9ed9/8: Difference between revisions

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==Intervals==
==Intervals==
{| class="wikitable"
{| class="wikitable right-2"
|-
|-
! | degree
! degree
! | cents value
! cents value
! | ratio
! ratio
|-
|-
| | 0
| 0
| | 0.0000
| 0.0000
| | '''[[1/1]]'''
| '''[[1/1]]'''
|-
|-
| | 1
| 1
| | 22.6567
| 22.6567
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">1/9</font>
| (9/8)<sup>1/9</sup>
|-
|-
| | 2
| 2
| | 45.3133
| 45.3133
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">2/9</font>
| (9/8)<sup>2/9</sup>
|-
|-
| | 3
| 3
| | 67.9700
| 67.9700
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">1/3</font>
| (9/8)<sup>1/3</sup>
|-
|-
| | 4
| 4
| | 90.6267
| 90.6267
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">4/9</font>
| (9/8)<sup>4/9</sup>
|-
|-
| | 5
| 5
| | 113.2833
| 113.2833
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">5/9</font>
| (9/8)<sup>5/9</sup>
|-
|-
| | 6
| 6
| | 135.9400
| 135.9400
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">2/3</font>
| (9/8)<sup>2/3</sup>
|-
|-
| | 7
| 7
| | 158.5967
| 158.5967
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">7/9</font>
| (9/8)<sup>7/9</sup>
|-
|-
| | 8
| 8
| | 181.2533
| 181.2533
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">8/9</font>
| (9/8)<sup>8/9</sup>
|-
|-
| | 9
| 9
| | 203.9100
| 203.9100
| | '''[[9/8]]'''
| '''[[9/8]]'''
|-
|-
| | 10
| 10
| | 226.5667
| 226.5667
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">10/9</font>
| (9/8)<sup>10/9</sup>
|-
|-
| | 11
| 11
| | 249.2233
| 249.2233
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">11/9</font>
| (9/8)<sup>11/9</sup>
|-
|-
| | 12
| 12
| | 271.8800
| 271.8800
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">4/3</font>
| (9/8)<sup>4/3</sup>
|-
|-
| | 13
| 13
| | 294.5367
| 294.5367
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">13/9</font>
| (9/8)<sup>13/9</sup>
|-
|-
| | 14
| 14
| | 317.1933
| 317.1933
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">14/9</font>
| (9/8)<sup>14/9</sup>
|-
|-
| | 15
| 15
| | 339.8500
| 339.8500
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">5/3</font>
| (9/8)<sup>5/3</sup>
|-
|-
| | 16
| 16
| | 362.5067
| 362.5067
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">16/9</font>
| (9/8)<sup>16/9</sup>
|-
|-
| | 17
| 17
| | 385.1633
| 385.1633
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">17/9</font>
| (9/8)<sup>17/9</sup>
|-
|-
| | 18
| 18
| | 407.8200
| 407.8200
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">2</font> = [[81/64]]
| (9/8)<sup>2</sup> = [[81/64]]
|-
|-
| | 19
| 19
| | 430.4767
| 430.4767
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">19/9</font>
| (9/8)<sup>19/9</sup>
|-
|-
| | 20
| 20
| | 453.1333
| 453.1333
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">20/9</font>
| (9/8)<sup>20/9</sup>
|-
|-
| | 21
| 21
| | 475.7900
| 475.7900
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">7/3</font>
| (9/8)<sup>7/3</sup>
|-
|-
| | 22
| 22
| | 498.4467
| 498.4467
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">22/9</font>
| (9/8)<sup>22/9</sup>
|-
|-
| | 23
| 23
| | 521.1033
| 521.1033
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">23/9</font>
| (9/8)<sup>23/9</sup>
|-
|-
| | 24
| 24
| | 543.7600
| 543.7600
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">8/3</font>
| (9/8)<sup>8/3</sup>
|-
|-
| | 25
| 25
| | 566.4167
| 566.4167
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">25/9</font>
| (9/8)<sup>25/9</sup>
|-
|-
| | 26
| 26
| | 589.0733
| 589.0733
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">26/9</font>
| (9/8)<sup>26/9</sup>
|-
|-
| | 27
| 27
| | 611.7300
| 611.7300
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">3</font> = [[729/512]]
| (9/8)<sup>3</sup> = [[729/512]]
|-
|-
| | 28
| 28
| | 634.3867
| 634.3867
