13ed5/2: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''13ed5/2''' is the equal division of the [[5/2]] interval into 13 parts of 122.024 [[cent]]s each. It roughly corresponds to [[10edo]].
'''13ed5/2''' is the equal division of the [[5/2]] interval into 13 parts of 122.024 [[cent]]s each. It roughly corresponds to [[10edo]], and their [[patent val]]s match up until the 7-limit.
 
== Theory ==
== Theory ==
Like 10edo, 13ed5/2 tempers out [[50/49]] in the no-threes 7-limit, [[support]]ing 5/2-equivalent jubilismic temperament (named "jub" by [[User:CompactStar|CompactStar]]) with a generator of ~[[7/5]].
Like 10edo, 13ed5/2 tempers out [[50/49]] in the no-threes 7-limit, [[support]]ing 5/2-equivalent jubilic temperament with a generator of ~[[7/5]]. In this regard, it could be considered a "no-threes cousin" of [[12edo]] and [[13edt]], having the basic tuning for the octatonic scale of 5/2-equivalent jubilic ([[5L 3s (5/2-equivalent)|5L 3s⟨5/2⟩]]). It also tempers out [[56/55]] in the 11-limit and [[26/25]], [[52/49]] and [[65/64]] in the 13-limit.
{{Harmonics in equal|13|5|2}}
 
== Harmonics ==
{{Harmonics in equal
| steps = 13
| num = 5
| denom = 2
}}
{{Harmonics in equal
| steps = 13
| num = 5
| denom = 2
| start = 12
| collapsed = 1
}}
 
== Intervals ==
== Intervals ==
{|class="wikitable article-table"
{|class="wikitable article-table"
Line 9: Line 24:
!#
!#
!Cents
!Cents
!Approximate ratios*
!Jubilic[8] notation
|-
|-
|0
|0
|0.000
|0.000
|[[1/1]]
|J
|-
|-
|1
|1
|122.024
|122.024
|[[14/13]], [[35/32]]
|J&, K@
|-
|-
|2
|2
|244.048
|244.048
|[[8/7]], [[28/25]]
|K
|-
|-
|3
|3
|366.072
|366.072
|[[5/4]], [[16/13]], [[49/40]]
|L
|-
|-
|4
|4
|488.096
|488.096
|[[32/25]], [[64/49]]
|L&, M@
|-
|-
|5
|5
|610.120
|610.120
|[[7/5]], [[10/7]]
|M
|-
|-
|6
|6
|732.144
|732.144
|[[20/13]], [[25/16]], [[49/32]]
|M&, N@
|-
|-
|7
|7
|854.168
|854.168
|[[8/5]], [[13/8]]
|N
|-
|-
|8
|8
|976.192
|976.192
|[[7/4]], [[25/14]]
|O
|-
|-
|9
|9
|1098.216
|1098.216
|[[13/7]], [[64/35]]
|O&, P@
|-
|-
|10
|10
|1220.240
|1220.240
|[[2/1]], [[49/25]], 52/25
|P
|-
|-
|11
|11
|1342.264
|1342.264
|35/16
|Q
|-
|-
|12
|12
|1464.288
|1464.288
|[[16/7]]
|Q&, J@
|-
|-
|13
|13
|1586.312
|1586.312
|[[5/2]]
|J
|}
|}
<nowiki>*</nowiki> Based on treating 13ed5/2 as a 5/2.5.7.13 subgroup temperament

Latest revision as of 08:42, 4 October 2024

← 12ed5/2 13ed5/2 14ed5/2 →
Prime factorization 13 (prime)
Step size 122.024 ¢ 
Octave 10\13ed5/2 (1220.24 ¢)
(semiconvergent)
Twelfth 16\13ed5/2 (1952.39 ¢)
Consistency limit 5
Distinct consistency limit 5

13ed5/2 is the equal division of the 5/2 interval into 13 parts of 122.024 cents each. It roughly corresponds to 10edo, and their patent vals match up until the 7-limit.

Theory

Like 10edo, 13ed5/2 tempers out 50/49 in the no-threes 7-limit, supporting 5/2-equivalent jubilic temperament with a generator of ~7/5. In this regard, it could be considered a "no-threes cousin" of 12edo and 13edt, having the basic tuning for the octatonic scale of 5/2-equivalent jubilic (5L 3s⟨5/2⟩). It also tempers out 56/55 in the 11-limit and 26/25, 52/49 and 65/64 in the 13-limit.

Harmonics

Approximation of harmonics in 13ed5/2
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +20.2 +50.4 +40.5 +20.2 -51.4 +47.8 +60.7 -21.2 +40.5 -2.5 -31.1
Relative (%) +16.6 +41.3 +33.2 +16.6 -42.1 +39.2 +49.8 -17.3 +33.2 -2.0 -25.5
Steps
(reduced) 		10
(10) 		16
(3) 		20
(7) 		23
(10) 		25
(12) 		28
(2) 		30
(4) 		31
(5) 		33
(7) 		34
(8) 		35
(9)
Approximation of harmonics in 13ed5/2
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -47.7 -53.9 -51.4 -41.1 -24.0 -0.9 +27.5 +60.7 -23.7 +17.7 -59.2
Relative (%) -39.1 -44.2 -42.1 -33.6 -19.7 -0.8 +22.5 +49.8 -19.5 +14.5 -48.5
Steps
(reduced) 		36
(10) 		37
(11) 		38
(12) 		39
(0) 		40
(1) 		41
(2) 		42
(3) 		43
(4) 		43
(4) 		44
(5) 		44
(5)

Intervals

# Cents Approximate ratios* Jubilic[8] notation
0 0.000 1/1 J
1 122.024 14/13, 35/32 J&, K@
2 244.048 8/7, 28/25 K
3 366.072 5/4, 16/13, 49/40 L
4 488.096 32/25, 64/49 L&, M@
5 610.120 7/5, 10/7 M
6 732.144 20/13, 25/16, 49/32 M&, N@
7 854.168 8/5, 13/8 N
8 976.192 7/4, 25/14 O
9 1098.216 13/7, 64/35 O&, P@
10 1220.240 2/1, 49/25, 52/25 P
11 1342.264 35/16 Q
12 1464.288 16/7 Q&, J@
13 1586.312 5/2 J

* Based on treating 13ed5/2 as a 5/2.5.7.13 subgroup temperament

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