9ed9/8: Difference between revisions
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Cleanup; -virtually duplicate stuff from 53edo that fails to undo octave reduction |
+subsets and supersets; +see also |
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{{Harmonics in equal|9|9|8}} | {{Harmonics in equal|9|9|8}} | ||
{{Harmonics in equal|9|9|8|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 9ed9/8 (continued)}} | {{Harmonics in equal|9|9|8|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 9ed9/8 (continued)}} | ||
=== Subsets and supersets === | |||
9ed9/8 is the first odd composite ed9/8, containing [[3ed9/8]] as a subset. | |||
== Intervals == | == Intervals == | ||
| Line 236: | Line 239: | ||
| (9/8)<sup>6</sup> = 531441/262144 | | (9/8)<sup>6</sup> = 531441/262144 | ||
|} | |} | ||
== See also == | |||
* [[31edf]] – relative edf | |||
* [[53edo]] – relative edo | |||
* [[84edt]] – relative edt | |||
* [[137ed6]] – relative ed6 | |||
Revision as of 10:54, 24 March 2025
| ← 8ed9/8 | 9ed9/8 | 10ed9/8 → |
(convergent)
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
| 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) | |
| 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 | 0.0000 | 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 |