33ed5: Difference between revisions

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'''33ED5''' is the [[Ed5|equal division of the 5th harmonic]] into 33 parts of 84.1004 [[cent|cents]] each (very close to the septimal chroma, [[21/20]]). It is related to the 128&270 temperament (''Septisemitonic'' temperament, named by [[User:Xenllium|Xenllium]]), which tempers out 102760448/102515625 and 200120949/200000000 in the 7-limit, supported by [[128edo|128]], [[142edo|142]], [[270edo|270]], 398, 412, 668, and 682 EDOs.
{{Infobox ET}}
'''33ED5''' is the [[Ed5|equal division of the 5th harmonic]] into 33 parts of 84.4337 [[cent|cents]] each (very close to the minor septimal semitone, [[21/20]], 84.4672 cents). It is related to the 128&270 temperament (''Septisemitonic'' temperament, named by [[User:Xenllium|Xenllium]]), which tempers out 102760448/102515625 and 200120949/200000000 in the 7-limit, supported by [[128edo|128]], [[142edo|142]], [[270edo|270]], [[398edo|398]], [[412edo|412]], [[668edo|668]], and [[682edo|682]] EDOs.


==Related regular temperaments==
==Related regular temperaments==
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POTE generator: ~21/20 = 84.4463
POTE generator: ~21/20 = 84.4463


Map: [<2 5 0 -1|, <0 -13 33 47|]
Mapping: [<2 5 0 -1|, <0 -13 33 47|]


EDOs: 128, 142, 270, 398, 412, 668, 682
EDOs: 128, 142, 270, 398, 412, 668, 682
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POTE generator: ~21/20 = 84.4430
POTE generator: ~21/20 = 84.4430


Map: [<2 5 0 -1 11|, <0 -13 33 47 -29|]
Mapping: [<2 5 0 -1 11|, <0 -13 33 47 -29|]


EDOs: 128, 142, 270, 398, 412, 668, 682
EDOs: 128, 142, 270, 398, 412, 668, 682
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Badness: 0.0583
Badness: 0.0583


[[Category:Ed5]]
== Intervals ==
[[Category:Edonoi]]
{{Interval table}}
 
== Harmonics ==
{{Harmonics in equal
| steps = 33
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 33
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}
 
{{todo|expand}}

Latest revision as of 19:21, 1 August 2025

← 32ed5 33ed5 34ed5 →
Prime factorization 3 × 11
Step size 84.4337 ¢ 
Octave 14\33ed5 (1182.07 ¢)
Twelfth 23\33ed5 (1941.98 ¢)
Consistency limit 2
Distinct consistency limit 2

33ED5 is the equal division of the 5th harmonic into 33 parts of 84.4337 cents each (very close to the minor septimal semitone, 21/20, 84.4672 cents). It is related to the 128&270 temperament (Septisemitonic temperament, named by Xenllium), which tempers out 102760448/102515625 and 200120949/200000000 in the 7-limit, supported by 128, 142, 270, 398, 412, 668, and 682 EDOs.

Related regular temperaments

7-limit 128&270

Commas: 102760448/102515625, 200120949/200000000

POTE generator: ~21/20 = 84.4463

Mapping: [<2 5 0 -1|, <0 -13 33 47|]

EDOs: 128, 142, 270, 398, 412, 668, 682

Badness: 0.1661

11-limit 128&270

Commas: 3025/3024, 131072/130977, 422576/421875

POTE generator: ~21/20 = 84.4430

Mapping: [<2 5 0 -1 11|, <0 -13 33 47 -29|]

EDOs: 128, 142, 270, 398, 412, 668, 682

Badness: 0.0583

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 84.4 20/19, 23/22
2 168.9 11/10
3 253.3 15/13, 22/19
4 337.7 17/14, 23/19, 28/23
5 422.2 14/11
6 506.6
7 591 7/5, 24/17
8 675.5 25/17, 28/19
9 759.9 17/11
10 844.3
11 928.8 12/7
12 1013.2
13 1097.6
14 1182.1
15 1266.5 25/12
16 1350.9 24/11
17 1435.4 23/10
18 1519.8 12/5
19 1604.2
20 1688.7
21 1773.1
22 1857.5
23 1942
24 2026.4
25 2110.8 17/5
26 2195.3 25/7
27 2279.7 26/7
28 2364.1
29 2448.6
30 2533 13/3
31 2617.4
32 2701.9 19/4
33 2786.3 5/1

Harmonics

Approximation of harmonics in 33ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -17.9 +40.0 -35.9 +0.0 +22.1 +8.5 +30.7 -4.4 -17.9 -14.1 +4.2
Relative (%) -21.2 +47.4 -42.5 +0.0 +26.2 +10.1 +36.3 -5.2 -21.2 -16.7 +4.9
Steps
(reduced) 		14
(14) 		23
(23) 		28
(28) 		33
(0) 		37
(4) 		40
(7) 		43
(10) 		45
(12) 		47
(14) 		49
(16) 		51
(18)
Approximation of harmonics in 33ed5
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +34.5 -9.4 +40.0 +12.7 -7.8 -22.3 -31.5 -35.9 -35.9 -32.0 -24.5
Relative (%) +40.8 -11.1 +47.4 +15.1 -9.2 -26.4 -37.3 -42.5 -42.5 -37.9 -29.0
Steps
(reduced) 		53
(20) 		54
(21) 		56
(23) 		57
(24) 		58
(25) 		59
(26) 		60
(27) 		61
(28) 		62
(29) 		63
(30) 		64
(31)
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