390edo: Difference between revisions

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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..."
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|390}}
{{ED intro}}


== 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" |[[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|-103 65}}
! rowspan="2" | [[Comma list]]
|{{mapping|390 618}}
! rowspan="2" | [[Mapping]]
| 0.1314
! rowspan="2" | Optimal<br />8ve stretch (¢)
| 0.1314
! colspan="2" | Tuning error
| 4.27
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
|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
|}
|}

Latest revision as of 12:28, 21 February 2025

← 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

390 equal divisions of the octave (abbreviated 390edo or 390ed2), also called 390-tone equal temperament (390tet) or 390 equal temperament (390et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 390 equal parts of about 3.08 ¢ each. Each step represents a frequency ratio of 21/390, or the 390th root of 2.

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
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