9ed9/8: Difference between revisions

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+subsets and supersets; +see also
→Intervals: reduce cent values to one decimal place, per discussion on Discord
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|-
|-
| 0
| 0
| 0.0000
| 0.0
| '''[[1/1]]'''
| '''[[1/1]]'''
|-
|-
| 1
| 1
| 22.6567
| 22.7
| (9/8)<sup>1/9</sup>
| (9/8)<sup>1/9</sup>
|-
|-
| 2
| 2
| 45.3133
| 45.3
| (9/8)<sup>2/9</sup>
| (9/8)<sup>2/9</sup>
|-
|-
| 3
| 3
| 67.9700
| 68.0
| (9/8)<sup>1/3</sup>
| (9/8)<sup>1/3</sup>
|-
|-
| 4
| 4
| 90.6267
| 90.6
| (9/8)<sup>4/9</sup>
| (9/8)<sup>4/9</sup>
|-
|-
| 5
| 5
| 113.2833
| 113.3
| (9/8)<sup>5/9</sup>
| (9/8)<sup>5/9</sup>
|-
|-
| 6
| 6
| 135.9400
| 135.9
| (9/8)<sup>2/3</sup>
| (9/8)<sup>2/3</sup>
|-
|-
| 7
| 7
| 158.5967
| 158.6
| (9/8)<sup>7/9</sup>
| (9/8)<sup>7/9</sup>
|-
|-
| 8
| 8
| 181.2533
| 181.3
| (9/8)<sup>8/9</sup>
| (9/8)<sup>8/9</sup>
|-
|-
| 9
| 9
| 203.9100
| 203.9
| '''[[9/8]]'''
| '''[[9/8]]'''
|-
|-
| 10
| 10
| 226.5667
| 226.6
| (9/8)<sup>10/9</sup>
| (9/8)<sup>10/9</sup>
|-
|-
| 11
| 11
| 249.2233
| 249.2
| (9/8)<sup>11/9</sup>
| (9/8)<sup>11/9</sup>
|-
|-
| 12
| 12
| 271.8800
| 271.9
| (9/8)<sup>4/3</sup>
| (9/8)<sup>4/3</sup>
|-
|-
| 13
| 13
| 294.5367
| 294.5
| (9/8)<sup>13/9</sup>
| (9/8)<sup>13/9</sup>
|-
|-
| 14
| 14
| 317.1933
| 317.2
| (9/8)<sup>14/9</sup>
| (9/8)<sup>14/9</sup>
|-
|-
| 15
| 15
| 339.8500
| 339.9
| (9/8)<sup>5/3</sup>
| (9/8)<sup>5/3</sup>
|-
|-
| 16
| 16
| 362.5067
| 362.5
| (9/8)<sup>16/9</sup>
| (9/8)<sup>16/9</sup>
|-
|-
| 17
| 17
| 385.1633
| 385.2
| (9/8)<sup>17/9</sup>
| (9/8)<sup>17/9</sup>
|-
|-
| 18
| 18
| 407.8200
| 407.8
| (9/8)<sup>2</sup> = [[81/64]]
| (9/8)<sup>2</sup> = [[81/64]]
|-
|-
| 19
| 19
| 430.4767
| 430.5
| (9/8)<sup>19/9</sup>
| (9/8)<sup>19/9</sup>
|-
|-
| 20
| 20
| 453.1333
| 453.1
| (9/8)<sup>20/9</sup>
| (9/8)<sup>20/9</sup>
|-
|-
| 21
| 21
| 475.7900
| 475.8
| (9/8)<sup>7/3</sup>
| (9/8)<sup>7/3</sup>
|-
|-
| 22
| 22
| 498.4467
| 498.4
| (9/8)<sup>22/9</sup>
| (9/8)<sup>22/9</sup>
|-
|-
| 23
| 23
| 521.1033
| 521.1
| (9/8)<sup>23/9</sup>
| (9/8)<sup>23/9</sup>
|-
|-
| 24
| 24
| 543.7600
| 543.8
| (9/8)<sup>8/3</sup>
| (9/8)<sup>8/3</sup>
|-
|-
| 25
| 25
| 566.4167
| 566.4
| (9/8)<sup>25/9</sup>
| (9/8)<sup>25/9</sup>
|-
|-
| 26
| 26
| 589.0733
| 589.1
| (9/8)<sup>26/9</sup>
| (9/8)<sup>26/9</sup>
|-
|-
