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'''29EDT''' is the [[Edt|equal division of the third harmonic]] into 29 parts of 65.5847 [[cent|cents]] each, corresponding to 18.2970 [[edo]]. It is related to the regular temperament supported by [[183edo]], [[311edo]], and [[494edo]].
{{Infobox ET}}
'''29EDT''' is the [[Edt|equal division of the third harmonic]] into 29 parts of 65.5847 [[cent|cents]] each, corresponding to 18.2970 [[edo]]. It is related to the [[Metric microtemperaments #Luminal|luminal temperament]].


{| class="wikitable"
While not possessing good approximations to most integer harmonics, it does have a few very good rational intervals: notably [[27/26]], [[17/10]], [[45/32]], and [[11/6]].
 
==Intervals==
{| class="wikitable right-2 right-3"
|-
|-
! | degree
! steps
! | cents value
! [[cent]]s
! | corresponding <br>JI intervals
! [[hekt]]s
! | comments
! corresponding <br>JI intervals
! comments
|-
|-
| | 0
! colspan="3" | 0
| | 0.0000
| [[1/1]]
| | '''exact [[1/1]]'''
|  
| |  
|-
|-
| | 1
| 1
| | 65.5847
| 65.5847
| | [[27/26]]
| 44.8276
| |  
| [[27/26]]
|  
|-
|-
| | 2
| 2
| | 131.1693
| 131.1693
| |  
| 89.6552
| |  
| 14/13, 27/25, 55/51
|  
|-
|-
| | 3
| 3
| | 196.7540
| 196.7540
| | [[28/25]]
| 134.4828
| |
| 9/8, [[28/25]]
| pseudo-[[10/9]]
|-
|-
| | 4
| 4
| | 262.3386
| 262.3386
| |  
| 179.3103
| |  
| 7/6
|  
|-
|-
| | 5
| 5
| | 327.9233
| 327.9233
| |  
| 224.1379
| |
|  
| pseudo-[[6/5]]
|-
|-
| | 6
| 6
| | 393.5079
| 393.5079
| | 64/51
| 268.9655
| |
| 64/51
| pseudo-[[5/4]]
|-
|-
| | 7
| 7
| | 459.0926
| 459.0926
| |  
| 313.7931
| |  
| 13/10
|  
|-
|-
| | 8
| 8
| | 524.6772
| 524.6772
| | 65/48
| 358.6206
| |  
| 65/48, 27/20
|  
|-
|-
| | 9
| 9
| | 590.2619
| 590.2619
| | [[45/32]]
| 403.4483
| |  
| [[45/32]]
|  
|-
|-
| | 10
| 10
| | 655.8466
| 655.8466
| |  
| 448.2759
| |  
| 16/11
|  
|-
|-
| | 11
| 11
| | 721.4312
| 721.4312
| |  
| 493.1034
| |
|  
| pseudo-[[3/2]]
|-
|-
| | 12
| 12
| | 787.0159
| 787.0159
| | 63/40, 52/33
| 537.9310
| |
| 63/40, 52/33
| pseudo-[[8/5]]
|-
|-
| | 13
| 13
| | 852.6005
| 852.6005
| | [[18/11]]
| 582.7586
| |
| [[18/11]]
| flat pseudo-[[5/3]]
|-
|-
| | 14
| 14
| | 918.1852
| 918.1852
| |  
| 627.5862
| |
| 17/10
| sharp pseudo-[[5/3]]
|-
|-
| | 15
| 15
| | 983.7698
| 983.7698
| |  
| 672.4138
| |
| 30/17
| flat pseudo-[[9/5]]
|-
|-
| | 16
| 16
| | 1049.3545
| 1049.3545
| | [[11/6]]
| 717.2414
| |
| [[11/6]]
| sharp pseudo-[[9/5]]
|-
|-
| | 17
| 17
| | 1114.9391
| 1114.9391
| | 99/52, 40/21
| 772.0690
| |
| 99/52, 40/21
| pseudo-[[15/8]]
|-
|-
| | 18
| 18
| | 1180.5238
| 1180.5238
| |  
| 806.8966
| |
|  
| pseudooctave
|-
|-
| | 19
| 19
| | 1246.1084
| 1246.1084
| |  
| 851.7241
| |  
| 33/16
|  
|-
|-
| | 20
| 20
| | 1311.6931
| 1311.6931
| | [[16/15|32/15]]
| 896.5517
| |  
| [[16/15|32/15]]
|  
|-
|-
| | 21
| 21
| | 1377.2778
| 1377.2778
| | 144/65
| 941.3794
| |  
| 144/65, 20/9
|  
|-
|-
| | 22
| 22
| | 1442.8624
| 1442.8624
| |  
| 986.2069
| |
| 30/13
| pseudo-7/3 (7/6 plus pseudooctave)
|-
|-
| | 23
| 23
| | 1508.4471
| 1508.4471
| | 153/64
| 1031.0345
| |
| 153/64
| pseudo-12/5
|-
|-
| | 24
| 24
| | 1574.0317
| 1574.0317
| |  
| 1075.8621
| |
|  
| pseudo-5/2
|-
|-
| | 25
| 25
| | 1639.6164
| 1639.6164
| |  
| 1120.6897
| |  
| 18/7
|  
|-
|-
| | 26
| 26
| | 1705.2010
| 1705.2010
| | 75/28
| 1165.5172
| |
| 8/3, 75/28
| pseudo-27/10
|-
|-
| | 27
| 27
| | 1770.7857
| 1770.7857
| |  
| 1210.3448
| |  
| 39/14, 25/9, 153/55
|  
|-
|-
| | 28
| 28
| | 1836.3703
| 1836.3703
| | [[13/9|26/9]]
| 1255.1724
| |  
| [[13/9|26/9]]
|  
|-
|-
| | 29
| 29
| | 1901.9550
| 1901.9550
| | '''exact [[3/1]]'''
| 1300.0000
| | [[3/2|just perfect fifth]] plus an octave
| [[3/1]]
| [[3/2|just perfect fifth]] plus an octave
|}
|}


==Related regular temperament==
==Harmonics==
===5-limit 183&amp;311===
{{Harmonics in equal
Comma: |145 -49 -29&gt;
| steps = 29
 
| num = 3
POTE generator: 65.585
| denom = 1
 
| intervals = integer
Map: [&lt;1 0 5|, &lt;0 29 -49|]
}}
 
{{Harmonics in equal
EDOs: 183, 311, 494, 677, 860, 1171, 1665, 1848, 2159, 2836, 3019, 4007
| steps = 29
 
| num = 3
Badness: 0.1949
| denom = 1
 
| start = 12
===7-limit 183&amp;311===
| collapsed = 1
Commas: 703125/702464, 26843545600/26795786661
| intervals = integer
 
}}
POTE generator: ~8505/8192 = 65.588
 
Map: [&lt;1 0 5 8|, &lt;0 29 -49 -95|]
 
EDOs: 128, 183, 311, 494, 677, 805, 1116, 1299
 
Badness: 0.1348
 
===11-limit 183&amp;311===
Commas: 3025/3024, 131072/130977, 703125/702464
 
POTE generator: ~80/77 = 65.588
 
Map: [&lt;1 0 5 8 1|, &lt;0 29 -49 -95 45|]
 
EDOs: 128, 183, 311, 494, 677, 805, 1299
 
Badness: 0.0328
 
===13-limit 183&amp;311===
Commas: 2080/2079, 3025/3024, 4096/4095, 31250/31213
 
POTE generator: ~27/26 = 65.588
 
Map: [&lt;1 0 5 8 1 -1|, &lt;0 29 -49 -95 45 86|]
 
EDOs: 128, 183, 311, 494, 677, 805, 1299
 
Badness: 0.0168
 
===17-limit 183&amp;311===
Commas: 1156/1155, 1275/1274, 2080/2079, 2431/2430, 4096/4095
 
