4327edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | |- | ||
! rowspan="2" |[[Comma list | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" |Optimal<br>8ve | ! rowspan="2" | [[Mapping]] | ||
! colspan="2" |Tuning | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
|- | ! colspan="2" | Tuning error | ||
![[TE error|Absolute]] (¢) | |- | ||
![[TE simple badness|Relative]] (%) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo|-6858 4327}} | | {{monzo|-6858 4327}} | ||
| {{mapping|4327 6858}} | | {{mapping|4327 6858}} | ||
| 0.0116 | | +0.0116 | ||
| 0.0116 | | 0.0116 | ||
| 4.18 | | 4.18 | ||
| Line 32: | Line 33: | ||
| {{monzo|-69 45 -1}}, {{monzo|96 83 -98}} | | {{monzo|-69 45 -1}}, {{monzo|96 83 -98}} | ||
| {{mapping|4327 6858 10047}} | | {{mapping|4327 6858 10047}} | ||
| 0.0071 | | +0.0071 | ||
| 0.0115 | | 0.0115 | ||
| 4.15 | | 4.15 | ||
| Line 39: | Line 40: | ||
| {{monzo|-9 5 -8 7}}, {{monzo|3 -24 3 10}}, {{monzo|-57 16 10 3}} | | {{monzo|-9 5 -8 7}}, {{monzo|3 -24 3 10}}, {{monzo|-57 16 10 3}} | ||
| {{mapping|4327 6858 10047 12147}} | | {{mapping|4327 6858 10047 12147}} | ||
| 0.0158 | | +0.0158 | ||
| 0.0181 | | 0.0181 | ||
| 6.53 | | 6.53 | ||
| Line 46: | Line 47: | ||
| 117649/117612, 50014503/50000000, 234365481/234256000, 369140625/369098752 | | 117649/117612, 50014503/50000000, 234365481/234256000, 369140625/369098752 | ||
| {{mapping|4327 6858 10047 12147 14969}} | | {{mapping|4327 6858 10047 12147 14969}} | ||
| 0.0120 | | +0.0120 | ||
| 0.0179 | | 0.0179 | ||
| 6.45 | | 6.45 | ||
| Line 53: | Line 54: | ||
| 123201/123200, 4100625/4100096, 1990656/1990625, 4917248/4915625, 3195731/3194880 | | 123201/123200, 4100625/4100096, 1990656/1990625, 4917248/4915625, 3195731/3194880 | ||
| {{mapping|4327 6858 10047 12147 14969 16012}} | | {{mapping|4327 6858 10047 12147 14969 16012}} | ||
| 0.0075 | | +0.0075 | ||
| 0.0191 | | 0.0191 | ||
| 6.89 | | 6.89 | ||
Latest revision as of 13:08, 21 February 2025
| ← 4326edo | 4327edo | 4328edo → |
4327 equal divisions of the octave (abbreviated 4327edo or 4327ed2), also called 4327-tone equal temperament (4327tet) or 4327 equal temperament (4327et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4327 equal parts of about 0.277 ¢ each. Each step represents a frequency ratio of 21/4327, or the 4327th root of 2.
Theory
4327edo is consistent to the 11-odd-limit, tempering out 117649/117612, 50014503/50000000, 234365481/234256000 and 369140625/369098752. It supports mikkolic.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.037 | +0.005 | -0.118 | -0.074 | +0.011 | +0.055 | -0.032 | -0.125 | +0.060 | +0.123 | -0.126 |
| Relative (%) | -13.3 | +1.7 | -42.5 | -26.5 | +3.9 | +19.7 | -11.6 | -45.2 | +21.8 | +44.3 | -45.3 | |
| Steps (reduced) |
6858 (2531) |
10047 (1393) |
12147 (3493) |
13716 (735) |
14969 (1988) |
16012 (3031) |
16905 (3924) |
17686 (378) |
18381 (1073) |
19006 (1698) |
19573 (2265) | |
Subsets and supersets
4327edo is the 591st prime EDO.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-6858 4327⟩ | [⟨4327 6858]] | +0.0116 | 0.0116 | 4.18 |
| 2.3.5 | [-69 45 -1⟩, [96 83 -98⟩ | [⟨4327 6858 10047]] | +0.0071 | 0.0115 | 4.15 |
| 2.3.5.7 | [-9 5 -8 7⟩, [3 -24 3 10⟩, [-57 16 10 3⟩ | [⟨4327 6858 10047 12147]] | +0.0158 | 0.0181 | 6.53 |
| 2.3.5.7.11 | 117649/117612, 50014503/50000000, 234365481/234256000, 369140625/369098752 | [⟨4327 6858 10047 12147 14969]] | +0.0120 | 0.0179 | 6.45 |
| 2.3.5.7.11.13 | 123201/123200, 4100625/4100096, 1990656/1990625, 4917248/4915625, 3195731/3194880 | [⟨4327 6858 10047 12147 14969 16012]] | +0.0075 | 0.0191 | 6.89 |