {{monzo| 51 -18 -8 }}
-|{{mapping|374 593 1050}}
+| {{mapping| 374 593 1050 }}
-| -0.1699
+| −0.1699
 | 0.2018
 | 6.29
 |-
-|2.3.7.11
+| 2.3.7.11
-|41503/41472, 1362944/1361367, 70493667328/70027449129
+| 41503/41472, 1362944/1361367, 70493667328/70027449129
-|{{mapping|374 593 1050 1294}}
+| {{mapping| 374 593 1050 1294 }}
-| -0.1675
+| −0.1675
 | 0.1748
 | 5.45
 |-
-|2.3.7.11.13
+| 2.3.7.11.13
-|41503/41472, 20449/20412, 652288/649539, 10648/10647
+| 10648/10647, 20449/20412, 41503/41472, 652288/649539
-|{{mapping|374 593 1050 1294 1384}}
+| {{mapping| 374 593 1050 1294 138 4}}
-| -0.1401
+| −0.1401
 | 0.1656
 | 5.16
 |-
-|2.3.7.11.13.17
+| 2.3.7.11.13.17
-|22528/22491, 2058/2057, 34816/34749, 8624/8619, 8281/8262
+| 2058/2057, 8281/8262, 8624/8619, 22528/22491, 34816/34749
-|{{mapping|374 593 1050 1294 1384 1529}}
+| {{mapping| 374 593 1050 1294 1384 1529 }}
-| -0.1546
+| −0.1546
 | 0.1546
 | 4.82
 |-
-|2.3.7.11.13.17.19
+| 2.3.7.11.13.17.19
-|1729/1728, 2912/2907, 22528/22491, 2058/2057, 5929/5928, 34816/34749
+| 1729/1728, 2058/2057, 2912/2907, 5929/5928, 22528/22491, 34816/34749
-|{{mapping|374 593 1050 1294 1384 1529 1589}}
+| {{mapping| 374 593 1050 1294 1384 1529 1589 }}
-| -0.1622
+| −0.1622
 | 0.1444
 | 4.50
 |}