14348edo: Difference between revisions
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14348edo is a strong 17-limit system, with a lower 17-limit [[relative error]] than any smaller edo aside from [[7033edo|7033]]. It is also distinctly [[consistent]] in the 29-odd-limit, and has a lower 23-limit [[relative error]] than any lower equal temperaments aside from [[2460edo|2460]], [[8269edo|8269]], [[8539edo|8539]] and [[11664edo|11664]]. Besides all that, it is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]], which has to do with its higher limit capability—it has lower relative errors than any smaller equal temperaments in the 41-limit and way beyond. The only inconsistent interval pair in the [[69-odd-limit]] is ([[31/29]], [[58/31]]) with 50.2% relative error. Thus the full [[67-limit]] interpretation using the patent val is obvious. | 14348edo is a strong 17-limit system, with a lower 17-limit [[relative error]] than any smaller edo aside from [[7033edo|7033]]. It is also distinctly [[consistent]] in the 29-odd-limit, and has a lower 23-limit [[relative error]] than any lower equal temperaments aside from [[2460edo|2460]], [[8269edo|8269]], [[8539edo|8539]] and [[11664edo|11664]]. Besides all that, it is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]], which has to do with its higher limit capability—it has lower relative errors than any smaller equal temperaments in the 41-limit and way beyond. The only inconsistent interval pair in the [[69-odd-limit]] is ([[31/29]], [[58/31]]) with 50.2% relative error. Thus the full [[67-limit]] interpretation using the patent val is obvious, though it is best to avoid using primes [[29/1|29]] and [[31/1|31]] at once. | ||
An edo of similar size with full consistency to a very high limit ([[57-odd-limit]]) is [[20567edo]]. | An edo of similar size with full consistency to a very high limit ([[57-odd-limit]]) is [[20567edo]]. | ||
Revision as of 23:22, 22 February 2026
| ← 14347edo | 14348edo | 14349edo → |
14348 equal divisions of the octave (abbreviated 14348edo or 14348ed2), also called 14348-tone equal temperament (14348tet) or 14348 equal temperament (14348et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 14348 equal parts of about 0.0836 ¢ each. Each step represents a frequency ratio of 21/14348, or the 14348th root of 2.
14348edo is a strong 17-limit system, with a lower 17-limit relative error than any smaller edo aside from 7033. It is also distinctly consistent in the 29-odd-limit, and has a lower 23-limit relative error than any lower equal temperaments aside from 2460, 8269, 8539 and 11664. Besides all that, it is a zeta peak, integral and gap edo, which has to do with its higher limit capability—it has lower relative errors than any smaller equal temperaments in the 41-limit and way beyond. The only inconsistent interval pair in the 69-odd-limit is (31/29, 58/31) with 50.2% relative error. Thus the full 67-limit interpretation using the patent val is obvious, though it is best to avoid using primes 29 and 31 at once.
An edo of similar size with full consistency to a very high limit (57-odd-limit) is 20567edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | -0.0035 | -0.0020 | +0.0060 | +0.0063 | +0.0076 | +0.0070 | -0.0221 | -0.0056 | -0.0260 | +0.0160 |
| Relative (%) | +0.0 | -4.2 | -2.4 | +7.2 | +7.5 | +9.1 | +8.3 | -26.4 | -6.7 | -31.1 | +19.1 | |
| Steps (reduced) |
14348 (0) |
22741 (8393) |
33315 (4619) |
40280 (11584) |
49636 (6592) |
53094 (10050) |
58647 (1255) |
60949 (3557) |
64904 (7512) |
69702 (12310) |
71083 (13691) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.0198 | -0.0131 | -0.0039 | -0.0201 | -0.0170 | -0.0136 | -0.0183 | -0.0207 | +0.0358 | +0.0332 | +0.0296 |
| Relative (%) | -23.7 | -15.6 | -4.7 | -24.1 | -20.3 | -16.2 | -21.9 | -24.8 | +42.8 | +39.7 | +35.4 | |
| Steps (reduced) |
74745 (3005) |
76870 (5130) |
77856 (6116) |
79697 (7957) |
82184 (10444) |
84404 (12664) |
85094 (13354) |
87036 (948) |
88237 (2149) |
88812 (2724) |
90447 (4359) | |
Subsets and supersets
14348 factors into primes as 22 × 17 × 211, so 17, 34, 68 and 422 are all subsets.