POTE generator: ~27/26 = 65.588
 
Map: [&lt;1 0 5 8 1 -1 6|, &lt;0 29 -49 -95 45 86 -35|]
 
EDOs: 128, 183, 311, 494, 677, 805
 
Badness: 0.0147


[[Category:Edt]]
{{todo|expand}}
[[Category:Edonoi]]

Latest revision as of 19:21, 1 August 2025

← 28edt 29edt 30edt →
Prime factorization 29 (prime)
Step size 65.5847 ¢ 
Octave 18\29edt (1180.52 ¢)
Consistency limit 3
Distinct consistency limit 3

29EDT is the equal division of the third harmonic into 29 parts of 65.5847 cents each, corresponding to 18.2970 edo. It is related to the luminal temperament.

While not possessing good approximations to most integer harmonics, it does have a few very good rational intervals: notably 27/26, 17/10, 45/32, and 11/6.

Intervals

steps cents hekts corresponding
JI intervals 		comments
0 1/1
1 65.5847 44.8276 27/26
2 131.1693 89.6552 14/13, 27/25, 55/51
3 196.7540 134.4828 9/8, 28/25 pseudo-10/9
4 262.3386 179.3103 7/6
5 327.9233 224.1379 pseudo-6/5
6 393.5079 268.9655 64/51 pseudo-5/4
7 459.0926 313.7931 13/10
8 524.6772 358.6206 65/48, 27/20
9 590.2619 403.4483 45/32
10 655.8466 448.2759 16/11
11 721.4312 493.1034 pseudo-3/2
12 787.0159 537.9310 63/40, 52/33 pseudo-8/5
13 852.6005 582.7586 18/11 flat pseudo-5/3
14 918.1852 627.5862 17/10 sharp pseudo-5/3
15 983.7698 672.4138 30/17 flat pseudo-9/5
16 1049.3545 717.2414 11/6 sharp pseudo-9/5
17 1114.9391 772.0690 99/52, 40/21 pseudo-15/8
18 1180.5238 806.8966 pseudooctave
19 1246.1084 851.7241 33/16
20 1311.6931 896.5517 32/15
21 1377.2778 941.3794 144/65, 20/9
22 1442.8624 986.2069 30/13 pseudo-7/3 (7/6 plus pseudooctave)
23 1508.4471 1031.0345 153/64 pseudo-12/5
24 1574.0317 1075.8621 pseudo-5/2
25 1639.6164 1120.6897 18/7
26 1705.2010 1165.5172 8/3, 75/28 pseudo-27/10
27 1770.7857 1210.3448 39/14, 25/9, 153/55
28 1836.3703 1255.1724 26/9
29 1901.9550 1300.0000 3/1 just perfect fifth plus an octave

Harmonics

Approximation of harmonics in 29edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -19.5 +0.0 +26.6 -31.8 -19.5 -24.0 +7.2 +0.0 +14.4 -19.5 +26.6
Relative (%) -29.7 +0.0 +40.6 -48.4 -29.7 -36.6 +10.9 +0.0 +21.9 -29.7 +40.6
Steps
(reduced) 		18
(18) 		29
(0) 		37
(8) 		42
(13) 		47
(18) 		51
(22) 		55
(26) 		58
(0) 		61
(3) 		63
(5) 		66
(8)
Approximation of harmonics in 29edt
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +19.2 +22.1 -31.8 -12.3 +13.9 -19.5 +18.1 -5.1 -24.0 +26.6 +15.3
Relative (%) +29.3 +33.7 -48.4 -18.8 +21.2 -29.7 +27.6 -7.8 -36.6 +40.6 +23.3
Steps
(reduced) 		68
(10) 		70
(12) 		71
(13) 		73
(15) 		75
(17) 		76
(18) 		78
(20) 		79
(21) 		80
(22) 		82
(24) 		83
(25)
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