22ed5/2: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit

FloraC (talk | contribs)

4 user groups
27,941 edits
VisualWikitext
CompactStar (talk | contribs)
autopatrolled
6,215 edits
Standardize layout; reduce decimal places
 
Line 4: Line 4:
== Theory ==
== Theory ==
22ed5/2 has a vague similarity to [[17edo]] but they cannot be considered truly equivalent due to the severe octave stretch of 25.8 cents. It does not approximate harmonics well, but it approximates several half-integer (or /2 in [[primodality]] terms) ratios well, in particular it has decent approximations of [[7/2]], [[11/2]] and [[13/2]], and could be analyzed as a 5/2.7/2.11/2.13/2 subgroup temperament. It supports [[frosthog]] temperament, equating [[11/7]] to the square root of 5/2.
22ed5/2 has a vague similarity to [[17edo]] but they cannot be considered truly equivalent due to the severe octave stretch of 25.8 cents. It does not approximate harmonics well, but it approximates several half-integer (or /2 in [[primodality]] terms) ratios well, in particular it has decent approximations of [[7/2]], [[11/2]] and [[13/2]], and could be analyzed as a 5/2.7/2.11/2.13/2 subgroup temperament. It supports [[frosthog]] temperament, equating [[11/7]] to the square root of 5/2.
=== Harmonics ===
{{Harmonics in equal|22|5|2|columns=11}}
{{Harmonics in equal|22|5|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22ed5/2 (continued)}}


== Intervals ==
== Intervals ==
{|class="wikitable"
{|class="wikitable center-1 right-2"
|-
|-
!Steps
! #
!Cents
! Cents
!Approximate ratios*
! Approximate ratios*
|-
|-
|0
| 0
|0.000
| 0.0
|[[1/1]]
| [[1/1]]
|-
|-
|1
| 1
|72.105
| 72.1
|[[26/25]]
| [[26/25]]
|-
|-
|2
| 2
|144.210
| 144.2
|[[625/572]], [[676/625]]
| [[625/572]], [[676/625]]
|-
|-
|3
| 3
|216.316
| 216.3
|[[25/22]]
| [[25/22]]
|-
|-
|4
| 4
|288.421
| 288.4
|[[13/11]]
| [[13/11]]
|-
|-
|5
| 5
|360.525
| 360.5
|[[154/125]]
| [[154/125]]
|-
|-
|6
| 6
|432.631
| 432.6
|[[125/98]], [[625/484]], [[875/676]]
| [[125/98]], [[625/484]], [[875/676]]
|-
|-
|7
| 7
|504.736
| 504.7
|[[35/26]], [[121/91]]
| [[35/26]], [[121/91]]
|-
|-
|8
| 8
|576.842
| 576.8
|[[7/5]]
| [[7/5]]
|-
|-
|9
| 9
|648.947
| 648.9
|[[182/125]]
| [[182/125]]
|-
|-
|10
| 10
|721.052
| 721.1
|[[275/182]], [[875/572]]
| [[275/182]], [[875/572]]
|-
|-
|11
| 11
|793.157
| 793.2
|[[11/7]], [[35/22]]
| [[11/7]], [[35/22]]
|-
|-
|12
| 12
|865.262
| 865.3
|[[91/55]]
| [[91/55]]
|-
|-
|13
| 13
|937.367
| 937.4
|[[625/364]]
| [[625/364]]
|-
|-
|14
| 14
|1009.473
| 1009.5
|[[25/14]]
| [[25/14]]
|-
|-
|15
| 15
|1081.578
| 1081.6
|[[13/7]]
| [[13/7]]
|-
|-
|16
| 16
|1153.683
| 1153.7
|[[49/25]], [[242/125]], [[338/175]]
| [[49/25]], [[242/125]], [[338/175]]
|-
|-
|17
| 17
|1225.788
| 1225.8
|[[625/308]]
| [[625/308]]
|-
|-
|18
| 18
|1297.893
| 1297.9
|[[55/26]]
| [[55/26]]
|-
|-
|19
| 19
|1369.998
| 1370.0
|[[11/5]]
| [[11/5]]
|-
|-
|20
| 20
|1442.104
| 1442.1
|[[286/125]], [[3125/1352]]
| [[286/125]], [[3125/1352]]
|-
|-
|21
| 21
|1514.209
| 1514.2
|[[125/52]]
| [[125/52]]
|-
|-
|22
| 22
|1586.314
| 1586.3
|[[5/2]]
| [[5/2]]
|}
|}
<nowiki>*</nowiki> Based on treating 22ed5/2 as a 5/2.7/2.11/2.13/2 subgroup temperament.
<nowiki/>* Based on treating 22ed5/2 as a 5/2.7/2.11/2.13/2-subgroup temperament.
 
