User:Francium/4597edo
| ← 4596edo | 4597edo | 4598edo → |
4597 equal divisions of the octave (abbreviated 4597edo or 4597ed2), also called 4597-tone equal temperament (4597tet) or 4597 equal temperament (4597et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4597 equal parts of about 0.261 ¢ each. Each step represents a frequency ratio of 21/4597, or the 4597th root of 2.
Theory
4597edo is consistent to the 5-limit, its harmonic 7 is about halfway its steps. It is strong in the 2.3.5.11.13.17.23.29 subgroup, tempering out 20736/20735, 359424/359375, 3408075/3407872, 13311/13310, 486243/486200, 2460375/2460172, 1069056/1068925. Using the 2.5.11.29.37 subgroup, it tempers out 9251/9250.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.019 | +0.025 | -0.107 | -0.002 | +0.021 | -0.017 | +0.072 | +0.048 | -0.036 | -0.115 |
| Relative (%) | +0.0 | -7.3 | +9.7 | -41.1 | -0.7 | +7.9 | -6.7 | +27.7 | +18.6 | -13.9 | -44.0 | |
| Steps (reduced) |
4597 (0) |
7286 (2689) |
10674 (1480) |
12905 (3711) |
15903 (2112) |
17011 (3220) |
18790 (402) |
19528 (1140) |
20795 (2407) |
22332 (3944) |
22774 (4386) | |
Subsets and supersets
4597edo is the 622nd prime edo. 9194edo, which doubles it, gives a good correction to its harmonic 7.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-7286 4597⟩ | [⟨4597 7286]] | 0.0060 | 0.0060 | 2.30 |
| 2.3.5 | [-16 35 -17⟩, [342 -81 -92⟩ | [⟨4597 7286 10674]] | 0.0004 | 0.0093 | 3.56 |
| 2.3.5.7 | 200120949/200000000, [-7 30 -9 -7⟩, [47 -18 -14 5⟩ | [⟨4597 7286 10674 12905]] | 0.0098 | 0.0183 | 7.01 |
| 2.3.5.7.11 | 117649/117612, 50014503/50000000, 14348907/14348180, 6576668672/6576582375 | [⟨4597 7286 10674 12905 15903]] | 0.0080 | 0.0167 | 6.40 |
| 2.3.5.7.11.13 | 123201/123200, 1990656/1990625, 117649/117612, 1574640/1574573, 3195731/3194880 | [⟨4597 7286 10674 12905 15903 17011]] | 0.0057 | 0.0161 | 6.17 |
| 2.3.5.7.11.13.17 | 12376/12375, 123201/123200, 1990656/1990625, 790272/790075, 637637/637500, 163863/163840 | [⟨4597 7286 10674 12905 15903 17011 18790]] | 0.0055 | 0.0149 | 5.71 |
| 2.3.5.7.11.13.17.19 | 12376/12375, 10241/10240, 5929/5928, 123201/123200, 1549184/1549125, 203148/203125, 1285956/1285625 | [⟨4597 7286 10674 12905 15903 17011 18790 19528]] | 0.0027 | 0.0158 | 4.44 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 699\4597 | 182.4668 | 10/9 | Minortone |