39ed5: Difference between revisions

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'''[[Ed5|Division of the 5th harmonic]] into 39 equal parts''' (39ed5) is a good [[hyperpyth]] tuning for the 5.7.9.11.13.19.23.31.41.43.47.51.53 subgroup. The step size about 71.4439 cents. It is similar to every fifth step of [[84edo]], but with the 5/1 rather than the 2/1 being just.
{{Infobox ET}}
'''[[Ed5|Division of the 5th harmonic]] into 39 equal parts''' (39ED5) is a good [[hyperpyth]] tuning. The step size about 71.4439 cents. It is compared to every fifth step of [[84edo|84EDO]], but with the 5/1 rather than the 2/1 being just.


{| class="wikitable"
== Intervals ==
{| class="wikitable mw-collapsible"
|+ Intervals of 39ed5
|-
|-
! | degree
! | degree
Line 15: Line 18:
| | 1
| | 1
| | 71.4439
| | 71.4439
| | [[25/24]], 24/23
| | [[25/24]], [[24/23]]
| |  
| |  
|-
|-
| | 2
| | 2
| | 142.8879
| | 142.8879
| | 38/35, 25/23
| | 38/35, [[25/23]]
| | +9.7 cents from 27/25
| |  
|-
|-
| | 3
| | 3
| | 214.3318
| | 214.3318
| | 26/23, 43/38, 60/53, [[17/15]]
| | [[26/23]], 43/38, 60/53, [[17/15]]
| |  
| |  
|-
|-
Line 36: Line 39:
| | 357.2197
| | 357.2197
| | [[16/13]]
| | [[16/13]]
| | -15.2 cents from 31/25
| |  
|-
|-
| | 6
| | 6
Line 46: Line 49:
| | 500.1076
| | 500.1076
| | [[4/3]]
| | [[4/3]]
| | +19.5 cents from 33/25
| |  
|-
|-
| | 8
| | 8
Line 66: Line 69:
| | 785.8834
| | 785.8834
| | [[11/7]]
| | [[11/7]]
| | +16.0 cents from 39/25
| |  
|-
|-
| | 12
| | 12
| | 857.3273
| | 857.3273
| | 41/25, 23/14
| | 41/25, [[23/14]]
| | +0.9 cents from 41/25
| |  
|-
|-
| | 13
| | 13
| | 928.7712
| | 928.7712
| | 70/41, 65/38
| | 70/41, 65/38
| | -10.1 cents from 43/25
| |  
|-
|-
| | 14
| | 14
Line 86: Line 89:
| | 1071.6591
| | 1071.6591
| | [[13/7]]
| | [[13/7]]
| | -21.2 cents from 47/25
| |  
|-
|-
| | 16
| | 16
| | 1143.1031
| | 1143.1031
| | 29/15, 31/16
| | 29/15, 31/16
| | -21.9 cents from 49/25
| |  
|-
|-
| | 17
| | 17
| | 1214.5470
| | 1214.5470
| | 125/62, 115/57, 105/52
| | 125/62, 115/57, 105/52
| | -19.7 cents from 51/25
| |  
|-
|-
| | 18
| | 18
| | 1285.9909
| | 1285.9909
| | [[21/20|21/10]]
| | [[21/20|21/10]]
| | -14.9 cents from 53/25
| |  
|-
|-
| | 19
| | 19
| | 1357.4349
| | 1357.4349
| | 35/16, 46/21, 125/57
| | [[35/32|35/16]], [[23/21|46/21]], 125/57
| | -7.6 cents from 11/5
| | -7.6 cents from 11/5
|-
|-
| | 20
| | 20
| | 1428.8788
| | 1428.8788
| | 57/25, 105/46, 16/7
| | 57/25, 105/46, [[16/7]]
| | +2.0 cents from 57/25
| |  
|-
|-
| | 21
| | 21
| | 1500.3228
| | 1500.3228
| | [[25/21|50/21]]
| | [[25/21|50/21]]
| | +13.8 cents from 59/25
| |  
|-
|-
| | 22
| | 22
| | 1571.7667
| | 1571.7667
| | 52/21, 57/23, 62/25
| | [[26/21|52/21]], 57/23, 62/25
| | -28.3 cents from 63/25
| |  
|-
|-
| | 23
| | 23
Line 131: Line 134:
| | 1714.6546
| | 1714.6546
| | 35/13
| | 35/13
| | +8.0 cents from 67/25
| |  
|-
|-
| | 25
| | 25
| | 1786.0985
| | 1786.0985
| | 115/41, 160/57
| | 115/41, 160/57
| | -21.0 cents from 71/25
| |  
|-
|-
| | 26
| | 26
| | 1857.5425
| | 1857.5425
| | [[19/13|38/13]], 41/14
| | [[19/13|38/13]], 41/14
| | +2.4 cents from 73/25
| |  
|-
|-
| | 27
| | 27
Line 195: Line 198:
| | 37
| | 37
| | 2643.4258
| | 2643.4258
| | 23/5
| | [[23/20|23/5]]
| | +1.5 cents from 23/5
| | +1.5 cents from 23/5
|-
|-
Line 209: Line 212:
|}
|}