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">28/9</font>
| (9/8)<sup>28/9</sup>
|-
|-
| | 29
| 29
| | 657.0433
| 657.0433
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">29/9</font>
| (9/8)<sup>29/9</sup>
|-
|-
| | 30
| 30
| | 679.7000
| 679.7000
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">10/3</font>
| (9/8)<sup>10/3</sup>
|-
|-
| | 31
| 31
| | 702.3567
| 702.3567
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">31/9</font>
| (9/8)<sup>31/9</sup>
|-
|-
| | 32
| 32
| | 725.0133
| 725.0133
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">32/9</font>
| (9/8)<sup>32/9</sup>
|-
|-
| | 33
| 33
| | 747.6700
| 747.6700
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">11/3</font>
| (9/8)<sup>11/3</sup>
|-
|-
| | 34
| 34
| | 770.3267
| 770.3267
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">34/9</font>
| (9/8)<sup>34/9</sup>
|-
|-
| | 35
| 35
| | 792.9833
| 792.9833
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">35/9</font>
| (9/8)<sup>35/9</sup>
|-
|-
| | 36
| 36
| | 815.6400
| 815.6400
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">4</font> = 6561/4096
| (9/8)<sup>4</sup> = 6561/4096
|-
|-
| | 37
| 37
| | 838.2967
| 838.2967
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">37/9</font>
| (9/8)<sup>37/9</sup>
|-
|-
| | 38
| 38
| | 860.9533
| 860.9533
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">38/9</font>
| (9/8)<sup>38/9</sup>
|-
|-
| | 39
| 39
| | 883.6100
| 883.6100
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">13/3</font>
| (9/8)<sup>13/3</sup>
|-
|-
| | 40
| 40
| | 906.2667
| 906.2667
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">40/9</font>
| (9/8)<sup>40/9</sup>
|-
|-
| | 41
| 41
| | 928.9233
| 928.9233
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">41/9</font>
| (9/8)<sup>41/9</sup>
|-
|-
| | 42
| 42
| | 951.5800
| 951.5800
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">14/3</font>
| (9/8)<sup>14/3</sup>
|-
|-
| | 43
| 43
| | 974.2367
| 974.2367
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">43/9</font>
| (9/8)<sup>43/9</sup>
|-
|-
| | 44
| 44
| | 996.8933
| 996.8933
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">44/9</font>
| (9/8)<sup>44/9</sup>
|-
|-
| | 45
| 45
| | 1019.5500
| 1019.5500
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">5</font> = 59049/32768
| (9/8)<sup>5</sup> = 59049/32768
|-
|-
| | 46
| 46
| | 1042.2067
| 1042.2067
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">46/9</font>
| (9/8)<sup>46/9</sup>
|-
|-
| | 47
| 47
| | 1064.8633
| 1064.8633
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">47/9</font>
| (9/8)<sup>47/9</sup>
|-
|-
| | 48
| 48
| | 1087.5200
| 1087.5200
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">16/3</font>
| (9/8)<sup>16/3</sup>
|-
|-
| | 49
| 49
| | 1110.1767
| 1110.1767
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">49/9</font>
| (9/8)<sup>49/9</sup>
|-
|-
| | 50
| 50
| | 1132.8333
| 1132.8333
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">50/9</font>
| (9/8)<sup>50/9</sup>
|-
|-
| | 51
| 51
| | 1155.4900
| 1155.4900
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">17/3</font>
| (9/8)<sup>17/3</sup>
|-
|-
| | 52
| 52
| | 1178.1467
| 1178.1467
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">52/9</font>
| (9/8)<sup>52/9</sup>
|-
|-
| | 53
| 53
| | 1200.8033
| 1200.8033
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">53/9</font>
| (9/8)<sup>53/9</sup>
|-
|-
| | 54
| 54
| | 1223.4600
| 1223.4600
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">6</font> = 531441/262144
| (9/8)<sup>6</sup> = 531441/262144
|}
|}


Revision as of 11:31, 4 December 2021

9ED9/8 is the equal division of the Pythagorean whole tone into nine parts of 22.6567 cents each, corresponding to 52.9645 edo. This tuning is used in Ottoman classical music theory, in which ninth root of the 9/8 whole tone is treated as the minimum interval.

