417edo: Difference between revisions

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== Theory ==

417et is only consistent to the [[5-odd-limit]]. Using the patent val, it tempers out [[589824/588245]], [[43046721/43025920]], [[33554432/33480783]] and [[65625/65536]] in the 7-limit; 78121827/77948684, 20155392/20131375, 10333575/10307264, 1019215872/1019046875, 46656/46585, 1366875/1362944, 78675968/78594219, [[536870912/535869675]], 7168000/7144929, 496125/495616, 514714375/514434888, 2359296/2358125, [[540/539]], 1265625/1261568, 17561600/17537553, 180224/180075, 1375/1372, 645922816/645700815, [[3025/3024]], 9453125/9437184 and 1362944/1361367 in the 11-limit. It [[support]]s [[familia]] and 5-limit [[fortune]].

417et is only consistent to the [[5-odd-limit]]. Using the patent val, it tempers out [[589824/588245]], [[43046721/43025920]], [[33554432/33480783]], and [[65625/65536]] in the 7-limit; 78121827/77948684, 20155392/20131375, 10333575/10307264, 1019215872/1019046875, 46656/46585, 1366875/1362944, 78675968/78594219, [[536870912/535869675]], 7168000/7144929, 496125/495616, 514714375/514434888, 2359296/2358125, [[540/539]], 1265625/1261568, 17561600/17537553, 180224/180075, 1375/1372, 645922816/645700815, [[3025/3024]], 9453125/9437184, and 1362944/1361367 in the 11-limit. It [[support]]s [[familia]] and 5-limit [[fortune]].

=== Prime harmonics ===

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=== Subsets and supersets ===

417 factors into ~~3 × 139~~, with [[3edo]] and [[139edo]] as its subset edos.

417 factors into {{factorisation|417}}, with [[3edo]] and [[139edo]] as its subset edos.

== Regular temperament properties ==

{~~{comma basis begin}}~~

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

|-

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

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

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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

|-

| 2.3

| {{monzo|661 -417}}

| {{mapping|417 661}}

| ~~−0~~.0641

| −0.0641

| 0.0641

| 2.23

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| 16875/16807, 65625/65536, 1600000/1594323

| {{mapping|417 661 968 1171}}

| ~~−0~~.0418

| −0.0418

| 0.2331

| 8.10

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| 540/539, 3025/3024, 496125/495616, 7168000/7144929

| {{mapping|417 661 968 1171 1443}}

| ~~−0~~.1029

| −0.1029

| 0.2416

| 8.40

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods<br />per 8ve

! Generator*

! Cents*

! Associated<br />ratio*

! Temperaments

|-

| 1

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| 16/15

| [[Tertiosec]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|417}}
+{{ED intro}}
 == Theory ==
-et is only consistent to the [[5-odd-limit]]. Using the patent val, it tempers out [[589824/588245]], [[43046721/43025920]], [[33554432/33480783]] and [[65625/65536]] in the 7-limit; 78121827/77948684, 20155392/20131375, 10333575/10307264, 1019215872/1019046875, 46656/46585, 1366875/1362944, 78675968/78594219, [[536870912/535869675]], 7168000/7144929, 496125/495616, 514714375/514434888, 2359296/2358125, [[540/539]], 1265625/1261568, 17561600/17537553, 180224/180075, 1375/1372, 645922816/645700815, [[3025/3024]], 9453125/9437184 and 1362944/1361367 in the 11-limit. It [[support]]s [[familia]] and 5-limit [[fortune]].
+et is only consistent to the [[5-odd-limit]]. Using the patent val, it tempers out [[589824/588245]], [[43046721/43025920]], [[33554432/33480783]], and [[65625/65536]] in the 7-limit; 78121827/77948684, 20155392/20131375, 10333575/10307264, 1019215872/1019046875, 46656/46585, 1366875/1362944, 78675968/78594219, [[536870912/535869675]], 7168000/7144929, 496125/495616, 514714375/514434888, 2359296/2358125, [[540/539]], 1265625/1261568, 17561600/17537553, 180224/180075, 1375/1372, 645922816/645700815, [[3025/3024]], 9453125/9437184, and 1362944/1361367 in the 11-limit. It [[support]]s [[familia]] and 5-limit [[fortune]].
 === Prime harmonics ===
@@ Line 9: / Line 9: @@
 === Subsets and supersets ===
-factors into 3 × 139, with [[3edo]] and [[139edo]] as its subset edos.
+factors into {{factorisation|417}}, with [[3edo]] and [[139edo]] as its subset edos.
 == Regular temperament properties ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
 | {{monzo|661 -417}}
 | {{mapping|417 661}}
-| &minus;0.0641
+| −0.0641
 | 0.0641
 | 2.23
@@ Line 31: / Line 40: @@
 | 16875/16807, 65625/65536, 1600000/1594323
 | {{mapping|417 661 968 1171}}
-| &minus;0.0418
+| −0.0418
 | 0.2331
 | 8.10
@@ Line 38: / Line 47: @@
 | 540/539, 3025/3024, 496125/495616, 7168000/7144929
 | {{mapping|417 661 968 1171 1443}}
-| &minus;0.1029
+| −0.1029
 | 0.2416
 | 8.40
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
 |-
 | 1
@@ Line 69: / Line 85: @@
 | 16/15
 | [[Tertiosec]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct