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417edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|417}}
{{ED intro}}


== Theory ==
== 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 ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== 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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|661 -417}}
! rowspan="2" | [[Comma list]]
|{{mapping|417 661}}
! rowspan="2" | [[Mapping]]
| -0.0641
! 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
| 2.23
|-
|-
|2.3.5
| 2.3.5
|1600000/1594323, {{monzo|-80 8 29}}
| 1600000/1594323, {{monzo|-80 8 29}}
|{{mapping|417 661 968}}
| {{mapping|417 661 968}}
| +0.0580
| +0.0580
| 0.1806
| 0.1806
| 6.28
| 6.28
|-
|-
|2.3.5.7
| 2.3.5.7
|16875/16807, 65625/65536, 1600000/1594323
| 16875/16807, 65625/65536, 1600000/1594323
|{{mapping|417 661 968 1171}}
| {{mapping|417 661 968 1171}}
| -0.0418
| −0.0418
| 0.2331
| 0.2331
| 8.10
| 8.10
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|540/539, 3025/3024, 496125/495616, 7168000/7144929
| 540/539, 3025/3024, 496125/495616, 7168000/7144929
|{{mapping|417 661 968 1171 1443}}
| {{mapping|417 661 968 1171 1443}}
| -0.1029
| −0.1029
| 0.2416
| 0.2416
| 8.40
| 8.40
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)*
! Periods<br />per 8ve
! Cents<br>(reduced)*
! Generator*
! Associated<br>Ratio*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
| 1
| 77\417
| 221.58
| 8388608/7381125
| [[Fortune]]
|-
| 1
| 118\417
| 339.57
| 243/200
| [[Amity]]
|-
| 1
| 121\417
| 348.20
| 60/49
| [[Eris]]
|-
|-
|1
| 3
|121\417
| 39\417
|348.20
| 112.23
|60/49
| 16/15
|[[Eris]]
| [[Tertiosec]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
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