6079edo: Difference between revisions
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6079edo is a very strong [[11-limit|11-]] and [[13-limit]] system, with a lower 11- and 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any smaller division. It is also a [[zeta peak edo]] and distinctly [[consistent]] through the [[29-odd-limit]]. | 6079edo is a very strong [[11-limit|11-]] and [[13-limit]] system, with a lower 11- and 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any smaller division. It is also a [[zeta peak edo]] and distinctly [[consistent]] through the [[29-odd-limit]]. | ||
We may note it is a [[pirate]], [[euzenius]] and [[starscape]] system. A basis for the 11-limit | We may note it is a [[pirate]], [[euzenius]] and [[starscape]] system. A basis for the 11-limit [[comma]]s is {3294225/3294172, 14348907/14348180, 35156250/35153041, 100663296/100656875}, and for the 13-limit commas, {[[123201/123200]], 1574640/1574573, 1664000/1663893, 1990656/1990625, 3294225/3294172}. | ||
The approximation to [[harmonic]]s [[17/1|17]] and [[23/1|23]] is weaker, though still quite impressive. It [[tempering out|tempers out]] [[14400/14399]], [[28561/28560]], [[31213/31212]], [[37180/37179]], [[194481/194480]], [[336141/336140]] in the 17-limit; 10830/10829, 43681/43680, 89376/89375, 104976/104975, 165376/165375, 228096/228095 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit. Its 2.3.5.7.11.13.19-subgroup is particularly strong, holding the record of lowest relative error until [[8269edo|8269]]. | The approximation to [[harmonic]]s [[17/1|17]] and [[23/1|23]] is weaker, though still quite impressive. It [[tempering out|tempers out]] [[14400/14399]], [[28561/28560]], [[31213/31212]], [[37180/37179]], [[194481/194480]], [[336141/336140]] in the 17-limit; 10830/10829, 43681/43680, 89376/89375, 104976/104975, 165376/165375, 228096/228095 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit. Its 2.3.5.7.11.13.19-subgroup is particularly strong, holding the record of lowest relative error until [[8269edo|8269]]. | ||
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=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|6079}} | {{Harmonics in equal|6079|columns=9}} | ||
{{Harmonics in equal|6079|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 6079edo (continued)}} | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
Revision as of 08:32, 7 March 2025
| ← 6078edo | 6079edo | 6080edo → |
6079 equal divisions of the octave (abbreviated 6079edo or 6079ed2), also called 6079-tone equal temperament (6079tet) or 6079 equal temperament (6079et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 6079 equal parts of about 0.197 ¢ each. Each step represents a frequency ratio of 21/6079, or the 6079th root of 2.
Theory
6079edo is a very strong 11- and 13-limit system, with a lower 11- and 13-limit relative error than any smaller division. It is also a zeta peak edo and distinctly consistent through the 29-odd-limit.
We may note it is a pirate, euzenius and starscape system. A basis for the 11-limit commas is {3294225/3294172, 14348907/14348180, 35156250/35153041, 100663296/100656875}, and for the 13-limit commas, {123201/123200, 1574640/1574573, 1664000/1663893, 1990656/1990625, 3294225/3294172}.
The approximation to harmonics 17 and 23 is weaker, though still quite impressive. It tempers out 14400/14399, 28561/28560, 31213/31212, 37180/37179, 194481/194480, 336141/336140 in the 17-limit; 10830/10829, 43681/43680, 89376/89375, 104976/104975, 165376/165375, 228096/228095 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit. Its 2.3.5.7.11.13.19-subgroup is particularly strong, holding the record of lowest relative error until 8269.
Since it tempers out 12168/12167, it allows vicetertismic chords in the 23-odd-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0026 | -0.0002 | +0.0177 | +0.0227 | +0.0053 | +0.0619 | -0.0299 | +0.0527 |
| Relative (%) | +0.0 | +1.3 | -0.1 | +8.9 | +11.5 | +2.7 | +31.3 | -15.1 | +26.7 | |
| Steps (reduced) |
6079 (0) |
9635 (3556) |
14115 (1957) |
17066 (4908) |
21030 (2793) |
22495 (4258) |
24848 (532) |
25823 (1507) |
27499 (3183) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0658 | +0.0870 | -0.0527 | +0.0871 | -0.0520 | -0.0682 | -0.0056 | +0.0815 | +0.0094 |
| Relative (%) | +33.4 | +44.1 | -26.7 | +44.1 | -26.3 | -34.6 | -2.8 | +41.3 | +4.8 | |
| Steps (reduced) |
29532 (5216) |
30117 (5801) |
31668 (1273) |
32569 (2174) |
32986 (2591) |
33766 (3371) |
34820 (4425) |
35761 (5366) |
36053 (5658) | |
Subsets and supersets
6079edo is the 793rd prime edo.
Music
- Threnody for the Victims of Wolfgang Amadeus Mozart (archived 2010) – 13-limit JI in 6079edo tuning