589edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
589edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. As every other step of [[1178edo]], the approximations to lower harmonics are not impressive, making it only suitable for a 2.9.15.21.19 [[subgroup]] interpretation, in which case it is identical to 1178edo. The full 17-limit [[patent val]], however, is plausible since all the harmonics from 3 to 17 are tuned sharp. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] 420175/419904 in the 7-limit; [[3025/3024]], 117649/117612, 422576/421875, 456533/455625, 644204/643125, 766656/765625, [[1953125/1948617]], 3294225/3294172, 4302592/4296875, 55296000/55240493, 85937500/85766121, 107495424/107421875 and 781258401/781250000 in the 11-limit. | |||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|589}} | {{Harmonics in equal|589}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
589 factors into | Since 589 factors into {{factorization|589}}, 589edo contains [[19edo]] and [[31edo]] as subsets. [[1178edo]], which doubles it, gives good corrections to harmonics 3, 5, 7, 11, 13, and 17. | ||
== 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 error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.9 | | 2.9 | ||
|{{monzo|-1867 589}} | | {{monzo| -1867 589 }} | ||
|{{ | | {{mapping| 589 1867 }} | ||
| +0.0276 | | +0.0276 | ||
| 0.0276 | | 0.0276 | ||
| 1.35 | | 1.35 | ||
|- | |- | ||
|2.9.5 | | 2.9.5 | ||
|{{monzo|-37 19 -10}}, {{monzo|72 0 -31}} | | {{monzo| -37 19 -10 }}, {{monzo| 72 0 -31 }} | ||
|{{ | | {{mapping| 589 1867 1368 }} | ||
| | | −0.0940 | ||
| 0.1734 | | 0.1734 | ||
| 8.51 | | 8.51 | ||
|} | |} | ||
== Scales == | == Scales == | ||
* [[Kryptonite10]] | * [[Kryptonite10]] | ||
* [[Kryptonite19]] | * [[Kryptonite19]] | ||
Latest revision as of 12:22, 21 February 2025
| ← 588edo | 589edo | 590edo → |
589 equal divisions of the octave (abbreviated 589edo or 589ed2), also called 589-tone equal temperament (589tet) or 589 equal temperament (589et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 589 equal parts of about 2.04 ¢ each. Each step represents a frequency ratio of 21/589, or the 589th root of 2.
Theory
589edo is consistent to the 7-odd-limit, but harmonic 3 is about halfway between its steps. As every other step of 1178edo, the approximations to lower harmonics are not impressive, making it only suitable for a 2.9.15.21.19 subgroup interpretation, in which case it is identical to 1178edo. The full 17-limit patent val, however, is plausible since all the harmonics from 3 to 17 are tuned sharp. Using the patent val, the equal temperament tempers out 420175/419904 in the 7-limit; 3025/3024, 117649/117612, 422576/421875, 456533/455625, 644204/643125, 766656/765625, 1953125/1948617, 3294225/3294172, 4302592/4296875, 55296000/55240493, 85937500/85766121, 107495424/107421875 and 781258401/781250000 in the 11-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.931 | +0.783 | +0.953 | -0.175 | +0.804 | +0.898 | -0.323 | +0.987 | -0.060 | -0.153 | -0.770 |
| Relative (%) | +45.7 | +38.4 | +46.8 | -8.6 | +39.5 | +44.1 | -15.9 | +48.4 | -2.9 | -7.5 | -37.8 | |
| Steps (reduced) |
934 (345) |
1368 (190) |
1654 (476) |
1867 (100) |
2038 (271) |
2180 (413) |
2301 (534) |
2408 (52) |
2502 (146) |
2587 (231) |
2664 (308) | |
Subsets and supersets
Since 589 factors into 19 × 31, 589edo contains 19edo and 31edo as subsets. 1178edo, which doubles it, gives good corrections to harmonics 3, 5, 7, 11, 13, and 17.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-1867 589⟩ | [⟨589 1867]] | +0.0276 | 0.0276 | 1.35 |
| 2.9.5 | [-37 19 -10⟩, [72 0 -31⟩ | [⟨589 1867 1368]] | −0.0940 | 0.1734 | 8.51 |