417edo
| ← 416edo | 417edo | 418edo → |
417 equal divisions of the octave (abbreviated 417edo or 417ed2), also called 417-tone equal temperament (417tet) or 417 equal temperament (417et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 417 equal parts of about 2.88 ¢ each. Each step represents a frequency ratio of 21/417, or the 417th root of 2.
Theory
417et is only consistent to the 5-odd-limit. Using the patent val, it tempers out 589824/588245, 43046721/43025920, 33554432/33480783 and 65625/65536 in the 7-limit; 78121827/77948684, 20155392/20131375, 10333575/10307264, 1019215872/1019046875, 46656/46585, 1366875/1362944, 78675968/78594219, 536870912/535869675, 7168000/7144929, 496125/495616, 514714375/514434888, 2359296/2358125, 540/539, 1265625/1261568, 17561600/17537553, 180224/180075, 1375/1372, 645922816/645700815, 3025/3024, 9453125/9437184 and 1362944/1361367 in the 11-limit. It supports familia and 5-limit fortune.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.20 | -0.70 | +0.96 | +1.20 | -0.24 | -1.36 | -1.11 | -0.94 | +0.64 | +0.29 |
| Relative (%) | +0.0 | +7.1 | -24.4 | +33.3 | +41.7 | -8.3 | -47.2 | -38.6 | -32.5 | +22.2 | +10.0 | |
| Steps (reduced) |
417 (0) |
661 (244) |
968 (134) |
1171 (337) |
1443 (192) |
1543 (292) |
1704 (36) |
1771 (103) |
1886 (218) |
2026 (358) |
2066 (398) | |
Subsets and supersets
417 factors into 3 × 139, with 3edo and 139edo as its subset edos.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [661 -417⟩ | [⟨417 661]] | −0.0641 | 0.0641 | 2.23 |
| 2.3.5 | 1600000/1594323, [-80 8 29⟩ | [⟨417 661 968]] | +0.0580 | 0.1806 | 6.28 |
| 2.3.5.7 | 16875/16807, 65625/65536, 1600000/1594323 | [⟨417 661 968 1171]] | −0.0418 | 0.2331 | 8.10 |
| 2.3.5.7.11 | 540/539, 3025/3024, 496125/495616, 7168000/7144929 | [⟨417 661 968 1171 1443]] | −0.1029 | 0.2416 | 8.40 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 77\417 | 221.58 | 8388608/7381125 | Fortune |
| 1 | 118\417 | 339.57 | 243/200 | Amity |
| 1 | 121\417 | 348.20 | 60/49 | Eris |
| 3 | 39\417 | 112.23 | 16/15 | Tertiosec |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct