390edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
Newer edit →
VisualWikitext
Francium (talk | contribs)
4,825 edits
Created page with "{{Infobox ET}} {{EDO intro|390}} == Theory == 390et is only consistent to the 3-odd-limit. It can be used in the 2.3.7.11.13.17.23.31.41 subgroup. Using the patent va..."
 
Review
Line 3: Line 3:


== Theory ==
== Theory ==
390et is only consistent to the [[3-odd-limit]]. It can be used in the 2.3.7.11.13.17.23.31.41 [[subgroup]]. Using the patent val, it tempers out [[32805/32768]] in the 5-limit; 283115520/282475249, 184528125/184473632, [[589824/588245]], 2460375/2458624, 67108864/66976875, [[6144/6125]], 102760448/102515625, [[3136/3125]], [[2401/2400]] and 5250987/5242880 in the 7-limit. It [[support]]s [[trilobite]].
390et is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[65edo]]. But its approximation to higher [[harmonic]]s are improved, so that it is suitable for use in the 2.3.7.11.13.17.23.31.41 [[subgroup]].  
 
Using the [[patent val]] nonetheless, it tempers out [[2401/2400]] and [[3136/3125]] in the 7-limit, [[support]]ing [[hemiwürschmidt]].  


=== Prime harmonics ===
=== Prime harmonics ===
Line 9: Line 11:


=== Subsets and supersets ===
=== Subsets and supersets ===
390 factors into 2 × 3 × 5 × 13 with subset edos {{EDOs|2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, and 195}}. [[780edo]], which doubles it, gives a good correction to the harmonic 5.
Since 390 factors into 2 × 3 × 5 × 13, 390edo has subset edos {{EDOs| 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, and 195 }}. [[780edo]], which doubles it, gives a good correction to the harmonic 5.


== 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" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo|-103 65}}
|{{mapping|390 618}}
| 0.1314
| 0.1314
| 4.27
|-
|-
|2.3.7
| 2.3.7
|118098/117649, 34451725707/34359738368
| 118098/117649, 34451725707/34359738368
|{{mapping|390 618 1095}}
| {{mapping| 390 618 1095 }}
| 0.0395
| 0.0395
| 0.1685
| 0.1685
| 5.48
| 5.48
|-
|-
|2.3.7.11
| 2.3.7.11
|118098/117649, 1362944/1361367, 235782657/234881024
| 118098/117649, 1362944/1361367, 235782657/234881024
|{{mapping|390 618 1095 1349}}
| {{mapping| 390 618 1095 1349 }}
| 0.0693
| 0.0693
| 0.1548
| 0.1548
| 5.03
| 5.03
|-
|-
|2.3.7.11.13
| 2.3.7.11.13
|729/728, 16848/16807, 10648/10647, 1574573/1572864
| 729/728, 10648/10647, 16848/16807, 1574573/1572864
|{{mapping|390 618 1095 1349 1443}}
| {{mapping| 390 618 1095 1349 1443 }}
| 0.0839
| 0.0839
| 0.1415
| 0.1415
| 4.60
| 4.60
|-
|-
|2.3.7.11.13.17
| 2.3.7.11.13.17
|729/728, 1089/1088, 16848/16807, 95823/95744, 65637/65536
| 729/728, 1089/1088, 16848/16807, 65637/65536, 95823/95744
|{{mapping|390 618 1095 1349 1443 1594}}
| {{mapping| 390 618 1095 1349 1443 1594 }}
| 0.0838
| 0.0838
| 0.1292
| 0.1292
| 4.20
| 4.20
|}
|}

Revision as of 14:36, 16 January 2024

← 389edo 390edo 391edo →
Prime factorization 2 × 3 × 5 × 13
Step size 3.07692 ¢ 
Fifth 228\390 (701.538 ¢) (→ 38\65)
Semitones (A1:m2) 36:30 (110.8 ¢ : 92.31 ¢)
Consistency limit 3
Distinct consistency limit 3

Template:EDO intro

Theory

390et is enfactored in the 5-limit, with the same tuning as 65edo. But its approximation to higher harmonics are improved, so that it is suitable for use in the 2.3.7.11.13.17.23.31.41 subgroup.

Using the patent val nonetheless, it tempers out 2401/2400 and 3136/3125 in the 7-limit, supporting hemiwürschmidt.

Prime harmonics

Approximation of prime harmonics in 390edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.42 +1.38 +0.40 -0.55 -0.53 -0.34 +0.95 -0.58 +1.19 -0.42
Relative (%) +0.0 -13.5 +44.8 +13.2 -17.8 -17.1 -11.1 +30.8 -18.9 +38.7 -13.7
Steps
(reduced) 		390
(0) 		618
(228) 		906
(126) 		1095
(315) 		1349
(179) 		1443
(273) 		1594
(34) 		1657
(97) 		1764
(204) 		1895
(335) 		1932
(372)

Subsets and supersets

Since 390 factors into 2 × 3 × 5 × 13, 390edo has subset edos 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, and 195. 780edo, which doubles it, gives a good correction to the harmonic 5.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3.7 118098/117649, 34451725707/34359738368 [⟨390 618 1095]] 0.0395 0.1685 5.48
2.3.7.11 118098/117649, 1362944/1361367, 235782657/234881024 [⟨390 618 1095 1349]] 0.0693 0.1548 5.03
2.3.7.11.13 729/728, 10648/10647, 16848/16807, 1574573/1572864 [⟨390 618 1095 1349 1443]] 0.0839 0.1415 4.60
2.3.7.11.13.17 729/728, 1089/1088, 16848/16807, 65637/65536, 95823/95744 [⟨390 618 1095 1349 1443 1594]] 0.0838 0.1292 4.20
Retrieved from "https://en.xen.wiki/index.php?title=390edo&oldid=131973"