21ed5/2: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
CompactStar (talk | contribs)
5,020 edits
No edit summary
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
18,215 edits
 
(2 intermediate revisions by one other user not shown)
Line 4: Line 4:
From a no-threes point of view, 21ed5/2 tempers out [[50/49]] in the 7-limit (being a jubilic system similar to [[13ed5/2]]), [[625/616]] and [[176/175]] in the 11-limit, and 143/140,  715/686 and [[847/845]] in the 13-limit. It is not particularly excellent as a no-threes system with the 5/4 and 7/4 being noticeably off, but can work for 5/2-equivalent jubilic.
From a no-threes point of view, 21ed5/2 tempers out [[50/49]] in the 7-limit (being a jubilic system similar to [[13ed5/2]]), [[625/616]] and [[176/175]] in the 11-limit, and 143/140,  715/686 and [[847/845]] in the 13-limit. It is not particularly excellent as a no-threes system with the 5/4 and 7/4 being noticeably off, but can work for 5/2-equivalent jubilic.


=== Harmonics ===
== Harmonics ==
{{Harmonics in equal|21|5|2}}
{{Harmonics in equal
| steps = 21
| num = 5
| denom = 2
}}
{{Harmonics in equal
| steps = 21
| num = 5
| denom = 2
| start = 12
| collapsed = 1
}}


== Interval table ==
== Interval table ==
Line 31: Line 42:
|-
|-
|3
|3
|
|226.616
|
|K
|[[8/7]], [[28/25]]
|[[8/7]], [[28/25]]
|-
|-
|4
|4
|
|302.155
|
|K&, L@
|[[13/11]], [[77/64]]
|[[13/11]], [[77/64]]
|-
|-
|5
|5
|
|377.694
|
|L
|[[5/4]], [[11/9]], [[16/13]], [[49/40]]
|[[5/4]], [[11/9]], [[16/13]], [[49/40]]
|-
|-
|6
|6
|
|453.233
|
|L&
|[[13/10]], [[32/25]]
|[[13/10]], [[32/25]]
|-
|-
|7
|7
|
|528.771
|
|M@
|[[11/8]], [[35/26]]
|[[11/8]], [[35/26]]
|-
|-
|8
|8
|
|604.310
|
|M
|[[7/5]], [[10/7]]
|[[7/5]], [[10/7]]
|-
|-
|9
|9
|
|679.849
|
|M&
|[[16/11]], [[52/35]]
|[[16/11]], [[52/35]]
|-
|-
|10
|10
|
|755.388
|
|N@
|[[11/7]], [[20/13]], [[25/16]], [[49/32]]
|[[11/7]], [[20/13]], [[25/16]], [[49/32]]
|-
|-
|11
|11
|
|830.926
|
|N
|[[8/5]], [[13/8]]
|[[8/5]], [[13/8]]
|-
|-
|12
|12
|
|906.465
|
|N&, O@
|[[22/13]], [[55/32]]
|[[22/13]], [[55/32]]
|-
|-
|13
|13
|
|982.004
|
|O
|[[7/4]], [[25/14]]
|[[7/4]], [[25/14]]
|-
|-
|14
|14
|
|1057.543
|
|O&
|[[13/7]], [[20/11]]
|[[13/7]], [[20/11]]
|-
|-
|15
|15
|
|1133.081
|
|P@
|[[25/13]]
|[[25/13]]
|-
|-
|16
|16
|
|1208.620
|
|P
|[[2/1]]
|[[2/1]]
|-
|-
|17
|17
|
|1284.159
|
|P&, Q@
|[[52/25]]
|[[52/25]]
|-
|-
|18
|18
|
|1359.698
|
|Q
|[[11/5]]
|[[11/5]]
|-
|-
|19
|19
|
|1435.236
|
|Q&
|[[16/7]]
|[[16/7]]
|-
|-
|20
|20
|
|1510.775
|
|J@
|[[26/11]]
|[[26/11]]
|-
|-
|21
|21
|
|1586.314
|
|J
|[[5/2]]
|[[5/2]]
|}
|}


<nowiki>*</nowiki> Based on treating 21ed5/2 as a no-threes 13-limit temperament
<nowiki>*</nowiki> Based on treating 21ed5/2 as a no-threes 13-limit temperament

Latest revision as of 08:36, 4 October 2024

← 20ed5/2 21ed5/2 22ed5/2 →
Prime factorization 3 × 7
Step size 75.5387 ¢ 
Octave 16\21ed5/2 (1208.62 ¢)
(semiconvergent)
Twelfth 25\21ed5/2 (1888.47 ¢)
(semiconvergent)
Consistency limit 6
Distinct consistency limit 6

21ed5/2 is the equal division of the 5/2 interval into 21 parts of approximately 75.539 cents each. It roughly corresponds to 16edo.

Theory

From a no-threes point of view, 21ed5/2 tempers out 50/49 in the 7-limit (being a jubilic system similar to 13ed5/2), 625/616 and 176/175 in the 11-limit, and 143/140, 715/686 and 847/845 in the 13-limit. It is not particularly excellent as a no-threes system with the 5/4 and 7/4 being noticeably off, but can work for 5/2-equivalent jubilic.

Harmonics

Approximation of harmonics in 21ed5/2
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +8.6 -13.5 +17.2 +8.6 -4.9 +30.4 +25.9 -27.0 +17.2 +3.3 +3.8
Relative (%) +11.4 -17.9 +22.8 +11.4 -6.4 +40.3 +34.2 -35.7 +22.8 +4.4 +5.0
Steps
(reduced) 		16
(16) 		25
(4) 		32
(11) 		37
(16) 		41
(20) 		45
(3) 		48
(6) 		50
(8) 		53
(11) 		55
(13) 		57
(15)
Approximation of harmonics in 21ed5/2
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +16.3 -36.5 -4.9 +34.5 +5.1 -18.4 -36.4 +25.9 +16.9 +11.9 +10.5
Relative (%) +21.5 -48.3 -6.4 +45.6 +6.7 -24.3 -48.2 +34.2 +22.4 +15.8 +13.9
Steps
(reduced) 		59
(17) 		60
(18) 		62
(20) 		64
(1) 		65
(2) 		66
(3) 		67
(4) 		69
(6) 		70
(7) 		71
(8) 		72
(9)

Interval table

Steps Cents Jubilic[8] notation Approximate ratios*
0 0.000 J 1/1
1 75.539 J& 26/25
2 151.078 K@ 35/32
3 226.616 K 8/7, 28/25
4 302.155 K&, L@ 13/11, 77/64
5 377.694 L 5/4, 11/9, 16/13, 49/40
6 453.233 L& 13/10, 32/25
7 528.771 M@ 11/8, 35/26
8 604.310 M 7/5, 10/7
9 679.849 M& 16/11, 52/35
10 755.388 N@ 11/7, 20/13, 25/16, 49/32
11 830.926 N 8/5, 13/8
12 906.465 N&, O@ 22/13, 55/32
13 982.004 O 7/4, 25/14
14 1057.543 O& 13/7, 20/11
15 1133.081 P@ 25/13
16 1208.620 P 2/1
17 1284.159 P&, Q@ 52/25
18 1359.698 Q 11/5
19 1435.236 Q& 16/7
20 1510.775 J@ 26/11
21 1586.314 J 5/2

* Based on treating 21ed5/2 as a no-threes 13-limit temperament

Retrieved from "https://en.xen.wiki/index.php?title=21ed5/2&oldid=157964"