72edo: Difference between revisions
Error table correction, +relative error |
+Temperament measures. Also moving "Z function" into "just approximation" section |
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| Line 633: | Line 633: | ||
== Just approximation == | == Just approximation == | ||
=== Selected just intervals === | |||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! colspan="2" | | ! colspan="2" | | ||
| Line 648: | Line 650: | ||
|- | |- | ||
! rowspan="2" |Error | ! rowspan="2" |Error | ||
!absolute (¢) | ! absolute (¢) | ||
| 0.000 | | 0.000 | ||
| -1.955 | | -1.955 | ||
| Line 661: | Line 663: | ||
| +4.964 | | +4.964 | ||
|- | |- | ||
![[Relative error|relative]] (%) | ! [[Relative error|relative]] (%) | ||
|0.0 | | 0.0 | ||
| -11.7 | | -11.7 | ||
| -17.9 | | -17.9 | ||
| Line 674: | Line 676: | ||
| +29.8 | | +29.8 | ||
|} | |} | ||
=== Temperament measures === | |||
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 72et. | |||
{| class="wikitable center-all" | |||
! colspan="2" | | |||
! 3-limit | |||
! 5-limit | |||
! 7-limit | |||
! 11-limit | |||
! 13-limit | |||
! 17-limit | |||
! 19-limit | |||
|- | |||
! colspan="2" |Octave stretch (¢) | |||
| +0.617 | |||
| +0.839 | |||
| +0.822 | |||
| +0.734 | |||
| +0.936 | |||
| +0.975 | |||
| +0.780 | |||
|- | |||
! rowspan="2" |Error | |||
! [[TE error|absolute]] (¢) | |||
| 0.617 | |||
| 0.594 | |||
| 0.515 | |||
| 0.493 | |||
| 0.638 | |||
| 0.599 | |||
| 0.762 | |||
|- | |||
! [[TE simple badness|relative]] (%) | |||
| 3.70 | |||
| 3.56 | |||
| 3.09 | |||
| 2.96 | |||
| 3.82 | |||
| 3.59 | |||
| 4.57 | |||
|} | |||
* 72et has a lower relative error than any previous ETs in the 7-, 11-, 13-, 17-, and 19-limit. The next ET that does better in these subgroups is 99, 270, 224, 494, and 217, respectively. | |||
=== Z function === | |||
72edo is the ninth [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[The_Riemann_Zeta_Function_and_Tuning #The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72. | |||
[[File:plot72.png|alt=plot72.png|plot72.png]] | |||
== Commas == | == Commas == | ||
| Line 1,108: | Line 1,157: | ||
| | | | | | ||
|} | |} | ||
== Music == | == Music == | ||
| Line 1,129: | Line 1,173: | ||
* [http://www.ekmelic-music.org/en/ Ekmelic Music Society/Gesellschaft für Ekmelische Musik], a group of composers and researchers dedicated to 72edo music | * [http://www.ekmelic-music.org/en/ Ekmelic Music Society/Gesellschaft für Ekmelische Musik], a group of composers and researchers dedicated to 72edo music | ||
* [http://72note.com/site/original.html Rick Tagawa's 72edo site], including theory and composers' list | * [http://72note.com/site/original.html Rick Tagawa's 72edo site], including theory and composers' list | ||
* [http://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo] | * [http://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo] [[Category:Edo]] [[Category:72edo| ]] <!-- main article --> | ||
[[Category:Listen]] | [[Category:Listen]] | ||
[[Category:Marvel]] | [[Category:Marvel]] | ||