== Harmonics ==
{{Harmonics in equal
| steps = 22
| num = 5
| denom = 2
}}
{{Harmonics in equal
| steps = 22
| num = 5
| denom = 2
| start = 12
| collapsed = 1
}}

Latest revision as of 07:38, 11 November 2026

← 21ed5/2 22ed5/2 23ed5/2 →
Prime factorization 2 × 11
Step size 72.1052 ¢ 
Octave 17\22ed5/2 (1225.79 ¢)
Twelfth 26\22ed5/2 (1874.73 ¢) (→ 13\11ed5/2)
Consistency limit 2
Distinct consistency limit 2

22 equal divisions of 5/2 (abbreviated 22ed5/2) is a nonoctave tuning system that divides the interval of 5/2 into 22 equal parts of about 72.1 ¢ each. Each step represents a frequency ratio of (5/2)1/22, or the 22nd root of 5/2.

Theory

22ed5/2 has a vague similarity to 17edo but they cannot be considered truly equivalent due to the severe octave stretch of 25.8 cents. It does not approximate harmonics well, but it approximates several half-integer (or /2 in primodality terms) ratios well, in particular it has decent approximations of 7/2, 11/2 and 13/2, and could be analyzed as a 5/2.7/2.11/2.13/2 subgroup temperament. It supports frosthog temperament, equating 11/7 to the square root of 5/2.

Harmonics

Approximation of harmonics in 22ed5/2
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +25.8 -27.2 -20.5 +25.8 -1.4 +20.1 +5.3 +17.7 -20.5 +30.8 +24.4
Relative (%) +35.8 -37.8 -28.5 +35.8 -2.0 +27.9 +7.3 +24.5 -28.5 +42.7 +33.8
Steps
(reduced) 		17
(17) 		26
(4) 		33
(11) 		39
(17) 		43
(21) 		47
(3) 		50
(6) 		53
(9) 		55
(11) 		58
(14) 		60
(16)
Approximation of harmonics in 22ed5/2 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +30.0 -26.2 -1.4 +31.0 -1.8 -28.7 +22.0 +5.3 -7.1 -15.5 -20.4 -22.0
Relative (%) +41.6 -36.3 -2.0 +43.1 -2.5 -39.7 +30.4 +7.3 -9.9 -21.5 -28.3 -30.5
Steps
(reduced) 		62
(18) 		63
(19) 		65
(21) 		67
(1) 		68
(2) 		69
(3) 		71
(5) 		72
(6) 		73
(7) 		74
(8) 		75
(9) 		76
(10)

Intervals

# Cents Approximate ratios*
0 0.0 1/1
1 72.1 26/25
2 144.2 625/572, 676/625
3 216.3 25/22
4 288.4 13/11
5 360.5 154/125
6 432.6 125/98, 625/484, 875/676
7 504.7 35/26, 121/91
8 576.8 7/5
9 648.9 182/125
10 721.1 275/182, 875/572
11 793.2 11/7, 35/22
12 865.3 91/55
13 937.4 625/364
14 1009.5 25/14
15 1081.6 13/7
16 1153.7 49/25, 242/125, 338/175
17 1225.8 625/308
18 1297.9 55/26
19 1370.0 11/5
20 1442.1 286/125, 3125/1352
21 1514.2 125/52
22 1586.3 5/2

* Based on treating 22ed5/2 as a 5/2.7/2.11/2.13/2-subgroup temperament.

Retrieved from "https://en.xen.wiki/index.php?title=22ed5/2&oldid=242446"