43ed7/4: Difference between revisions

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'''43ED7/4''' is the [[Equal-step tuning|equal division]] of the [[7/4|harmonic seventh]] into 43 parts of 22.5308 [[cent|cents]] each, corresponding to 53.2603 [[EDO]].  
{{stub}}
 
 
{{Infobox ET}}
{{ED intro}} It corresponds to 53.2603edo, which is closely related to [[53edo]] but with 7/4 tuned pure instead of the octave.  


== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | degree
! #
! | cents value
! Cents Value
! | ratio
! Ratio
|-
|-
| | 0
| 0
| | 0.0000
| 0.0000
| | '''[[1/1]]'''
| '''[[1/1]]'''
 
|-
|-
| | 1
| 1
| | 22.5308
| 22.5308
| | (7/4)<sup>1/43</sup>
| (7/4)<sup>1/43</sup>
 
|-
|-
| | 2
| 2
| | 45.0617
| 45.0617
| | (7/4)<sup>2/43</sup>
| (7/4)<sup>2/43</sup>
 
|-
|-
| | 3
| 3
| | 67.5925
| 67.5925
| | (7/4)<sup>3/43</sup>
| (7/4)<sup>3/43</sup>
 
|-
|-
| | 4
| 4
| | 90.1233
| 90.1233
| | (7/4)<sup>4/43</sup>
| (7/4)<sup>4/43</sup>
 
|-
|-
| | 5
| 5
| | 112.6542
| 112.6542
| | (7/4)<sup>5/43</sup>
| (7/4)<sup>5/43</sup>
 
|-
|-
| | 6
| 6
| | 135.1850
| 135.1850
| | (7/4)<sup>6/43</sup>
| (7/4)<sup>6/43</sup>
 
|-
|-
| | 7
| 7
| | 157.7158
| 157.7158
| | (7/4)<sup>7/43</sup>
| (7/4)<sup>7/43</sup>
 
|-
|-
| | 8
| 8
| | 180.2467
| 180.2467
| | (7/4)<sup>8/43</sup>
| (7/4)<sup>8/43</sup>
 
|-
|-
| | 9
| 9
| | 202.7775
| 202.7775
| | (7/4)<sup>9/43</sup>
| (7/4)<sup>9/43</sup>
 
|-
|-
| | 10
| 10
| | 225.3084
| 225.3084
| | (7/4)<sup>10/43</sup>
| (7/4)<sup>10/43</sup>
 
|-
|-
| | 11
| 11
| | 247.8392
| 247.8392
| | (7/4)<sup>11/43</sup>
| (7/4)<sup>11/43</sup>
 
|-
|-
| | 12
| 12
| | 270.3700
| 270.3700
| | (7/4)<sup>12/43</sup>
| (7/4)<sup>12/43</sup>
 
|-
|-
| | 13
| 13
| | 292.9009
| 292.9009
| | (7/4)<sup>13/43</sup>
| (7/4)<sup>13/43</sup>
 
|-
|-
| | 14
| 14
| | 315.4317
| 315.4317
| | (7/4)<sup>14/43</sup>
| (7/4)<sup>14/43</sup>
 
|-
|-
| | 15
| 15
| | 337.9625
| 337.9625
| | (7/4)<sup>15/43</sup>
| (7/4)<sup>15/43</sup>
 
|-
|-
| | 16
| 16
| | 360.4934
| 360.4934
| | (7/4)<sup>16/43</sup>
| (7/4)<sup>16/43</sup>
 
|-
|-
| | 17
| 17
| | 383.0242
| 383.0242
| | (7/4)<sup>17/43</sup>
| (7/4)<sup>17/43</sup>
 
|-
|-
| | 18
| 18
| | 405.5550
| 405.5550
| | (7/4)<sup>18/43</sup>
| (7/4)<sup>18/43</sup>
 
|-
|-
| | 19
| 19
| | 428.0859
| 428.0859
| | (7/4)<sup>19/43</sup>
| (7/4)<sup>19/43</sup>
 