| 27
| 27
| 611.7300
| 611.7
| (9/8)<sup>3</sup> = [[729/512]]
| (9/8)<sup>3</sup> = [[729/512]]
|-
|-
| 28
| 28
| 634.3867
| 634.4
| (9/8)<sup>28/9</sup>
| (9/8)<sup>28/9</sup>
|-
|-
| 29
| 29
| 657.0433
| 657.0
| (9/8)<sup>29/9</sup>
| (9/8)<sup>29/9</sup>
|-
|-
| 30
| 30
| 679.7000
| 679.7
| (9/8)<sup>10/3</sup>
| (9/8)<sup>10/3</sup>
|-
|-
| 31
| 31
| 702.3567
| 702.4
| (9/8)<sup>31/9</sup>
| (9/8)<sup>31/9</sup>
|-
|-
| 32
| 32
| 725.0133
| 725.0
| (9/8)<sup>32/9</sup>
| (9/8)<sup>32/9</sup>
|-
|-
| 33
| 33
| 747.6700
| 747.7
| (9/8)<sup>11/3</sup>
| (9/8)<sup>11/3</sup>
|-
|-
| 34
| 34
| 770.3267
| 770.3
| (9/8)<sup>34/9</sup>
| (9/8)<sup>34/9</sup>
|-
|-
| 35
| 35
| 792.9833
| 792.0
| (9/8)<sup>35/9</sup>
| (9/8)<sup>35/9</sup>
|-
|-
| 36
| 36
| 815.6400
| 815.6
| (9/8)<sup>4</sup> = [[6561/4096]]
| (9/8)<sup>4</sup> = [[6561/4096]]
|-
|-
| 37
| 37
| 838.2967
| 838.3
| (9/8)<sup>37/9</sup>
| (9/8)<sup>37/9</sup>
|-
|-
| 38
| 38
| 860.9533
| 861.0
| (9/8)<sup>38/9</sup>
| (9/8)<sup>38/9</sup>
|-
|-
| 39
| 39
| 883.6100
| 883.6
| (9/8)<sup>13/3</sup>
| (9/8)<sup>13/3</sup>
|-
|-
| 40
| 40
| 906.2667
| 906.3
| (9/8)<sup>40/9</sup>
| (9/8)<sup>40/9</sup>
|-
|-
| 41
| 41
| 928.9233
| 928.9
| (9/8)<sup>41/9</sup>
| (9/8)<sup>41/9</sup>
|-
|-
| 42
| 42
| 951.5800
| 951.6
| (9/8)<sup>14/3</sup>
| (9/8)<sup>14/3</sup>
|-
|-
| 43
| 43
| 974.2367
| 974.2
| (9/8)<sup>43/9</sup>
| (9/8)<sup>43/9</sup>
|-
|-
| 44
| 44
| 996.8933
| 996.9
| (9/8)<sup>44/9</sup>
| (9/8)<sup>44/9</sup>
|-
|-
| 45
| 45
| 1019.5500
| 1019.6
| (9/8)<sup>5</sup> = 59049/32768
| (9/8)<sup>5</sup> = 59049/32768
|-
|-
| 46
| 46
| 1042.2067
| 1042.2
| (9/8)<sup>46/9</sup>
| (9/8)<sup>46/9</sup>
|-
|-
| 47
| 47
| 1064.8633
| 1064.9
| (9/8)<sup>47/9</sup>
| (9/8)<sup>47/9</sup>
|-
|-
| 48
| 48
| 1087.5200
| 1087.5
| (9/8)<sup>16/3</sup>
| (9/8)<sup>16/3</sup>
|-
|-
| 49
| 49
| 1110.1767
| 1110.2
| (9/8)<sup>49/9</sup>
| (9/8)<sup>49/9</sup>
|-
|-
| 50
| 50
| 1132.8333
| 1132.8
| (9/8)<sup>50/9</sup>
| (9/8)<sup>50/9</sup>
|-
|-
| 51
| 51
| 1155.4900
| 1155.5
| (9/8)<sup>17/3</sup>
| (9/8)<sup>17/3</sup>
|-
|-
| 52
| 52
| 1178.1467
| 1178.1
| (9/8)<sup>52/9</sup>
| (9/8)<sup>52/9</sup>
|-
|-
| 53
| 53
| 1200.8033
| 1200.8
| (9/8)<sup>53/9</sup>
| (9/8)<sup>53/9</sup>
|-
|-
| 54
| 54
| 1223.4600
| 1223.5
| (9/8)<sup>6</sup> = 531441/262144
| (9/8)<sup>6</sup> = 531441/262144
|}
|}

