4191814edo
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4191814 equal divisions of the octave (abbreviated 4191814edo or 4191814ed2), also called 4191814-tone equal temperament (4191814tet) or 4191814 equal temperament (4191814et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4191814 equal parts of about 0.000286 ¢ each. Each step represents a frequency ratio of 21/4191814, or the 4191814th root of 2.
This edo has a consistency limit of 21, which is the most impressive out of all the 3-2 telic multiples of 190537edo. It tempers out the archangelic comma in the 3-limit, and though this system's 5-limit and 7-limit are rather lackluster for an edo this size, the representation of the prime 11 is a bit better, and the representations of the 13, 17, and 19 are excellent, all which help to bridge the lackluster 5 and 7. Thus, this system is worthy of a great deal of further exploration in the 19-limit.
In this system, the perfect fifth at 2452054\4191814 is divisible by the prime factors of 2, 11, 227 and 491. However, the perfect fourth, at 1739760\4191814, has more prime divisors, namely the prime factors of 24, 3, 5, 11 and 659. The latter means that just as in 159edo, the perfect fourth is divisible by 33, and thus, this system can offer not only a more accurate version of Ozan Yarman's original 79-tone system, but also metatemperaments to yarman I and yarman II.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000000 | +0.000000 | +0.000087 | +0.000096 | +0.000031 | -0.000005 | +0.000011 | -0.000006 | -0.000097 |
| Relative (%) | +0.0 | +0.0 | +30.5 | +33.5 | +10.9 | -1.7 | +3.8 | -2.2 | -33.7 | |
| Steps (reduced) |
4191814 (0) |
6643868 (2452054) |
9733091 (1349463) |
11767910 (3384282) |
14501294 (1925852) |
15511555 (2936113) |
17133884 (366628) |
17806522 (1039266) |
18961930 (2194674) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000072 | -0.000130 | +0.000129 | -0.000143 | +0.000124 | -0.000090 | -0.000044 | -0.000141 | +0.000057 |
| Relative (%) | +25.3 | -45.3 | +45.0 | -49.9 | +43.4 | -31.3 | -15.2 | -49.1 | +19.8 | |
| Steps (reduced) |
20363753 (3596497) |
20767069 (3999813) |
21837060 (877990) |
22457861 (1498791) |
22745893 (1786823) |
23283803 (2324733) |
24010377 (3051307) |
24658945 (3699875) |
24860548 (3901478) | |