|-
|-
| | 20
| 20
| | 450.6167
| 450.6167
| | (7/4)<sup>20/43</sup>
| (7/4)<sup>20/43</sup>
 
|-
|-
| | 21
| 21
| | 473.1475
| 473.1475
| | (7/4)<sup>21/43</sup>
| (7/4)<sup>21/43</sup>
 
|-
|-
| | 22
| 22
| | 495.6784
| 495.6784
| | (7/4)<sup>22/43</sup>
| (7/4)<sup>22/43</sup>
 
|-
|-
| | 23
| 23
| | 518.2092
| 518.2092
| | (7/4)<sup>23/43</sup>
| (7/4)<sup>23/43</sup>
 
|-
|-
| | 24
| 24
| | 540.7400
| 540.7400
| | (7/4)<sup>24/43</sup>
| (7/4)<sup>24/43</sup>
 
|-
|-
| | 25
| 25
| | 563.2709
| 563.2709
| | (7/4)<sup>25/43</sup>
| (7/4)<sup>25/43</sup>
 
|-
|-
| | 26
| 26
| | 585.8017
| 585.8017
| | (7/4)<sup>26/43</sup>
| (7/4)<sup>26/43</sup>
 
|-
|-
| | 27
| 27
| | 608.3325
| 608.3325
| | (7/4)<sup>27/43</sup>
| (7/4)<sup>27/43</sup>
 
|-
|-
| | 28
| 28
| | 630.8634
| 630.8634
| | (7/4)<sup>28/43</sup>
| (7/4)<sup>28/43</sup>
 
|-
|-
| | 29
| 29
| | 653.3942
| 653.3942
| | (7/4)<sup>29/43</sup>
| (7/4)<sup>29/43</sup>
 
|-
|-
| | 30
| 30
| | 675.9251
| 675.9251
| | (7/4)<sup>30/43</sup>
| (7/4)<sup>30/43</sup>
 
|-
|-
| | 31
| 31
| | 698.4559
| 698.4559
| | (7/4)<sup>31/43</sup>
| (7/4)<sup>31/43</sup>
 
|-
|-
| | 32
| 32
| | 720.9867
| 720.9867
| | (7/4)<sup>32/43</sup>
| (7/4)<sup>32/43</sup>
 
|-
|-
| | 33
| 33
| | 743.5176
| 743.5176
| | (7/4)<sup>33/43</sup>
| (7/4)<sup>33/43</sup>
 
|-
|-
| | 34
| 34
| | 766.0484
| 766.0484
| | (7/4)<sup>34/43</sup>
| (7/4)<sup>34/43</sup>
 
|-
|-
| | 35
| 35
| | 788.5792
| 788.5792
| | (7/4)<sup>35/43</sup>
| (7/4)<sup>35/43</sup>
 
|-
|-
| | 36
| 36
| | 811.1101
| 811.1101
| | (7/4)<sup>36/43</sup>
| (7/4)<sup>36/43</sup>
 
|-
|-
| | 37
| 37
| | 833.6409
| 833.6409
| | (7/4)<sup>37/43</sup>
| (7/4)<sup>37/43</sup>
 
|-
|-
| | 38
| 38
| | 856.1717
| 856.1717
| | (7/4)<sup>38/43</sup>
| (7/4)<sup>38/43</sup>
 
|-
|-
| | 39
| 39
| | 878.7026
| 878.7026
| | (7/4)<sup>39/43</sup>
| (7/4)<sup>39/43</sup>
 
|-
|-
| | 40
| 40
| | 901.2334
| 901.2334
| | (7/4)<sup>40/43</sup>
| (7/4)<sup>40/43</sup>
 
|-
|-
| | 41
| 41
| | 923.7642
| 923.7642
| | (7/4)<sup>41/43</sup>
| (7/4)<sup>41/43</sup>
 
|-
|-
| | 42
| 42
| | 946.2951
| 946.2951
| | (7/4)<sup>42/43</sup>
| (7/4)<sup>42/43</sup>
 
