34edf: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,264 edits
m Same as 35edf
→Theory: +subsets and supersets
 
(3 intermediate revisions by the same user not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 34 equal parts''' (34EDF) is related to [[58edo]], but with the [[3/2]] rather than the [[2/1]] being [[just]]. The [[octave]] is [[Octave shrinking|compressed]] by about 2.5474 [[cents]] and the step size is about 20.6457 cents (corresponding to 58.1234 edo).
{{ED intro}}


The [[patent val]] has a generally flat tendency for [[harmonic]]s up to 16, with the exception for 5. Unlike 58edo, it is only consistent up to the 15-[[integer-limit]], with discrepancy for the 16th harmonic (four octaves).
== Theory ==
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]].  


Lookalikes: [[58edo]], [[92edt]]
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 ===
{{Harmonics in equal|34|3|2|intervals=prime}}
{{Harmonics in equal|34|3|2|intervals=integer}}
{{Harmonics in equal|34|3|2|intervals=prime|start=12|collapsed=1}}
{{Harmonics in equal|34|3|2|intervals=integer|columns=12|start=12|collapsed=1|title=Approximation of harmonics in 34edf (continued)}}


==Intervals==
=== Subsets and supersets ===
{| class="wikitable mw-collapsible"
Since 34 factors into primes as {{nowrap| 2 × 17 }}, 34edf contains [[2edf]] and [[17edf]] as subset edfs.
|+ Intervals of 34edf
 
