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'''[[EDF|Division of the just perfect fifth]] into 34 equal parts''' (34EDF) is related to [[58edo|58 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 2.5474 cents compressed and the step size is about 20.6457 cents (corresponding to 58.1234 edo). The patent val has a generally flat tendency for harmonics up to 16, with the exception for 5. Unlike 58edo, it is only consistent up to the [[15-odd-limit|15-integer-limit]], with discrepancy for the 16th harmonic (four octaves).
{{Infobox ET}}
{{ED intro}}


Lookalikes: [[58edo]], [[92edt]]
== Theory ==
==Intervals==
34edf corresponds to 58.1234…edo. It is related to [[58edo]], but with the [[3/2]] rather than the [[2/1]] being [[just]]. The octave is [[stretched and compressed tuning|compressed]] by about 2.5474 [[cents]].
{| class="wikitable"
 
The [[patent val]] has a generally flat tendency for [[harmonic]]s up to [[16/1|16]] (four octaves), with the exception for [[5/1|5]]. Unlike 58edo, it is only consistent up to the [[integer limit|15-integer-limit]], with discrepancy for the 16th harmonic.
 
=== Harmonics ===
{{Harmonics in equal|34|3|2|intervals=integer}}
{{Harmonics in equal|34|3|2|intervals=integer|columns=12|start=12|collapsed=1|title=Approximation of harmonics in 34edf (continued)}}
 
=== Subsets and supersets ===
Since 34 factors into primes as {{nowrap| 2 × 17 }}, 34edf contains [[2edf]] and [[17edf]] as subset edfs.
 
== Intervals ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! Approximate ratios
|-
| 0
| 0.0
| 1/1
|-
| 1
| 20.6
| ''56/55'', 64/63, 81/80, 91/90, 105/104
|-
|-
! |Degree
| 2
! |Cents
| 41.3
! |Approx. ratios
| 36/35, 40/39, 45/44, 49/48, 50/49, 55/54
|-
|-
| style="text-align:center;" |0
| 3
| style="text-align:center;" |0
| 61.9
| style="text-align:center;" |1/1
| 26/25, 27/26, 28/27, 33/32
|-
|-
|1
| 4
|20.6457
| 82.6
|56/55, 64/63, 81/80, 128/125
| 21/20, 22/21, ''25/24''
|-
|-
| style="text-align:center;" |2
| 5
| style="text-align:center;" |41.2915
| 103.2
| style="text-align:center;" |36/35, 49/48, 50/49, 55/54
| 16/15, 17/16, 18/17
|-
|-
|3
| 6
|61.9372
| 123.9
|25/24, 26/25, 27/26, 28/27, 33/32
| 14/13, 15/14
|-
|-
| style="text-align:center;" |4
| 7·
| style="text-align:center;" |82.5829
| 144.5
| style="text-align:center;" |21/20, 22/21
| 12/11, 13/12
|-
|-
|5
| 8
|103.2287
| 165.2
|16/15, 17/16, 18/17
| 11/10
|-
|-
| style="text-align:center;" |6
| 9
| style="text-align:center;" |123.8744
| 185.8
| style="text-align:center;" |15/14, 14/13
| 10/9
|-
|-
|7·
| 10
|144.52015
| 206.5
|12/11, 13/12
| 9/8
|-
|-
| style="text-align:center;" |8
| 11
| style="text-align:center;" |165.1659
| 227.1
| style="text-align:center;" |11/10
| 8/7
|-
|-
|9
| 12·
|185.8116
| 248.7
|10/9
| 15/13
|-
|-
| style="text-align:center;" |10
| 13
| style="text-align:center;" |206.45735
| 268.4
| style="text-align:center;" |9/8
| 7/6
|-
|-
|11
| 14
|227.1031
| 289.0
|8/7
| 13/11, 20/17
|-
|-
| style="text-align:center;" |12·
| 15
| style="text-align:center;" |248.7488
| 309.7
| style="text-align:center;" |15/13
| 6/5
|-
|-
|13
| 16
|268.3946
| 330.3
|7/6
| 17/14, 40/33
|-
|-
| style="text-align:center;" |14
| 17·
| style="text-align:center;" |289.0403
| 351.0
| style="text-align:center;" |13/11, 20/17
| 11/9, 16/13
|-
|-
|15
| 18
|309.686
| 371.6
|6/5
| 21/17, 26/21
|-
|-
| style="text-align:center;" |16
| 19
| style="text-align:center;" |330.3318
| 392.3
| style="text-align:center;" |17/14
| 5/4
|-
|-
|17·
| 20
|350.9775
| 412.9
|11/9, 16/13
| 14/11
|-
|-
| style="text-align:center;" |18
| 21
| style="text-align:center;" |371.6232
| 433.6
| style="text-align:center;" |21/17
| 9/7
|-
|-
|19
| 22·
|392.269
| 455.2
|5/4
| 13/10, 17/13, 22/17
|-
|-
| style="text-align:center;" |20
| 23
| style="text-align:center;" |412.9147
| 474.9
| style="text-align:center;" |14/11
| 21/16
|-
|-
|21
| 24
|433.5604
| 495.5
|9/7
| 4/3
|-
|-
| style="text-align:center;" |22·
| 25
| style="text-align:center;" |455.2062
| 516.1
| style="text-align:center;" |13/10, 17/13, 22/17
| 27/20
|-
|-
|23
| 26
|474.8519
| 536.8
|21/16
| 15/11
|-
|-
| style="text-align:center;" |24
| 27
| style="text-align:center;" |495.49765
| 557.4
| style="text-align:center;" |4/3
| 11/8, 18/13
|-
|-
|25
| 28
|516.1434
| 578.1
|27/20
| 7/5
|-
|-
| style="text-align:center;" |26
| 29
| style="text-align:center;" |536.7891
| 598.7
| style="text-align:center;" |15/11
| 17/12, 24/17
|-
|-
|27
| 30
|557.43485
| 619.4
|11/8, 18/13
| 10/7
|-
|-
| style="text-align:center;" |28
| 31
| style="text-align:center;" |578.0806
| 640.0
| style="text-align:center;" |7/5
| 13/9, 16/11
|-
|-
|29
| 32
|598.7263
| 660.7
|17/12, 24/17
| 22/15
|-
|-
| style="text-align:center;" |30
| 33
| style="text-align:center;" |619.3721
| 681.3
| style="text-align:center;" |10/7
| 40/27
|-
|-
|31
| 34
|640.0178
| 702.0
|13/9, 16/11
| 3/2
|-
|-
| style="text-align:center;" |32
| 35
| style="text-align:center;" |660.6635
| 722.6
| style="text-align:center;" |22/15
| 32/21
|-
|-
|33
| 36
|681.3093
| 743.2
|40/27
| 17/11, 20/13, 26/17
|-
|-
| style="text-align:center;" |34
| 37
| style="text-align:center;" |701.955
| 763.9
| style="text-align:center;" |3/2
| 14/9
|-
|-
|35
| 38
|722.6007
| 784.5
|32/21
| 11/7
|-
|-
| style="text-align:center;" |36
| 39
| style="text-align:center;" |743.2465
| 805.2
| style="text-align:center;" |20/13, 26/17, 17/11
| 8/5
|-
|-
|37
| 40
|763.8922
| 825.8
|14/9
| 21/13, 34/21
|-
|-
| style="text-align:center;" |38
| 41
| style="text-align:center;" |784.5379
| 846.5
| style="text-align:center;" |11/7
| 13/8, 18/11
|-
|-
|39
| 42
|805.1837
| 867.1
|8/5
| 28/17, 33/20
|-
|-
| style="text-align:center;" |40
| 43
| style="text-align:center;" |825.8294
| 887.8
| style="text-align:center;" |34/21
| 5/3
|-
|-
|41
| 44
|846.47515
| 908.4
|13/8, 18/11
| 17/10, 22/13
|-
|-
| style="text-align:center;" |42
| 45
| style="text-align:center;" |867.1209
| 929.1
| style="text-align:center;" |28/17
| 12/7
|-
|-
|43
| 46
|887.7666
| 949.7
|5/3,
| 26/15
|-
|-
| style="text-align:center;" |44
| 47
| style="text-align:center;" |908.41235
| 970.3
| style="text-align:center;" |22/13, 17/10
| 7/4
|-
|-
|45
| 48
|929.0581
| 991.0
|12/7
| 16/9
|-
|-
| style="text-align:center;" |46
| 49
| style="text-align:center;" |949.7038
| 1011.7
| style="text-align:center;" |26/15
| 9/5
|-
|-
|47
| 50
|970.35
| 1032.3
|7/4
| 20/11
|-
|-
| style="text-align:center;" |48
| 51
| style="text-align:center;" |990.9952
| 1052.9
| style="text-align:center;" |16/9
| 11/6
|-
|-
|49
| 52
|1011.641
| 1073.6
|9/5
| 13/7
|-
|-
| style="text-align:center;" |50
| 53
| style="text-align:center;" |1032.32868
| 1094.2
| style="text-align:center;" |20/11
| 15/8, 17/9
|-
|-
|51
| 54
|1052.9235
| 1114.9
|11/6, 24/13
| 21/11
|-
|-
| style="text-align:center;" |52
| 55
| style="text-align:center;" |1073.5782
| 1135.5
| style="text-align:center;" |28/15, 13/7,
| 25/13, 27/14
|-
|-
|53
| 56
|1094.224
| 1156.2
|15/8, 32/17, 17/9
| 35/18, 39/20, 49/25
|-
|-
| style="text-align:center;" |54
| 57
| style="text-align:center;" |1114.8697
| 1176.8
| style="text-align:center;" |40/21, 21/11
| 55/28, 63/32
|-
|-
|55
| 58
|1135.5154
| 1197.5
|48/25, 25/13, 52/27, 27/14, 64/33
| 2/1
|-
|-
| style="text-align:center;" |56
| 59
| style="text-align:center;" |1156.1612
| 1218.1
| style="text-align:center;" |35/18, 96/49, 49/25, 108/55
| 81/40, 91/45, 105/52
|-
|-
|57
| 60
|1176.8069
| 1238.7
|55/28, 63/32, 160/81, 125/64
| 45/22, 49/24, 55/27
|-
|-
| style="text-align:center;" |58
| 61
| style="text-align:center;" |1197.45265
| 1259.4
| style="text-align:center;" |2/1
| 27/13, 33/16
|-
| 62
| 1280.0
| 21/10, 25/12
|-
| 63
| 1300.7
| 17/8
|-
| 64
| 1321.3
| 15/7
|-
| 65
| 1342.0
| 13/6
|-
| 66
| 1362.6
| 11/5
|-
| 67
| 1383.4
| 20/9
|-
| 68
| 1403.9
| 9/4
|}
|}
[[Category:Edf]]
 
[[Category:Edonoi]]
== See also ==
* [[58edo]] – relative edo
* [[92edt]] – relative edt
* [[150ed6]] – relative ed6
* [[163ed7]] – relative ed7
Retrieved from "https://en.xen.wiki/w/34edf"