Intervals

degree cents value ratio
0 0.0000 1/1
1 22.6567 (9/8)1/9
2 45.3133 (9/8)2/9
3 67.9700 (9/8)1/3
4 90.6267 (9/8)4/9
5 113.2833 (9/8)5/9
6 135.9400 (9/8)2/3
7 158.5967 (9/8)7/9
8 181.2533 (9/8)8/9
9 203.9100 9/8
10 226.5667 (9/8)10/9
11 249.2233 (9/8)11/9
12 271.8800 (9/8)4/3
13 294.5367 (9/8)13/9
14 317.1933 (9/8)14/9
15 339.8500 (9/8)5/3
16 362.5067 (9/8)16/9
17 385.1633 (9/8)17/9
18 407.8200 (9/8)2 = 81/64
19 430.4767 (9/8)19/9
20 453.1333 (9/8)20/9
21 475.7900 (9/8)7/3
22 498.4467 (9/8)22/9
23 521.1033 (9/8)23/9
24 543.7600 (9/8)8/3
25 566.4167 (9/8)25/9
26 589.0733 (9/8)26/9
27 611.7300 (9/8)3 = 729/512
28 634.3867 (9/8)28/9
29 657.0433 (9/8)29/9
30 679.7000 (9/8)10/3
31 702.3567 (9/8)31/9
32 725.0133 (9/8)32/9
33 747.6700 (9/8)11/3
34 770.3267 (9/8)34/9
35 792.9833 (9/8)35/9
36 815.6400 (9/8)4 = 6561/4096
37 838.2967 (9/8)37/9
38 860.9533 (9/8)38/9
39 883.6100 (9/8)13/3
40 906.2667 (9/8)40/9
41 928.9233 (9/8)41/9
42 951.5800 (9/8)14/3
43 974.2367 (9/8)43/9
44 996.8933 (9/8)44/9
45 1019.5500 (9/8)5 = 59049/32768
46 1042.2067 (9/8)46/9
47 1064.8633 (9/8)47/9
48 1087.5200 (9/8)16/3
49 1110.1767 (9/8)49/9
50 1132.8333 (9/8)50/9
51 1155.4900 (9/8)17/3
52 1178.1467 (9/8)52/9
53 1200.8033 (9/8)53/9
54 1223.4600 (9/8)6 = 531441/262144

Just approximation

15-odd-limit mappings

The following table shows how 15-odd-limit intervals are represented in 9ed9/8 (ordered by absolute error).

Direct mapping (even if inconsistent)
Interval(s) Error (abs, ¢)
9/8 0.0
3/2, 4/3 0.402
26/15 0.679
15/8, 5/3 0.749
16/9 0.803
13/10 1.081
5/4, 10/9 1.15
15/13 1.482
6/5, 16/15 1.552
20/13 1.884
9/5, 8/5 1.954
13/8, 13/9 2.231
13/12 2.633
16/13, 18/13 3.034
24/13 3.436
12/7 4.206
22/13 4.524
9/7, 8/7 4.607
7/6 5.009
13/11 5.327
7/4, 14/9 5.411
10/7 5.758
22/15 6.006
15/14 6.159
11/10 6.408
7/5 6.561
15/11 6.809
13/7 6.838
28/15 6.963
11/6 7.156
20/11 7.211
11/9, 11/8 7.558
14/13 7.642
12/11 7.96
18/11, 16/11 8.361
14/11 9.688
11/7 10.491

See also

Retrieved from "https://en.xen.wiki/index.php?title=9ed9/8&oldid=81450"