Revision as of 10:57, 24 March 2025

← 8ed9/8 9ed9/8 10ed9/8 →
Prime factorization 32
Step size 22.6567 ¢ 
Octave 53\9ed9/8 (1200.8 ¢)
(convergent)
Twelfth 84\9ed9/8 (1903.16 ¢) (→ 28\3ed9/8)
Consistency limit 10
Distinct consistency limit 10

9 equal divisions of 9/8 (abbreviated 9ed9/8) is a nonoctave tuning system that divides the interval of 9/8 into 9 equal parts of about 22.7 ¢ each. Each step represents a frequency ratio of (9/8)1/9, or the 9th root of 9/8.

Theory

9ed9/8 corresponds to 52.9645…edo, which is closely related to 53edo but with the whole tone instead of the octave tuned pure. Like 53edo, 9ed9/8 is consistent to the 10-integer-limit.

Harmonics

Approximation of harmonics in 9ed9/8
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.80 +1.21 +1.61 +0.46 +2.01 +7.02 +2.41 +2.41 +1.26 -5.15 +2.81
Relative (%) +3.5 +5.3 +7.1 +2.0 +8.9 +31.0 +10.6 +10.6 +5.6 -22.7 +12.4
Steps
(reduced) 		53
(8) 		84
(3) 		106
(7) 		123
(6) 		137
(2) 		149
(5) 		159
(6) 		168
(6) 		176
(5) 		183
(3) 		190
(1)
Approximation of harmonics in 9ed9/8 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +0.18 +7.82 +1.66 +3.21 -11.12 +3.21 +0.24 +2.06 +8.22 -4.34 +9.33 +3.62
Relative (%) +0.8 +34.5 +7.3 +14.2 -49.1 +14.2 +1.0 +9.1 +36.3 -19.2 +41.2 +16.0
Steps
(reduced) 		196
(7) 		202
(4) 		207
(0) 		212
(5) 		216
(0) 		221
(5) 		225
(0) 		229
(4) 		233
(8) 		236
(2) 		240
(6) 		243
(0)

Subsets and supersets

9ed9/8 is the first odd composite ed9/8, containing 3ed9/8 as a subset.

Intervals

# Cents Ratio
0 0.0 1/1
1 22.7 (9/8)1/9
2 45.3 (9/8)2/9
3 68.0 (9/8)1/3
4 90.6 (9/8)4/9
5 113.3 (9/8)5/9
6 135.9 (9/8)2/3
7 158.6 (9/8)7/9
8 181.3 (9/8)8/9
9 203.9 9/8
10 226.6 (9/8)10/9
11 249.2 (9/8)11/9
12 271.9 (9/8)4/3
13 294.5 (9/8)13/9
14 317.2 (9/8)14/9
15 339.9 (9/8)5/3
16 362.5 (9/8)16/9
17 385.2 (9/8)17/9
18 407.8 (9/8)2 = 81/64
19 430.5 (9/8)19/9
20 453.1 (9/8)20/9
21 475.8 (9/8)7/3
22 498.4 (9/8)22/9
23 521.1 (9/8)23/9
24 543.8 (9/8)8/3
25 566.4 (9/8)25/9
26 589.1 (9/8)26/9
27 611.7 (9/8)3 = 729/512
28 634.4 (9/8)28/9
29 657.0 (9/8)29/9
30 679.7 (9/8)10/3
31 702.4 (9/8)31/9
32 725.0 (9/8)32/9
33 747.7 (9/8)11/3
34 770.3 (9/8)34/9
35 792.0 (9/8)35/9
36 815.6 (9/8)4 = 6561/4096
37 838.3 (9/8)37/9
38 861.0 (9/8)38/9
39 883.6 (9/8)13/3
40 906.3 (9/8)40/9
41 928.9 (9/8)41/9
42 951.6 (9/8)14/3
43 974.2 (9/8)43/9
44 996.9 (9/8)44/9
45 1019.6 (9/8)5 = 59049/32768
46 1042.2 (9/8)46/9
47 1064.9 (9/8)47/9
48 1087.5 (9/8)16/3
49 1110.2 (9/8)49/9
50 1132.8 (9/8)50/9
51 1155.5 (9/8)17/3
52 1178.1 (9/8)52/9
53 1200.8 (9/8)53/9
54 1223.5 (9/8)6 = 531441/262144

See also

Retrieved from "https://en.xen.wiki/index.php?title=9ed9/8&oldid=188001"