== Intervals ==
{| class="wikitable center-1 right-2"
|-
|-
! |Degree
! #
! |Cents
! Cents
! |Approx. ratios
! Approximate ratios
|-
|-
| colspan="2" style="text-align:center;" |0
| 0
| style="text-align:left;" |1/1
| 0.0
| 1/1
|-
|-
|1
| 1
|20.6457
| 20.6
|56/55, 64/63, 81/80, 128/125
| ''56/55'', 64/63, 81/80, 91/90, 105/104
|-
|-
|2
| 2
|41.2915
| 41.3
|6/35, 49/48, 50/49, 55/54
| 36/35, 40/39, 45/44, 49/48, 50/49, 55/54
|-
|-
|3
| 3
|61.9372
| 61.9
|25/24, 26/25, 27/26, 28/27, 33/32
| 26/25, 27/26, 28/27, 33/32
|-
|-
|4
| 4
|82.5829
| 82.6
|21/20, 22/21
| 21/20, 22/21, ''25/24''
|-
|-
|5
| 5
|103.2287
| 103.2
|16/15, 17/16, 18/17
| 16/15, 17/16, 18/17
|-
|-
|6
| 6
|123.8744
| 123.9
|15/14, 14/13
| 14/13, 15/14
|-
|-
|7·
| 7·
|144.52015
| 144.5
|12/11, 13/12
| 12/11, 13/12
|-
|-
|8
| 8
|165.1659
| 165.2
|11/10
| 11/10
|-
|-
|9
| 9
|185.8116
| 185.8
|10/9
| 10/9
|-
|-
|10
| 10
|206.45735
| 206.5
|9/8
| 9/8
|-
|-
|11
| 11
|227.1031
| 227.1
|8/7
| 8/7
|-
|-
|12·
| 12·
|248.7488
| 248.7
|15/13
| 15/13
|-
|-
|13
| 13
|268.3946
| 268.4
|7/6
| 7/6
|-
|-
|14
| 14
|289.0403
| 289.0
|13/11, 20/17
| 13/11, 20/17
|-
|-
|15
| 15
|309.686
| 309.7
|6/5
| 6/5
|-
|-
|16
| 16
|330.3318
| 330.3
|17/14
| 17/14, 40/33
|-
|-
|17·
| 17·
|350.9775
| 351.0
|11/9, 16/13
| 11/9, 16/13
|-
|-
|18
| 18
|371.6232
| 371.6
|21/17
| 21/17, 26/21
|-
|-
|19
| 19
|392.269
| 392.3
|5/4
| 5/4
|-
|-
|20
| 20
|412.9147
| 412.9
|14/11
| 14/11
|-
|-
|21
| 21
|433.5604
| 433.6
|9/7
| 9/7
|-
|-
|22·
| 22·
|455.2062
| 455.2
|13/10, 17/13, 22/17
| 13/10, 17/13, 22/17
|-
|-
|23
| 23
|474.8519
| 474.9
|21/16
| 21/16
|-
|-
|24
| 24
|495.49765
| 495.5
|4/3
| 4/3
|-
|-
|25
| 25
|516.1434
| 516.1
|27/20
| 27/20
|-
|-
|26
| 26
|536.7891
| 536.8
|15/11
| 15/11
|-
|-
|27
| 27
|557.43485
| 557.4
|11/8, 18/13
| 11/8, 18/13
|-
|-
|28
| 28
|578.0806
| 578.1
|7/5  
| 7/5  
|-
|-
|29
| 29
|598.7263
| 598.7
|17/12, 24/17
| 17/12, 24/17
|-
|-
|30
| 30
|619.3721
| 619.4
|10/7
| 10/7
|-
|-
|31
| 31
|640.0178
| 640.0
|13/9, 16/11
| 13/9, 16/11
|-
|-
|32
| 32
|660.6635
| 660.7
|22/15
| 22/15
|-
|-
|33
| 33
|681.3093
| 681.3
|40/27
| 40/27
|-
|-
|34
| 34
|701.955
| 702.0
|3/2
| 3/2
|-
|-
|35
| 35
|722.6007
| 722.6
|32/21
| 32/21
|-
|-
|36
| 36
|743.2465
| 743.2
|20/13, 26/17, 17/11
| 17/11, 20/13, 26/17
|-
|-
|37
| 37
|763.8922
| 763.9
|14/9
| 14/9
|-
|-
|38
| 38
|784.5379
| 784.5
|11/7
| 11/7
|-
|-
|39
| 39
|805.1837
| 805.2
|8/5
| 8/5
|-
|-
|40
| 40
|825.8294
| 825.8
|34/21
| 21/13, 34/21
|-
|-
|41
| 41
|846.47515
| 846.5
|13/8, 18/11
| 13/8, 18/11
|-
|-
|42
| 42
|867.1209
| 867.1
|28/17
| 28/17, 33/20
|-
|-
|43
| 43
|887.7666
| 887.8
|5/3,
| 5/3
|-
|-
|44
| 44
|908.41235
| 908.4
|22/13, 17/10
| 17/10, 22/13
|-
|-
|45
| 45
|929.0581
| 929.1
|12/7
| 12/7
|-
|-
|46
| 46
|949.7038
| 949.7
|26/15
| 26/15
|-
|-
|47
| 47
|970.35
| 970.3
|7/4
| 7/4
|-
|-
|48
| 48
|990.9952
| 991.0
|16/9
| 16/9
|-
|-
|49
| 49
|1011.641
| 1011.7
|9/5
| 9/5
|-
|-
|50
| 50
|1032.32868
| 1032.3
|20/11
| 20/11
|-
|-
|51
| 51
|1052.9235
| 1052.9
|11/6, 24/13
| 11/6
|-
|-
|52
| 52
|1073.5782
| 1073.6
|28/15, 13/7  
| 13/7
|-
|-
|53
| 53
|1094.224
| 1094.2
|15/8, 32/17, 17/9
| 15/8, 17/9
|-
|-
|54
| 54
|1114.8697
| 1114.9
|40/21, 21/11
| 21/11
|-
|-
|55
| 55
|1135.5154
| 1135.5
|48/25, 25/13, 52/27, 27/14, 64/33
| 25/13, 27/14
|-
|-
|56
| 56
|1156.1612
| 1156.2
|35/18, 96/49, 49/25, 108/55
| 35/18, 39/20, 49/25
|-
|-
|57
| 57
|1176.8069
| 1176.8
|55/28, 63/32, 160/81, 125/64
| 55/28, 63/32
|-
|-
|58
| 58
|1197.45265
| 1197.5
|2/1
| 2/1
|-
|-
|59
| 59
|1218.0984
| 1218.1
|112/55, 128/63, 81/40, 256/125
| 81/40, 91/45, 105/52
|-
|-
|60
| 60
|1238.7441
| 1238.7
|72/35, 49/24, 100/49, 55/27
| 45/22, 49/24, 55/27
|-
|-
|61
| 61
|1259.38985
| 1259.4
|25/12, 52/25, 27/13, 56/27, 33/16
| 27/13, 33/16
|-
|-
|62
| 62
|1280.0356
| 1280.0
|21/10, 44/21
| 21/10, 25/12
|-
|-
|63
| 63
|1300.6813
| 1300.7
|32/15, 17/8, 36/17
| 17/8
|-
|-
|64
| 64
|1321.3271
| 1321.3
|15/7, 28/13
| 15/7
|-
|-
|65
| 65
|1341.9728
| 1342.0
|24/11, 13/5
| 13/6
|-
|-
|66
| 66
|1362.6185
| 1362.6
|11/10
| 11/5
|-
|-
|67
| 67
|1383.3643
| 1383.4
|20/9
| 20/9
|-
|-
|68
| 68
|1403.91
| 1403.9
|9/4
| 9/4
|}
|}


