212edo: Difference between revisions

Line 3:

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.

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]]. 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 ==

{| class="wikitable center-all"

! 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:Edo]]

@@ Line 3: / Line 3: @@
 = 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.
-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]]. 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 ==
+{| class="wikitable center-all"
+! 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:Edo]]