422edo: Difference between revisions
+infobox and improve intro |
+RTT table |
||
| Line 8: | Line 8: | ||
The '''422 equal divisions of the octave''' ('''422edo'''), or the '''422(-tone) equal temperament''' ('''422tet''', '''422et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 422 [[equal]] parts of 2.84 [[cent]]s each. | The '''422 equal divisions of the octave''' ('''422edo'''), or the '''422(-tone) equal temperament''' ('''422tet''', '''422et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 422 [[equal]] parts of 2.84 [[cent]]s each. | ||
422edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is distinctly [[consistent]] through the [[27-odd-limit]], with harmonics of 3 through 23 all tuned sharp. In the 5-limit it tempers out the [[vishnuzma]], {{monzo| 23 6 -14 }}; and in the 7-limit [[4375/4374]] and 589824/588245 so that it [[support]]s the [[gamera]] temperament, and provides its [[optimal patent val]], and also supports the [[vishnu]] temperament | == Theory == | ||
422edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is distinctly [[consistent]] through the [[27-odd-limit]], with harmonics of 3 through 23 all tuned sharp. In the 5-limit it tempers out the [[vishnuzma]], {{monzo| 23 6 -14 }}; and in the 7-limit [[4375/4374]] and 589824/588245 so that it [[support]]s the [[gamera]] temperament, and provides its [[optimal patent val]], and also supports the [[vishnu]] temperament. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|422|columns=11}} | {{Harmonics in equal|422|columns=11}} | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | Subgroup | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| 669 -422 }} | |||
| [{{val| 422 669 }}] | |||
| -0.1308 | |||
| 0.1308 | |||
| 4.60 | |||
|- | |||
| 2.3.5 | |||
| {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }} | |||
| [{{val| 422 669 980 }}] | |||
| -0.1469 | |||
| 0.1092 | |||
| 3.84 | |||
|- | |||
| 2.3.5.7 | |||
| 4375/4374, 589824/588245, 29360128/29296875 | |||
| [{{val| 422 669 980 1185 }}] | |||
| -0.1852 | |||
| 0.1155 | |||
| 4.06 | |||
|- | |||
| 2.3.5.7.11 | |||
| 3025/3024, 4375/4374, 5632/5625, 589824/588245 | |||
| [{{val| 422 669 980 1185 1460 }}] | |||
| -0.1679 | |||
| 0.1090 | |||
| 3.83 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625 | |||
| [{{val| 422 669 980 1185 1460 1562 }}] | |||
| -0.1930 | |||
| 0.1142 | |||
| 4.02 | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430 | |||
| [{{val| 422 669 980 1185 1460 1562 1725 }}] | |||
| -0.1744 | |||
| 0.1151 | |||
| 4.05 | |||
|- | |||
| 2.3.5.7.11.13.17.19 | |||
| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197 | |||
| [{{val| 422 669 980 1185 1460 1562 1725 1793 }}] | |||
| -0.1839 | |||
| 0.1106 | |||
| 3.89 | |||
|} | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:Gamera]] | [[Category:Gamera]] | ||
[[Category:Vishnu]] | [[Category:Vishnu]] | ||