{{todo|expand}}
== See also ==
[[Category:Edf]]
* [[58edo]] – relative edo
[[Category:Edonoi]]
* [[92edt]] – relative edt
* [[150ed6]] – relative ed6
* [[163ed7]] – relative ed7

Latest revision as of 16:24, 18 March 2025

← 33edf 34edf 35edf →
Prime factorization 2 × 17
Step size 20.6457 ¢ 
Octave 58\34edf (1197.45 ¢) (→ 29\17edf)
Twelfth 92\34edf (1899.41 ¢) (→ 46\17edf)
Consistency limit 15
Distinct consistency limit 12

34 equal divisions of the perfect fifth (abbreviated 34edf or 34ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 34 equal parts of about 20.6 ¢ each. Each step represents a frequency ratio of (3/2)1/34, or the 34th root of 3/2.

Theory

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 compressed by about 2.5474 cents.

The patent val has a generally flat tendency for harmonics up to 16 (four octaves), with the exception for 5. Unlike 58edo, it is only consistent up to the 15-integer-limit, with discrepancy for the 16th harmonic.

Harmonics

Approximation of harmonics in 34edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.55 -2.55 -5.09 +0.86 -5.09 -3.57 -7.64 -5.09 -1.69 -1.53 -7.64
Relative (%) -12.3 -12.3 -24.7 +4.2 -24.7 -17.3 -37.0 -24.7 -8.2 -7.4 -37.0
Steps
(reduced) 		58
(24) 		92
(24) 		116
(14) 		135
(33) 		150
(14) 		163
(27) 		174
(4) 		184
(14) 		193
(23) 		201
(31) 		208
(4)
Approximation of harmonics in 34edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -1.69 -6.12 -1.69 -10.19 +8.73 -7.64 +1.98 -4.23 -6.12 -4.07 +1.55 -10.19
Relative (%) -8.2 -29.6 -8.2 -49.4 +42.3 -37.0 +9.6 -20.5 -29.6 -19.7 +7.5 -49.4
Steps
(reduced) 		215
(11) 		221
(17) 		227
(23) 		232
(28) 		238
(0) 		242
(4) 		247
(9) 		251
(13) 		255
(17) 		259
(21) 		263
(25) 		266
(28)

Subsets and supersets

Since 34 factors into primes as 2 × 17, 34edf contains 2edf and 17edf as subset edfs.

Intervals

# Cents Approximate ratios
0 0.0 1/1
1 20.6 56/55, 64/63, 81/80, 91/90, 105/104
2 41.3 36/35, 40/39, 45/44, 49/48, 50/49, 55/54
3 61.9 26/25, 27/26, 28/27, 33/32
4 82.6 21/20, 22/21, 25/24
5 103.2 16/15, 17/16, 18/17
6 123.9 14/13, 15/14
7· 144.5 12/11, 13/12
8 165.2 11/10
9 185.8 10/9
10 206.5 9/8
11 227.1 8/7
12· 248.7 15/13
13 268.4 7/6
14 289.0 13/11, 20/17
15 309.7 6/5
16 330.3 17/14, 40/33
17· 351.0 11/9, 16/13
18 371.6 21/17, 26/21
19 392.3 5/4
20 412.9 14/11
21 433.6 9/7
22· 455.2 13/10, 17/13, 22/17
23 474.9 21/16
24 495.5 4/3
25 516.1 27/20
26 536.8 15/11
27 557.4 11/8, 18/13
28 578.1 7/5
29 598.7 17/12, 24/17
30 619.4 10/7
31 640.0 13/9, 16/11
32 660.7 22/15
33 681.3 40/27
34 702.0 3/2
35 722.6 32/21
36 743.2 17/11, 20/13, 26/17
37 763.9 14/9
38 784.5 11/7
39 805.2 8/5
40 825.8 21/13, 34/21
41 846.5 13/8, 18/11
42 867.1 28/17, 33/20
43 887.8 5/3
44 908.4 17/10, 22/13
45 929.1 12/7
46 949.7 26/15
47 970.3 7/4
48 991.0 16/9
49 1011.7 9/5
50 1032.3 20/11
51 1052.9 11/6
52 1073.6 13/7
53 1094.2 15/8, 17/9
54 1114.9 21/11
55 1135.5 25/13, 27/14
56 1156.2 35/18, 39/20, 49/25
57 1176.8 55/28, 63/32
58 1197.5 2/1
59 1218.1 81/40, 91/45, 105/52
60 1238.7 45/22, 49/24, 55/27
61 1259.4 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

See also

Retrieved from "https://en.xen.wiki/index.php?title=34edf&oldid=186952"