4ed5/4: Difference between revisions

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'''4ED5/4''' is the [[Equal-step tuning|equal division]] of the [[5/4|just major third]] into four parts of 96.5784 [[cent|cents]] each, corresponding to 12.4251 [[edo]]. It is related to the temperament which tempers out 48828125/48771072 and 67108864/66976875 in the 7-limit, which is supported by [[87edo]], [[323edo]], [[410edo]], and [[733edo]] among others.
'''4ED5/4''' is the [[Equal-step tuning|equal division]] of the [[5/4|just major third]] into four parts of 96.5784 [[cent|cents]] each, corresponding to 12.4251 [[EDO]]. It is related to the temperament which tempers out 48828125/48771072 and 67108864/66976875 in the 7-limit (''hemiluna'' temperament), which is supported by [[87edo|87EDO]], [[323edo|323EDO]], [[410edo|410EDO]], and [[733edo|733EDO]] among others.


==Intervals==
== Intervals ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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|}
|}


==Related regular temperament==
== Related regular temperament ==
===7-limit 87&323===
4ED5/4 tuning is closely related to the ''hemiluna'' temperament (87&323), which tempers out 48828125/48771072 (''neptunisma'', laruruleyo) and 67108864/66976875 (''decovulture'' comma, sasabirugugu) in the 7-limit. It splits a hemithird interval into two equal parts.
Commas: 48828125/48771072, 67108864/66976875


POTE generator: ~200/189 = 96.591
; [[Luna family #Hemiluna|Hemiluna]] (87 & 323)
 
'''7-limit'''<br>
Map: [&lt;1 4 2 -5|, &lt;0 -30 4 97|]
Comma list: 48828125/48771072, 67108864/66976875<br>
 
Mapping: [{{val|1 4 2 -5}}, {{val|0 -30 4 97}}]<br>
EDOs: 87, 323, 410, 733
POTE generator: ~200/189 = 96.591<br>
 
Vals: 87, 236, 323, 410, 733<br>
Badness: 0.1793
Badness: 0.179304<br><br>
 
'''11-limit'''<br>
===11-limit 87&amp;323===
Comma list: 5632/5625, 14641/14580, 131072/130977<br>
Commas: 5632/5625, 14641/14580, 131072/130977
Mapping: [{{val|1 4 2 -5 7}}, {{val|0 -30 4 97 -44}}]<br>
 
POTE generator: ~200/189 = 96.587<br>
POTE generator: ~200/189 = 96.587
Vals: 87, 236, 323, 410<br>
 
Badness: 0.100077<br><br>
Map: [&lt;1 4 2 -5 7|, &lt;0 -30 4 97 -44|]
'''13-limit'''<br>
 
Comma list: 676/675, 1001/1000, 4096/4095, 14641/14580<br>
EDOs: 87, 323, 410, 733e
Mapping: [{{val|1 4 2 -5 7 7}}, {{val|0 -30 4 97 -44 -41}}]<br>
 
POTE generator: ~143/135 = 96.586<br>
Badness: 0.1001
Vals: 87, 236, 323, 410<br>
 
Badness: 0.046151<br><br>
===13-limit 87&amp;323===
Commas: 676/675, 1001/1000, 4096/4095, 14641/14580
 
POTE generator: ~143/135 = 96.586
 
Map: [&lt;1 4 2 -5 7 7|, &lt;0 -30 4 97 -44 -41|]
 
EDOs: 87, 323, 410, 733e
 
Badness: 0.0462


[[Category:Major third]]
[[Category:Major third]]
[[Category:Equal-step tuning]]
[[Category:Equal-step tuning]]
[[Category:Edonoi]]
[[Category:Edonoi]]

Revision as of 22:40, 14 September 2021

4ED5/4 is the equal division of the just major third into four parts of 96.5784 cents each, corresponding to 12.4251 EDO. It is related to the temperament which tempers out 48828125/48771072 and 67108864/66976875 in the 7-limit (hemiluna temperament), which is supported by 87EDO, 323EDO, 410EDO, and 733EDO among others.

Intervals

degree cents value ratio
0 0.0000 1/1
1 96.5784 (5/4)1/4
2 193.1569 (5/4)1/2
3 289.7353 (5/4)3/4
4 386.3137 5/4
5 482.8921 (5/4)5/4
6 579.4706 (5/4)3/2
7 676.0490 (5/4)7/4
8 772.6274 (5/4)2 = 25/16
9 869.2059 (5/4)9/4
10 965.7843 (5/4)5/2
11 1062.3627 (5/4)11/4
12 1158.9411 (5/4)3 = 125/64
13 1255.5196 (5/4)13/4
14 1352.0980 (5/4)7/2
15 1448.6764 (5/4)15/4
16 1545.2549 (5/4)4 = 625/256
17 1641.8333 (5/4)17/4
18 1738.4117 (5/4)9/2
19 1834.9901 (5/4)19/4
20 1931.5686 (5/4)5 = 3125/1024
21 2028.1470 (5/4)21/4
22 2124.7254 (5/4)11/2
23 2221.3039 (5/4)23/4
24 2317.8823 (5/4)6 = 15625/4096
25 2414.4607 (5/4)25/4
26 2511.0391 (5/4)13/2
27 2607.6176 (5/4)27/4
28 2704.1960 (5/4)7 = 78125/16384
29 2800.7744 (5/4)29/4
30 2897.3529 (5/4)15/2
31 2993.9313 (5/4)31/4
32 3090.5097 (5/4)8 = 390625/65536
33 3187.0881 (5/4)33/4
34 3283.6666 (5/4)17/2
35 3380.2450 (5/4)35/4
36 3476.8234 (5/4)9 = 1953125/262144
37 3573.4019 (5/4)37/4
38 3669.9803 (5/4)19/2
39 3766.5587 (5/4)39/4
40 3863.1371 (5/4)10 = 9765625/1048576
41 3959.7156 (5/4)41/4
42 4056.2940 (5/4)21/2
43 4152.8724 (5/4)43/4
44 4249.4509 (5/4)11 = 48828125/4194304
45 4346.0293 (5/4)45/4
46 4442.6077 (5/4)23/2
47 4539.1861 (5/4)47/4
48 4635.7646 (5/4)12 = 244140625/16777216
49 4732.3430 (5/4)49/4
50 4828.9214 (5/4)25/2
51 4925.4999 (5/4)51/4
52 5022.0783 (5/4)13 = 1220703125/67108864

Related regular temperament

4ED5/4 tuning is closely related to the hemiluna temperament (87&323), which tempers out 48828125/48771072 (neptunisma, laruruleyo) and 67108864/66976875 (decovulture comma, sasabirugugu) in the 7-limit. It splits a hemithird interval into two equal parts.

Hemiluna (87 & 323)

7-limit
Comma list: 48828125/48771072, 67108864/66976875
Mapping: [⟨1 4 2 -5], ⟨0 -30 4 97]]
POTE generator: ~200/189 = 96.591
Vals: 87, 236, 323, 410, 733
Badness: 0.179304

11-limit
Comma list: 5632/5625, 14641/14580, 131072/130977
Mapping: [⟨1 4 2 -5 7], ⟨0 -30 4 97 -44]]
POTE generator: ~200/189 = 96.587
Vals: 87, 236, 323, 410
Badness: 0.100077

13-limit
Comma list: 676/675, 1001/1000, 4096/4095, 14641/14580
Mapping: [⟨1 4 2 -5 7 7], ⟨0 -30 4 97 -44 -41]]
POTE generator: ~143/135 = 96.586
Vals: 87, 236, 323, 410
Badness: 0.046151

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