398edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|398}} == Theory == 398et is only consistent to the 5-odd-limit. Using the patent val, it tempers out 1220703125/1219784832, 1280000000/12759898..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|398}}
{{ED intro}}


== Theory ==
== Theory ==
398et is only consistent to the [[5-odd-limit]]. Using the patent val, it tempers out 1220703125/1219784832, 1280000000/1275989841, [[65625/65536]], 102760448/102515625 and 200120949/200000000 in the 7-limit; 1073741824/1071794405, 100663296/100656875, 161280/161051, 35156250/35153041, 2097152/2096325, [[4000/3993]], 2734375/2725888, 496125/495616, [[131072/130977]], [[6250/6237]], 9765625/9732096, 4302592/4296875, 352947/352000, 422576/421875, 184877/184320, [[3025/3024]], 9453125/9437184 and 456533/455625 in the 11-limit. It [[support]]s [[quartonic]], [[yarman]], [[bisupermajor]] and [[semiquindromeda]].  
398edo is only [[consistent]] to the [[5-odd-limit]], though it has a reasonable approximation to the full 13-limit using the [[patent val]], which [[tempering out|tempers out]] [[10976/10935]], [[65625/65536]], 1500625/1492992, 102760448/102515625, 102942875/102036672, and 200120949/200000000 in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], 59290/59049, 117649/117128, and [[131072/130977]] in the 11-limit; and [[625/624]], [[1575/1573]], [[2080/2079]], [[2200/2197]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It [[support]]s [[yarman I]], [[bisupermajor]] and [[semiquindromeda]].  


=== Prime harmonics ===
=== Prime harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
398 factors into 2 × 199, with [[2edo]] and [[199edo]] as its subset edos.
Since 398 factors into {{factorisation|398}}, 398edo has [[2edo]] and [[199edo]] as its subsets.


== 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|631 -398}}
! rowspan="2" | [[Comma list]]
|{{mapping|398 631}}
! rowspan="2" | [[Mapping]]
| -0.1759
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 631 -398 }}
| {{mapping| 398 631 }}
| −0.1759
| 0.1759
| 0.1759
| 5.83
| 5.83
|-
|-
|2.3.5
| 2.3.5
|390625000/387420489, {{monzo|-53 10 16}}
| 390625000/387420489, {{monzo| -53 10 16 }}
|{{mapping|398 631 924}}
| {{mapping| 398 631 924 }}
| -0.0622
| −0.0622
| 0.2157
| 0.2157
| 7.15
| 7.15
|-
|-
|2.3.5.7
| 2.3.5.7
|10976/10935, 65625/65536, 1280000000/1275989841
| 10976/10935, 65625/65536, 200120949/200000000
|{{mapping|398 631 924 1117}}
| {{mapping| 398 631 924 1117 }}
| +0.0412
| +0.0412
| 0.2588
| 0.2588
| 8.58
| 8.58
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|3025/3024, 6250/6237, 10976/10935, 496125/495616
| 3025/3024, 4000/3993, 10976/10935, 65625/65536
|{{mapping|398 631 924 1117 1377}}
| {{mapping| 398 631 924 1117 1377 }}
| +0.0075
| +0.0075
| 0.2411
| 0.2411
| 8.00
| 8.00
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|2080/2079, 625/624, 3025/3024, 4096/4095, 10976/10935
| 625/624, 1575/1573, 2080/2079, 2200/2197, 10976/10935
|{{mapping|398 631 924 1117 1377 1473}}
| {{mapping| 398 631 924 1117 1377 1473 }}
| -0.0243
| −0.0243
| 0.2313
| 0.2313
| 7.67
| 7.67
Line 60: Line 61:
=== 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
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|5\398
| 5\398
|15.08
| 15.08
|126/125
| 126/125
|[[Yarman]]
| [[Yarman I]]
|-
|-
|1
| 1
|183\398
| 183\398
|551.76
| 551.76
|11/8
| 11/8
|[[Emka]] / [[Emkay]]
| [[Emkay]]
|-
|-
|2
| 2
|54\398
| 54\398
|162.81
| 162.81
|11/10
| 11/10
|[[Kwazy]] / [[Bisupermajor]]
| [[Bisupermajor]]
|}
|}
 
<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
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Latest revision as of 13:31, 13 March 2026

← 397edo 398edo 399edo →
Prime factorization 2 × 199
Step size 3.01508 ¢ 
Fifth 233\398 (702.513 ¢)
Semitones (A1:m2) 39:29 (117.6 ¢ : 87.44 ¢)
Consistency limit 5
Distinct consistency limit 5

398 equal divisions of the octave (abbreviated 398edo or 398ed2), also called 398-tone equal temperament (398tet) or 398 equal temperament (398et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 398 equal parts of about 3.02 ¢ each. Each step represents a frequency ratio of 21/398, or the 398th root of 2.

Theory

398edo is only consistent to the 5-odd-limit, though it has a reasonable approximation to the full 13-limit using the patent val, which tempers out 10976/10935, 65625/65536, 1500625/1492992, 102760448/102515625, 102942875/102036672, and 200120949/200000000 in the 7-limit; 3025/3024, 4000/3993, 6250/6237, 59290/59049, 117649/117128, and 131072/130977 in the 11-limit; and 625/624, 1575/1573, 2080/2079, 2200/2197, 4096/4095, and 4225/4224 in the 13-limit. It supports yarman I, bisupermajor and semiquindromeda.

Prime harmonics

Approximation of prime harmonics in 398edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.56 -0.38 -0.99 +0.44 +0.68 +0.57 +0.98 -1.14 -1.44 +0.69
Relative (%) +0.0 +18.5 -12.7 -32.7 +14.6 +22.5 +19.0 +32.5 -37.8 -47.6 +23.0
Steps
(reduced) 		398
(0) 		631
(233) 		924
(128) 		1117
(321) 		1377
(183) 		1473
(279) 		1627
(35) 		1691
(99) 		1800
(208) 		1933
(341) 		1972
(380)

Subsets and supersets

Since 398 factors into 2 × 199, 398edo has 2edo and 199edo as its subsets.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [631 -398⟩ [⟨398 631]] −0.1759 0.1759 5.83
2.3.5 390625000/387420489, [-53 10 16⟩ [⟨398 631 924]] −0.0622 0.2157 7.15
2.3.5.7 10976/10935, 65625/65536, 200120949/200000000 [⟨398 631 924 1117]] +0.0412 0.2588 8.58
2.3.5.7.11 3025/3024, 4000/3993, 10976/10935, 65625/65536 [⟨398 631 924 1117 1377]] +0.0075 0.2411 8.00
2.3.5.7.11.13 625/624, 1575/1573, 2080/2079, 2200/2197, 10976/10935 [⟨398 631 924 1117 1377 1473]] −0.0243 0.2313 7.67

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 5\398 15.08 126/125 Yarman I
1 183\398 551.76 11/8 Emkay
2 54\398 162.81 11/10 Bisupermajor

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

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