|-
|-
| | 43
| 43
| | 968.8259
| 968.8259
| | '''[[7/4]]'''
| '''[[7/4]]'''
 
|-
|-
| | 44
| 44
| | 991.3567
| 991.3567
| | (7/4)<sup>44/43</sup>
| (7/4)<sup>44/43</sup>
 
|-
|-
| | 45
| 45
| | 1013.8876
| 1013.8876
| | (7/4)<sup>45/43</sup>
| (7/4)<sup>45/43</sup>
 
|-
|-
| | 46
| 46
| | 1036.4184
| 1036.4184
| | (7/4)<sup>46/43</sup>
| (7/4)<sup>46/43</sup>
 
|-
|-
| | 47
| 47
| | 1058.9492
| 1058.9492
| | (7/4)<sup>47/43</sup>
| (7/4)<sup>47/43</sup>
 
|-
|-
| | 48
| 48
| | 1081.4801
| 1081.4801
| | (7/4)<sup>48/43</sup>
| (7/4)<sup>48/43</sup>
 
|-
|-
| | 49
| 49
| | 1104.0109
| 1104.0109
| | (7/4)<sup>49/43</sup>
| (7/4)<sup>49/43</sup>
 
|-
|-
| | 50
| 50
| | 1126.5418
| 1126.5418
| | (7/4)<sup>50/43</sup>
| (7/4)<sup>50/43</sup>
 
|-
|-
| | 51
| 51
| | 1149.0726
| 1149.0726
| | (7/4)<sup>51/43</sup>
| (7/4)<sup>51/43</sup>
 
|-
|-
| | 52
| 52
| | 1171.6034
| 1171.6034
| | (7/4)<sup>52/43</sup>
| (7/4)<sup>52/43</sup>
 
|-
|-
| | 53
| 53
| | 1194.1343
| 1194.1343
| | (7/4)<sup>53/43</sup>
| (7/4)<sup>53/43</sup>
 
|-
|-
| | 54
| 54
| | 1216.6651
| 1216.6651
| | (7/4)<sup>54/43</sup>
| (7/4)<sup>54/43</sup>
|}
|}


