171edo: Difference between revisions
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171edo is much less accurate in the 11-limit, but still quite useful as it is a good tuning (emphasizing accuracy in the 7-limit) for the important rank-3 temperament [[jove]], which tempers out [[243/242]] (rastma) and [[441/440]], not to mention [[540/539]] and 2401/2400. Jove can be extended by adding [[364/363]] for the 13-limit and 595/594 for the 17-limit, which 171edo also supports. Alternatively, the 171e val can be used, which tempers out [[385/384]]. | 171edo is much less accurate in the 11-limit, but still quite useful as it is a good tuning (emphasizing accuracy in the 7-limit) for the important rank-3 temperament [[jove]], which tempers out [[243/242]] (rastma) and [[441/440]], not to mention [[540/539]] and 2401/2400. Jove can be extended by adding [[364/363]] for the 13-limit and 595/594 for the 17-limit, which 171edo also supports. Alternatively, the 171e val can be used, which tempers out [[385/384]]. | ||
171edo is an excellent | 171edo is an excellent edo for the [[Carlos Gamma]] scale, since the difference between 5 steps of 171edo and 1 step of Carlos Gamma is only -0.010823 cents. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{ | {{Harmonics in equal|171|columns=11}} | ||
== Intervals == | == Intervals == | ||
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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" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[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 51: | Line 51: | ||
| 0.87 | | 0.87 | ||
|- | |- | ||
| 2.3.5.7.11 | | style="border-top: double;" | 2.3.5.7.11 | ||
| 243/242, 441/440, 4375/4356, 16384/16335 | | style="border-top: double;" | 243/242, 441/440, 4375/4356, 16384/16335 | ||
| [{{val| 171 271 397 480 592 }}] (171) | | style="border-top: double;" | [{{val| 171 271 397 480 592 }}] (171) | ||
| -0.093 | | style="border-top: double;" | -0.093 | ||
| 0.401 | | style="border-top: double;" | 0.401 | ||
| 5.71 | | style="border-top: double;" | 5.71 | ||
|- | |- | ||
| 2.3.5.7.11 | | style="border-top: double;" | 2.3.5.7.11 | ||
| 385/384, 1331/1323, 1375/1372, 4375/4374 | | style="border-top: double;" | 385/384, 1331/1323, 1375/1372, 4375/4374 | ||
| [{{val| 171 271 397 480 591 }}] (171e) | | style="border-top: double;" | [{{val| 171 271 397 480 591 }}] (171e) | ||
| +0.312 | | style="border-top: double;" | +0.312 | ||
| 0.418 | | style="border-top: double;" | 0.418 | ||
| 5.96 | | style="border-top: double;" | 5.96 | ||
|} | |} | ||
* 171et is lower in relative error than any previous | * 171et is lower in relative error than any previous equal temperaments in the 7-limit. Not until [[441edo|441]] do we find a better equal temperaments in terms of absolute error, and not until [[3125edo|3125]] 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 | ||
|- | |- | ||