342edo: Difference between revisions
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== Theory == | == Theory == | ||
342edo is a very strong 11-limit system. It is, as one would expect, distinctly consistent through the 11-odd-limit, but goes no higher; nonetheless, it is a [[The Riemann | 342edo is a very strong 11-limit system. It is, as one would expect, distinctly [[consistent]] through the 11-odd-limit, but goes no higher; nonetheless, it is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. A basis for the 11-limit commas is 2401/2400, 3025/3024, 4375/4374 and 32805/32768. It is the optimal patent val for 11-limit [[Breedsmic temperaments #Hemitert|hemitert]] temperament, and [[support]]s hemiennealimmal. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|342|columns=11}} | |||
=== Miscellany === | |||
342 factors as 2 × 3<sup>2</sup> × 19, with subset edos {{EDOs| 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171 }}. | 342 factors as 2 × 3<sup>2</sup> × 19, with subset edos {{EDOs| 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171 }}. | ||
== 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]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 28: | Line 29: | ||
| 1.59 | | 1.59 | ||
|- | |- | ||
| 2.3.5.7.11.13 | | style="border-top: double;" | 2.3.5.7.11.13 | ||
| 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712 | | style="border-top: double;" | 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712 | ||
| [{{val| 342 542 794 960 1183 1265 }}] (342f) | | style="border-top: double;" | [{{val| 342 542 794 960 1183 1265 }}] (342f) | ||
| +0.178 | | style="border-top: double;" | +0.178 | ||
| 0.1618 | | style="border-top: double;" | 0.1618 | ||
| 4.61 | | style="border-top: double;" | 4.61 | ||
|- | |- | ||
| 2.3.5.7.11.13 | | style="border-top: double;" | 2.3.5.7.11.13 | ||
| 625/624, 729/728, 847/845, 1575/1573, 4096/4095 | | style="border-top: double;" | 625/624, 729/728, 847/845, 1575/1573, 4096/4095 | ||
| [{{val| 342 542 794 960 1183 1266 }}] (342) | | style="border-top: double;" | [{{val| 342 542 794 960 1183 1266 }}] (342) | ||
| +0.020 | | style="border-top: double;" | +0.020 | ||
| 0.2061 | | style="border-top: double;" | 0.2061 | ||
| 5.87 | | style="border-top: double;" | 5.87 | ||
|} | |} | ||
* 342et is lower in relative error than any previous | * 342et is lower in relative error than any previous equal temperaments in the 11-limit. Not until [[612edo|612]] do we find a better equal temperament in terms of absolute error, and not until [[1848edo|1848]] do we find one in terms of relative error. | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per | ! Periods<br>per 8ve | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||