== Just approximation ==
== Approximation to JI ==
Several intervals like the [[6/5|just minor third]] and the [[9/8|whole tone]] are well approximated by 43ed7/4.
Several intervals like the [[6/5|just minor third]] and the [[9/8|whole tone]] are well approximated by 43ed7/4.
=== 15-odd-limit mappings ===
=== 15-odd-limit mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 43ed7/4 (can be ordered by absolute error).
The following table shows how [[15-odd-limit intervals]] are represented in 43ed7/4 (can be ordered by absolute error).
Line 290: Line 241:
{| class="wikitable sortable"
{| class="wikitable sortable"
|-
|-
|+ Direct mapping (even if inconsistent)
|+ Direct approximation (even if inconsistent)
|-
|-
! Interval(s)
! Interval(s)
Line 297: Line 248:
|-
|-
| [[7/4]]
| [[7/4]]
|0.0
| 0.0
|-
|-
| [[2/1]]
| [[2/1]]
|5.866
| 5.866
|-
|-
| [[3/2]]
| [[3/2]]
|3.499
| 3.499
|-
|-
| [[5/4]]
| [[5/4]]
|3.29
| 3.29
|-
|-
| [[9/8]]
| [[9/8]]
|1.132
| 1.132
|-
|-
| [[11/8]]
| [[11/8]]
|10.578
| 10.578
|-
|-
| [[13/8]]
| [[13/8]]
|6.887
| 6.887
|-
|-
| [[15/8]]
| [[15/8]]
|6.789
| 6.789
|-
|-
| [[14/9]]
| [[14/9]]
|1.132
| 1.132
|-
|-
| [[28/15]]
| [[28/15]]
|0.923
| 0.923
|-
|-
| [[10/7]]
| [[10/7]]
|9.155
| 9.155
|-
|-
| [[16/11]]
| [[16/11]]
|4.712
| 4.712
|-
|-
| [[13/10]]
| [[13/10]]
|3.597
| 3.597
|-
|-
| [[9/5]]
| [[9/5]]
|3.709
| 3.709
|-
|-
| [[10/9]]
| [[10/9]]
|2.157
| 2.157
|-
|-
| [[26/15]]
| [[26/15]]
|5.964
| 5.964
|-
|-
| [[13/11]]
| [[13/11]]
|3.691
| 3.691
|-
|-
| [[13/7]]
| [[13/7]]
|9.778
| 9.778
|-
|-
| [[16/13]]
| [[16/13]]
|1.021
| 1.021
|-
|-
| [[7/6]]
| [[7/6]]
|3.499
| 3.499
|-
|-
| [[5/3]]
| [[5/3]]
|5.656
| 5.656
|-
|-
| [[20/13]]
| [[20/13]]
|2.268
| 2.268
|-
|-
| [[11/10]]
| [[11/10]]
|7.288
| 7.288
|-
|-
| [[8/5]]
| [[8/5]]
|2.576
| 2.576
|-
|-
| [[9/7]]
| [[9/7]]
|6.998
| 6.998
|-
|-
| [[11/9]]
| [[11/9]]
|9.445
| 9.445
|-
|-
| [[18/11]]
| [[18/11]]
|3.58
| 3.58
|-
|-
| [[24/13]]
| [[24/13]]
|2.478
| 2.478
|-
|-
| [[22/15]]
| [[22/15]]
|9.655
| 9.655
|-
|-
| [[15/13]]
| [[15/13]]
|0.098
| 0.098
|-
|-
| [[15/11]]
| [[15/11]]
|3.789
| 3.789
|-
|-
| [[16/9]]
| [[16/9]]
|4.733
| 4.733
|-
|-
| [[12/7]]
| [[12/7]]
|9.365
| 9.365
|-
|-
| [[7/5]]
| [[7/5]]
|3.29
| 3.29
|-
|-
| [[12/11]]
| [[12/11]]
|7.079
| 7.079
|-
|-
| [[4/3]]
| [[4/3]]
|2.367
| 2.367
|-
|-
| [[11/6]]
| [[11/6]]
|9.586
| 9.586
|-
|-
| [[13/12]]
| [[13/12]]
|3.388
| 3.388
|-
|-
| [[8/7]]
| [[8/7]]
|5.866
| 5.866
|-
|-
| [[20/11]]
| [[20/11]]
|1.423
| 1.423
|-
|-
| [[14/13]]
| [[14/13]]
|6.887
| 6.887
|-
|-
| [[6/5]]
| [[6/5]]
|0.21
| 0.21
|-
|-
| [[18/13]]
| [[18/13]]
|0.111
| 0.111
|-
|-
| [[15/14]]
| [[15/14]]
|6.789
| 6.789
|-
|-
| [[11/7]]
| [[11/7]]
|6.087
| 6.087
|-
|-
| [[13/9]]
| [[13/9]]
|5.754
| 5.754
|-
|-
| [[14/11]]
| [[14/11]]
|10.578
| 10.578
|-
|-
| [[22/13]]
| [[22/13]]
|9.557
| 9.557
|-
|-
| [[16/15]]
| [[16/15]]
|0.923
| 0.923
|}
|}
[[Category:Subminor seventh]]
[[Category:Equal-step tuning]]
[[Category:Edonoi]]

Latest revision as of 19:23, 1 August 2025

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← 42ed7/4 43ed7/4 44ed7/4 →
Prime factorization 43 (prime)
Step size 22.5308 ¢ 
Octave 53\43ed7/4 (1194.13 ¢)
Twelfth 84\43ed7/4 (1892.59 ¢)
Consistency limit 3
Distinct consistency limit 3

43 equal divisions of 7/4 (abbreviated 43ed7/4) is a nonoctave tuning system that divides the interval of 7/4 into 43 equal parts of about 22.5 ¢ each. Each step represents a frequency ratio of (7/4)1/43, or the 43rd root of 7/4. It corresponds to 53.2603edo, which is closely related to 53edo but with 7/4 tuned pure instead of the octave.

