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212edo: Difference between revisions

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'''212 equal temperament''' divides the octave into 212 equal parts of 5.660 cents each.  
'''212 equal temperament''' divides the octave into 212 equal parts of 5.660 cents each.  


212 = 4 × 53, and it shares the 3rd, 5th, and 13th [[Overtone series|harmonics]] with [[53edo]], but the mapping differs for 7 and 11. It tempers out the same commas ([[32805/32768]], [[15625/15552]] and [[amity comma|1600000/1594323]]) as 53 in the [[5-limit]]. In the [[7-limit]] it tempers out [[2401/2400]], [[dimcomp comma|390625/388962]] and [[canousma|4802000/4782969]]; in the [[11-limit]] [[385/384]], 1375/1372, 6250/6237, [[9801/9800]] and [[semicanousma|14641/14580]]; and in the [[13-limit]] [[325/324]], 625/624, 676/675 and 1001/1000.  
== Theory ==
 
212 = 4 × 53, and it shares the 3rd, 5th, and 13th [[Overtone series|harmonics]] with [[53edo]], but the mapping differs for 7 and 11.  
 
It tempers out the same commas ([[15625/15552]], [[32805/32768]], and [[amity comma|1600000/1594323]]) as 53edo in the [[5-limit]].  
 
In the [[7-limit]], it tempers out 2401/2400 ([[breedsma]]), 390625/388962 ([[dimcomp comma|dimcomp comma]]), and 4802000/4782969 ([[canousma|canousma]]).
 
In the [[11-limit]], 385/384 ([[keenanisma]]), 1375/1372 ([[moctdel comma]]), 6250/6237 ([[liganellus comma]]), 9801/9800 ([[kalisma]]) and 14641/14580 ([[semicanousma]]).
 
In the [[13-limit]], 325/324 ([[marveltwin comma]]), 625/624 ([[tunbarsma]]), 676/675 ([[island comma]]), 1001/1000 ([[sinbadma]]), 1716/1715 ([[lummic comma]]), 2080/2079 ([[ibnsinma]]).  


It is distinctly [[consistent]] in the [[15-odd-limit]] with harmonics of 3 through 13 all tuned flat. It is the [[optimal patent val]] for 7 and 13 limit [[Kleismic family #Quadritikleismic|quadritikleismic temperament]], and the 13-limit rank three [[Breed family #Agni|agni temperament]]. 212gh val shows some potential beyond 15-odd-limit. Also, using 212bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning almost identical to the POTE tuning for 5-limit meantone.
It is distinctly [[consistent]] in the [[15-odd-limit]] with harmonics of 3 through 13 all tuned flat. It is the [[optimal patent val]] for 7 and 13 limit [[Kleismic family #Quadritikleismic|quadritikleismic temperament]], and the 13-limit rank three [[Breed family #Agni|agni temperament]]. 212gh val shows some potential beyond 15-odd-limit. Also, using 212bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning almost identical to the POTE tuning for 5-limit meantone.


== Just approximation ==
=== Primary intervals ===
{| class="wikitable center-all"
{{Primes in edo|212|prec=2|columns=11}}
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
! prime 23
! prime 29
! prime 31
|-
! rowspan="2" |Error
! absolute (¢)
| 0.00
| -0.07
| -1.41
| -0.90
| -2.26
| -2.79
| +2.59
| +2.49
| +0.03
| +0.61
| -1.64
|-
! [[Relative error|relative]] (%)
| 0.0
| -1.2
| -24.9
| -15.9
| -40.0
| -49.3
| +45.8
| +43.9
| +0.5
| +10.8
| -29.0
|}


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
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