[[Category:Ed5]]
== Harmonics ==
[[Category:Edonoi]]
{{Harmonics in equal
| steps = 39
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 39
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}
 
[[Category:Hyperpyth]]
{{todo|expand}}

Latest revision as of 19:21, 1 August 2025

← 38ed5 39ed5 40ed5 →
Prime factorization 3 × 13
Step size 71.4439 ¢ 
Octave 17\39ed5 (1214.55 ¢)
Twelfth 27\39ed5 (1928.99 ¢) (→ 9\13ed5)
Consistency limit 5
Distinct consistency limit 5

Division of the 5th harmonic into 39 equal parts (39ED5) is a good hyperpyth tuning. The step size about 71.4439 cents. It is compared to every fifth step of 84EDO, but with the 5/1 rather than the 2/1 being just.

Intervals

Intervals of 39ed5
degree cents value corresponding
JI intervals 		comments
0 0.0000 exact 1/1
1 71.4439 25/24, 24/23
2 142.8879 38/35, 25/23
3 214.3318 26/23, 43/38, 60/53, 17/15
4 285.7758 33/28, 46/39, 13/11
5 357.2197 16/13
6 428.6636 32/25, 41/32, 50/39
7 500.1076 4/3
8 571.5515 25/18 -11.0 cents from 7/5
9 642.9955 45/31
10 714.4394 80/53, 77/51, 68/45, 65/43
11 785.8834 11/7
12 857.3273 41/25, 23/14
13 928.7712 70/41, 65/38
14 1000.2152 57/32, 98/55, 41/23 -17.4 cents from 9/5
15 1071.6591 13/7
16 1143.1031 29/15, 31/16
17 1214.5470 125/62, 115/57, 105/52
18 1285.9909 21/10
19 1357.4349 35/16, 46/21, 125/57 -7.6 cents from 11/5
20 1428.8788 57/25, 105/46, 16/7
21 1500.3228 50/21
22 1571.7667 52/21, 57/23, 62/25
23 1643.2107 80/31, 75/29 -11.0 cents from 13/5
24 1714.6546 35/13
25 1786.0985 115/41, 160/57
26 1857.5425 38/13, 41/14
27 1928.9864 70/23 +27.0 cents from 3/1
28 2000.4304 35/11
29 2071.8743 43/13, 53/16
30 2143.3182 31/9 +24.7 cents from 17/5
31 2214.7622 18/5
32 2286.2061 15/4 -25.0 cents from 19/5
33 2357.6501 39/10
34 2429.0940 65/16
35 2500.5379 55/13 +16.1 cents from 21/5
36 2571.9819 75/17, 53/12, 190/43, 115/26
37 2643.4258 23/5 +1.5 cents from 23/5
38 2714.8698 24/5
39 2786.3137 exact 5/1 just major third plus two octaves

Harmonics

Approximation of harmonics in 39ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +14.5 +27.0 +29.1 +0.0 -29.9 -11.0 -27.8 -17.4 +14.5 -7.6 -15.3
Relative (%) +20.4 +37.8 +40.7 +0.0 -41.8 -15.3 -38.9 -24.3 +20.4 -10.6 -21.4
Steps
(reduced) 		17
(17) 		27
(27) 		34
(34) 		39
(0) 		43
(4) 		47
(8) 		50
(11) 		53
(14) 		56
(17) 		58
(19) 		60
(21)
Approximation of harmonics in 39ed5
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -11.0 +3.6 +27.0 -13.3 +24.7 -2.8 -25.0 +29.1 +16.1 +7.0 +1.5
Relative (%) -15.4 +5.0 +37.8 -18.6 +34.5 -4.0 -35.0 +40.7 +22.5 +9.8 +2.1
Steps
(reduced) 		62
(23) 		64
(25) 		66
(27) 		67
(28) 		69
(30) 		70
(31) 		71
(32) 		73
(34) 		74
(35) 		75
(36) 		76
(37)
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