422edo: Difference between revisions
Jump to navigation
Jump to search
+prime error table |
+infobox and improve intro |
||
| Line 1: | Line 1: | ||
The '''422 equal | {{Infobox ET | ||
| Prime factorization = 2 × 211 | |||
| Step size = 2.84360¢ | |||
| Fifth = 247\422 (702.37¢) | |||
| Semitones = 41:31 (116.59¢ : 88.15¢) | |||
| Consistency = 27 | |||
}} | |||
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. A basis for the 11-limit is 3025/3024, 4375/4374, 5632/5625 and 825000/823543. | 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. A basis for the 11-limit is 3025/3024, 4375/4374, 5632/5625 and 825000/823543. | ||
Revision as of 13:55, 28 January 2022
| ← 421edo | 422edo | 423edo → |
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 cents each.
422edo is a 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, [23 6 -14⟩; and in the 7-limit 4375/4374 and 589824/588245 so that it supports the gamera temperament, and provides its optimal patent val, and also supports the vishnu temperament. A basis for the 11-limit is 3025/3024, 4375/4374, 5632/5625 and 825000/823543.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.41 | +0.42 | +0.84 | +0.34 | +1.18 | +0.26 | +1.07 | +0.16 | -0.19 | +0.94 |
| Relative (%) | +0.0 | +14.6 | +14.6 | +29.6 | +12.0 | +41.4 | +9.1 | +37.5 | +5.7 | -6.8 | +32.9 | |
| Steps (reduced) |
422 (0) |
669 (247) |
980 (136) |
1185 (341) |
1460 (194) |
1562 (296) |
1725 (37) |
1793 (105) |
1909 (221) |
2050 (362) |
2091 (403) | |