Intervals

# Cents Value Ratio
0 0.0000 1/1
1 22.5308 (7/4)1/43
2 45.0617 (7/4)2/43
3 67.5925 (7/4)3/43
4 90.1233 (7/4)4/43
5 112.6542 (7/4)5/43
6 135.1850 (7/4)6/43
7 157.7158 (7/4)7/43
8 180.2467 (7/4)8/43
9 202.7775 (7/4)9/43
10 225.3084 (7/4)10/43
11 247.8392 (7/4)11/43
12 270.3700 (7/4)12/43
13 292.9009 (7/4)13/43
14 315.4317 (7/4)14/43
15 337.9625 (7/4)15/43
16 360.4934 (7/4)16/43
17 383.0242 (7/4)17/43
18 405.5550 (7/4)18/43
19 428.0859 (7/4)19/43
20 450.6167 (7/4)20/43
21 473.1475 (7/4)21/43
22 495.6784 (7/4)22/43
23 518.2092 (7/4)23/43
24 540.7400 (7/4)24/43
25 563.2709 (7/4)25/43
26 585.8017 (7/4)26/43
27 608.3325 (7/4)27/43
28 630.8634 (7/4)28/43
29 653.3942 (7/4)29/43
30 675.9251 (7/4)30/43
31 698.4559 (7/4)31/43
32 720.9867 (7/4)32/43
33 743.5176 (7/4)33/43
34 766.0484 (7/4)34/43
35 788.5792 (7/4)35/43
36 811.1101 (7/4)36/43
37 833.6409 (7/4)37/43
38 856.1717 (7/4)38/43
39 878.7026 (7/4)39/43
40 901.2334 (7/4)40/43
41 923.7642 (7/4)41/43
42 946.2951 (7/4)42/43
43 968.8259 7/4
44 991.3567 (7/4)44/43
45 1013.8876 (7/4)45/43
46 1036.4184 (7/4)46/43
47 1058.9492 (7/4)47/43
48 1081.4801 (7/4)48/43
49 1104.0109 (7/4)49/43
50 1126.5418 (7/4)50/43
51 1149.0726 (7/4)51/43
52 1171.6034 (7/4)52/43
53 1194.1343 (7/4)53/43
54 1216.6651 (7/4)54/43

Approximation to JI

Several intervals like the just minor third and the whole tone are well approximated by 43ed7/4.

15-odd-limit mappings

The following table shows how 15-odd-limit intervals are represented in 43ed7/4 (can be ordered by absolute error).

Direct approximation (even if inconsistent)
Interval(s) Error (abs, ¢)
7/4 0.0
2/1 5.866
3/2 3.499
5/4 3.29
9/8 1.132
11/8 10.578
13/8 6.887
15/8 6.789
14/9 1.132
28/15 0.923
10/7 9.155
16/11 4.712
13/10 3.597
9/5 3.709
10/9 2.157
26/15 5.964
13/11 3.691
13/7 9.778
16/13 1.021
7/6 3.499
5/3 5.656
20/13 2.268
11/10 7.288
8/5 2.576
9/7 6.998
11/9 9.445
18/11 3.58
24/13 2.478
22/15 9.655
15/13 0.098
15/11 3.789
16/9 4.733
12/7 9.365
7/5 3.29
12/11 7.079
4/3 2.367
11/6 9.586
13/12 3.388
8/7 5.866
20/11 1.423
14/13 6.887
6/5 0.21
18/13 0.111
15/14 6.789
11/7 6.087
13/9 5.754
14/11 10.578
22/13 9.557
16/15 0.923
Retrieved from "https://en.xen.wiki/index.php?title=43ed7/4&oldid=206019"