337edo: Difference between revisions
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Created page with "{{Infobox ET}} {{EDO intro|337}} == Theory == 337et tempers out 16875/16807, 1280000000/1275989841, 14348907/14336000, 5250987/5242880, 420175/419904 and 201768035/201..." |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|337}} | {{EDO intro|337}} | ||
== Theory == | == Theory == | ||
337edo is [[consistent]] to the [[9-odd-limit]], but the error of [[harmonic]] [[5/1|5]] is quite large. If the harmonic is used at all, it tends very flat. The equal temperament [[tempering out|tempers out]] [[16875/16807]], 420175/419904, and 5250987/5242880 in the 7-limit. It [[support]]s the [[kleirtismic]] temperament. | |||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|337}} | {{Harmonics in equal|337}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
337edo is the 68th [[prime | 337edo is the 68th [[prime edo]]. | ||
== 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" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|-534 337}} | | {{monzo| -534 337 }} | ||
|{{ | | {{mapping| 337 534 }} | ||
| 0.1487 | | 0.1487 | ||
| 0.1487 | | 0.1487 | ||
| 4.18 | | 4.18 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|15625/15552, {{monzo|-88 57 -1}} | | 15625/15552, {{monzo| -88 57 -1 }} | ||
|{{ | | {{mapping| 337 534 782 }} | ||
| 0.3495 | | 0.3495 | ||
| 0.3089 | | 0.3089 | ||
| 8.67 | | 8.67 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|15625/15552, 16875/16807, 7381125/7340032 | | 15625/15552, 16875/16807, 7381125/7340032 | ||
|{{ | | {{mapping| 337 534 782 946 }} | ||
| 0.2870 | | 0.2870 | ||
| 0.2886 | | 0.2886 | ||
| Line 43: | Line 47: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|67\337 | | 67\337 | ||
|238.58 | | 238.58 | ||
|147/128 | | 147/128 | ||
|[[Tokko]] | | [[Tokko]] | ||
|- | |- | ||
|1 | | 1 | ||
|89\337 | | 89\337 | ||
|316.91 | | 316.91 | ||
|6/5 | | 6/5 | ||
|[[Hanson]] | | [[Hanson]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
Revision as of 08:10, 16 November 2023
| ← 336edo | 337edo | 338edo → |
Theory
337edo is consistent to the 9-odd-limit, but the error of harmonic 5 is quite large. If the harmonic is used at all, it tends very flat. The equal temperament tempers out 16875/16807, 420175/419904, and 5250987/5242880 in the 7-limit. It supports the kleirtismic temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.47 | -1.74 | -0.28 | -0.94 | +0.61 | -0.17 | +1.35 | -1.69 | +1.60 | -0.75 | -1.57 |
| Relative (%) | -13.2 | -49.0 | -7.9 | -26.5 | +17.2 | -4.8 | +37.8 | -47.5 | +44.8 | -21.1 | -44.0 | |
| Steps (reduced) |
534 (197) |
782 (108) |
946 (272) |
1068 (57) |
1166 (155) |
1247 (236) |
1317 (306) |
1377 (29) |
1432 (84) |
1480 (132) |
1524 (176) | |
Subsets and supersets
337edo is the 68th prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-534 337⟩ | [⟨337 534]] | 0.1487 | 0.1487 | 4.18 |
| 2.3.5 | 15625/15552, [-88 57 -1⟩ | [⟨337 534 782]] | 0.3495 | 0.3089 | 8.67 |
| 2.3.5.7 | 15625/15552, 16875/16807, 7381125/7340032 | [⟨337 534 782 946]] | 0.2870 | 0.2886 | 8.10 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 67\337 | 238.58 | 147/128 | Tokko |
| 1 | 89\337 | 316.91 | 6/5 | Hanson |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct