User:Francium/3709edo
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Prime factorization
3709 (prime)
Step size
0.323537 ¢
Fifth
2170\3709 (702.076 ¢)
Semitones (A1:m2)
354:277 (114.5 ¢ : 89.62 ¢)
Dual sharp fifth
2170\3709 (702.076 ¢)
Dual flat fifth
2169\3709 (701.752 ¢)
Dual major 2nd
630\3709 (203.829 ¢)
Consistency limit
5
Distinct consistency limit
5
| ← 3708edo | 3709edo | 3710edo → |
3709 equal divisions of the octave (abbreviated 3709edo or 3709ed2), also called 3709-tone equal temperament (3709tet) or 3709 equal temperament (3709et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3709 equal parts of about 0.324 ¢ each. Each step represents a frequency ratio of 21/3709, or the 3709th root of 2.
Theory
3709edo is consistent to the 5-limit and the error of its harmonic 3 is very high. It is strong in the 2.5.21.11.13.23.31 subgroup, tempering out 213003/212992. 1024023/1024000, 5767168/5767125, 78375843/78369280, 44925111/44921875 and 4178302734375/4177455546368.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.121 | -0.010 | -0.155 | -0.081 | -0.010 | +0.022 | +0.111 | -0.129 | +0.141 | -0.034 | +0.035 |
| Relative (%) | +37.4 | -3.1 | -47.9 | -25.2 | -3.2 | +6.9 | +34.3 | -40.0 | +43.7 | -10.5 | +10.9 | |
| Steps (reduced) |
5879 (2170) |
8612 (1194) |
10412 (2994) |
11757 (630) |
12831 (1704) |
13725 (2598) |
14491 (3364) |
15160 (324) |
15756 (920) |
16291 (1455) |
16778 (1942) | |
Subsets and supersets
3709edo is the 518th prime edo. 7418edo, which doubles it, gives a good correction to its harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-11757 3709⟩ | [⟨3709 11757]] | 0.0129 | 0.0129 | 3.99 |
| 2.9.5 | [-53 5 16⟩, [-175 179 -169⟩ | [⟨3709 11757 8612]] | 0.0100 | 0.0112 | 3.46 |