5809edo: Difference between revisions
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{{ | {{ED intro}} | ||
5809edo is a fairly strong [[19-limit]] system, [[consistent]] to the [[21-odd-limit]], | 5809edo is a fairly strong [[19-limit]] system, [[consistent]] to the [[21-odd-limit]]. Moreover, it is a strong no-23 [[43-limit]] system, consistent to the no-23 45-odd-limit. Its full 43-limit interpretation using the [[patent val]] is also obvious, as [[23/22]], [[23/13]], [[37/23]] and their [[octave complement]]s exhaust the inconsistently mapped intervals in the full 45-odd-limit. | ||
We may note that it is an [[egads]] and [[euzenius]] system, [[support]]ing [[hemiegads]]. Some simpler | We may note that it is an [[egads]] and [[euzenius]] system, [[support]]ing [[hemiegads]]. Some simpler [[comma]]s it [[tempering out|tempers out]] in the higher limits include [[123201/123200]] in the [[13-limit]]; [[14400/14399]], [[194481/194480]], and [[336141/336140]] in the [[17-limit]]; 10830/10829, 23409/23408, 28900/28899, 43681/43680, and 89376/89375 in the 19-limit; 7866/7865, 8625/8624, 21505/21054, and [[25921/25920]] in the [[23-limit]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|5809|columns= | {{Harmonics in equal|5809|columns=11}} | ||
{{Harmonics in equal|5809|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 5809edo (continued)}} | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 5809 factors into 37 × 157, 5809edo contains [[37edo]] and [[157edo]] as subsets. | Since 5809 factors into primes as {{nowrap| 37 × 157 }}, 5809edo contains [[37edo]] and [[157edo]] as subsets. | ||
Latest revision as of 16:08, 2 October 2025
| ← 5808edo | 5809edo | 5810edo → |
5809 equal divisions of the octave (abbreviated 5809edo or 5809ed2), also called 5809-tone equal temperament (5809tet) or 5809 equal temperament (5809et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 5809 equal parts of about 0.207 ¢ each. Each step represents a frequency ratio of 21/5809, or the 5809th root of 2.
5809edo is a fairly strong 19-limit system, consistent to the 21-odd-limit. Moreover, it is a strong no-23 43-limit system, consistent to the no-23 45-odd-limit. Its full 43-limit interpretation using the patent val is also obvious, as 23/22, 23/13, 37/23 and their octave complements exhaust the inconsistently mapped intervals in the full 45-odd-limit.
We may note that it is an egads and euzenius system, supporting hemiegads. Some simpler commas it tempers out in the higher limits include 123201/123200 in the 13-limit; 14400/14399, 194481/194480, and 336141/336140 in the 17-limit; 10830/10829, 23409/23408, 28900/28899, 43681/43680, and 89376/89375 in the 19-limit; 7866/7865, 8625/8624, 21505/21054, and 25921/25920 in the 23-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | -0.0097 | -0.0166 | +0.0155 | +0.0334 | +0.0301 | -0.0148 | -0.0436 | -0.0767 | -0.0024 | +0.0152 |
| Relative (%) | +0.0 | -4.7 | -8.0 | +7.5 | +16.2 | +14.6 | -7.2 | -21.1 | -37.1 | -1.2 | +7.4 | |
| Steps (reduced) |
5809 (0) |
9207 (3398) |
13488 (1870) |
16308 (4690) |
20096 (2669) |
21496 (4069) |
23744 (508) |
24676 (1440) |
26277 (3041) |
28220 (4984) |
28779 (5543) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0590 | -0.0040 | -0.0355 | +0.0813 | -0.1012 | -0.0565 | +0.0716 | +0.0182 | +0.0246 | +0.0639 | +0.0697 |
| Relative (%) | +28.5 | -2.0 | -17.2 | +39.3 | -49.0 | -27.3 | +34.7 | +8.8 | +11.9 | +30.9 | +33.8 | |
| Steps (reduced) |
30262 (1217) |
31122 (2077) |
31521 (2476) |
32267 (3222) |
33273 (4228) |
34172 (5127) |
34452 (5407) |
35238 (384) |
35724 (870) |
35957 (1103) |
36619 (1765) | |
Subsets and supersets
Since 5809 factors into primes as 37 × 157, 5809edo contains 37edo and 157edo as subsets.