509edo
| ← 508edo | 509edo | 510edo → |
509 equal divisions of the octave (abbreviated 509edo or 509ed2), also called 509-tone equal temperament (509tet) or 509 equal temperament (509et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 509 equal parts of about 2.36 ¢ each. Each step represents a frequency ratio of 21/509, or the 509th root of 2.
Theory
509edo has a sharp tendency in lower harmonics. The 13-limit optimal tuning of this temperament is consistent to the 15-integer-limit, so one might want to keep the octave compression tight.
The equal temperament tempers out 1600000/1594323 (amity comma) in the 5-limit; 2401/2400 and 29360128/29296875 in the 7-limit; and 3025/3024, 5632/5625, 41503/41472, 42592/42525, 151263/151250, 172032/171875, 180224/180075, 322102/321489, 422576/421875, 456533/455625, and 1953125/1948617 in the 11-limit. It provides the optimal patent val for petrtri, the 2.11/5.13/5 subgroup temperament tempering out 2200/2197.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.60 | +0.33 | +0.13 | -1.16 | +0.35 | +1.12 | +0.93 | +1.13 | -0.46 | +0.73 | -1.16 |
| Relative (%) | +25.4 | +13.9 | +5.6 | -49.2 | +14.9 | +47.6 | +39.3 | +48.1 | -19.5 | +31.0 | -49.3 | |
| Steps (reduced) |
807 (298) |
1182 (164) |
1429 (411) |
1613 (86) |
1761 (234) |
1884 (357) |
1989 (462) |
2081 (45) |
2162 (126) |
2236 (200) |
2302 (266) | |
Subsets and supersets
509edo is the 97th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [807 -509⟩ | [⟨509 807]] | −0.1890 | 0.1889 | 8.01 |
| 2.3.5 | [9 -13 5⟩, [93 -3 -38⟩ | [⟨509 807 1182]] | −0.1729 | 0.1559 | 6.61 |
| 2.3.5.7 | 2401/2400, 1600000/1594323, 29360128/29296875 | [⟨509 807 1182 1429]] | −0.1415 | 0.1456 | 6.18 |
| 2.3.5.7.11 | 2401/2400, 3025/3024, 5632/5625, 1600000/1594323 | [⟨509 807 1182 1429 1761]] | −0.1335 | 0.1312 | 5.57 |
| 2.3.5.7.11.13 | 2080/2079, 2200/2197, 2401/2400, 3025/3024, 5632/5625 | [⟨509 807 1182 1429 1761 1884]] | −0.1618 | 0.1354 | 5.74 |
| 2.3.5.7.11.13.17 | 1225/1224, 2080/2079, 2200/2197, 2401/2400, 2431/2430, 4914/4913 | [⟨509 807 1182 1429 1761 1884 2081]] | −0.1784 | 0.1318 | 5.59 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 18\509 | 42.44 | 40/39 | Humorous |
| 1 | 36\509 | 84.87 | 21/20 | Amicable |
| 1 | 115\509 | 271.12 | 1024/875 | Quasiorwell |
| 1 | 144\509 | 339.49 | 243/200 | Amity |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct