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 ==
{{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
| 2.3
| {{monzo|661 -417}}
| {{monzo|661 -417}}
| {{mapping|417 661}}
| {{mapping|417 661}}
| &minus;0.0641
| −0.0641
| 0.0641
| 0.0641
| 2.23
| 2.23
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| 16875/16807, 65625/65536, 1600000/1594323
| 16875/16807, 65625/65536, 1600000/1594323
| {{mapping|417 661 968 1171}}
| {{mapping|417 661 968 1171}}
| &minus;0.0418
| −0.0418
| 0.2331
| 0.2331
| 8.10
| 8.10
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| 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}}
| &minus;0.1029
| −0.1029
| 0.2416
| 0.2416
| 8.40
| 8.40
{{comma basis end}}
|}


=== Rank-2 temperaments ===
=== 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
| 1
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| 16/15
| 16/15
| [[Tertiosec]]
| [[Tertiosec]]
{{rank-2 end}}
|}
{{orf}}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Latest revision as of 13:33, 13 March 2026

← 416edo 417edo 418edo →
Prime factorization 3 × 139
Step size 2.8777 ¢ 
Fifth 244\417 (702.158 ¢)
Semitones (A1:m2) 40:31 (115.1 ¢ : 89.21 ¢)
Consistency limit 5
Distinct consistency limit 5

417 equal divisions of the octave (abbreviated 417edo or 417ed2), also called 417-tone equal temperament (417tet) or 417 equal temperament (417et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 417 equal parts of about 2.88 ¢ each. Each step represents a frequency ratio of 21/417, or the 417th root of 2.

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 supports familia and 5-limit fortune.

Prime harmonics

Approximation of prime harmonics in 417edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.20 -0.70 +0.96 +1.20 -0.24 -1.36 -1.11 -0.94 +0.64 +0.29
Relative (%) +0.0 +7.1 -24.4 +33.3 +41.7 -8.3 -47.2 -38.6 -32.5 +22.2 +10.0
Steps
(reduced) 		417
(0) 		661
(244) 		968
(134) 		1171
(337) 		1443
(192) 		1543
(292) 		1704
(36) 		1771
(103) 		1886
(218) 		2026
(358) 		2066
(398)

Subsets and supersets

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

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [661 -417⟩ [⟨417 661]] −0.0641 0.0641 2.23
2.3.5 1600000/1594323, [-80 8 29⟩ [⟨417 661 968]] +0.0580 0.1806 6.28
2.3.5.7 16875/16807, 65625/65536, 1600000/1594323 [⟨417 661 968 1171]] −0.0418 0.2331 8.10
2.3.5.7.11 540/539, 3025/3024, 496125/495616, 7168000/7144929 [⟨417 661 968 1171 1443]] −0.1029 0.2416 8.40

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		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
3 39\417 112.23 16/15 Tertiosec

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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