31edo: Difference between revisions

Line 8:

| Prime factorization = 31 (prime)

| Step size = 38.710¢

| Fifth ~~type~~ = 13\31 = 696.77¢

| Fifth = 13\31 = 696.77¢

| Major 2nd = 5\31 = 194¢

| Minor 2nd = 3\31 = 116¢

Line 18:

== Theory ==

{| class="wikitable center-all"

! colspan="2" |

! colspan="2" |

!prime 2

! prime 2

!prime 3

! prime 3

!prime 5

! prime 5

!prime 7

! prime 7

!prime 11

! prime 11

!prime 13

! prime 13

!prime 17

! prime 17

!prime 19

! prime 19

!prime 23

! prime 23

|-

! rowspan="2" |Error

! rowspan="2" | Error

!absolute (¢)

! absolute (¢)

| 0

| -5.18

| +0.8

| -1.1

| -9.4

| +11.1

| +11.2

| +12.2

| -8.9

|-

![[Relative error|relative]] (%)

! [[Relative error|relative]] (%)

| 0

| -13

| +2

| -3

| -24

| +29

| +31

| -23

|-

! colspan="2" |[[nearest edomapping]]

! colspan="2" | [[nearest edomapping]]

|31

| 31

|18

| 18

|10

| 10

|25

| 25

|14

| 14

|22

| 22

|3

| 3

|8

| 8

|16

| 16

|-

! colspan="2" |[[fifthspan]]

! colspan="2" | [[fifthspan]]

| 0

| +1

| +4

| +10

| -13

| +15

| -5

| -3

| -6

|}Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit]] fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

|}

Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit]] fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

31edo's 12\31 generator (an approximate 21/16) supports [[A-Team]] and yields [[13edo#Modes_and_Harmony_in_the_Oneirotonic_Scale|8-note "oneirotonic" scales similar to those in 13edo]] but with the 9/8 and 5/4 better in tune; this temperament is also represented by [[13edo]], [[18edo]] and [[44edo]].

@@ Line 8: / Line 8: @@
 | Prime factorization = 31 (prime)
 | Step size = 38.710¢
-| Fifth type = 13\31 = 696.77¢
+| Fifth = 13\31 = 696.77¢
 | Major 2nd = 5\31 = 194¢
 | Minor 2nd = 3\31 = 116¢
@@ Line 18: / Line 18: @@
 == Theory ==
 {| class="wikitable center-all"
-! colspan="2" |
+! colspan="2" | <!-- empty cell -->
-!prime 2
+! prime 2
-!prime 3
+! prime 3
-!prime 5
+! prime 5
-!prime 7
+! prime 7
-!prime 11
+! prime 11
-!prime 13
+! prime 13
-!prime 17
+! prime 17
-!prime 19
+! prime 19
-!prime 23
+! prime 23
 |-
-! rowspan="2" |Error
+! rowspan="2" | Error
-!absolute (¢)
+! absolute (¢)
 | 0
-|  -5.18
+| -5.18
-|  +0.8
+| +0.8
-|  -1.1
+| -1.1
-|  -9.4
+| -9.4
-|  +11.1
+| +11.1
-|  +11.2
+| +11.2
-|  +12.2
+| +12.2
-|  -8.9
+| -8.9
 |-
-![[Relative error|relative]] (%)
+! [[Relative error|relative]] (%)
 | 0
-|  -13
+| -13
-|  +2
+| +2
-|  -3
+| -3
-|  -24
+| -24
-|  +29
+| +29
-|  +29
+| +29
-|  +31
+| +31
-|  -23
+| -23
 |-
-! colspan="2" |[[nearest edomapping]]
+! colspan="2" | [[nearest edomapping]]
-|31
+| 31
-|18
+| 18
-|10
+| 10
-|25
+| 25
-|14
+| 14
-|22
+| 22
-|3
+| 3
-|8
+| 8
-|16
+| 16
 |-
-! colspan="2" |[[fifthspan]]
+! colspan="2" | [[fifthspan]]
 | 0
-|  +1
+| +1
-|  +4
+| +4
-|  +10
+| +10
-|  -13
+| -13
-|  +15
+| +15
-|  -5
+| -5
-|  -3
+| -3
-|  -6
+| -6
-|}Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit]] fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.
+|}
+Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit]] fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.
 edo's 12\31 generator (an approximate 21/16) supports [[A-Team]] and yields [[13edo#Modes_and_Harmony_in_the_Oneirotonic_Scale|8-note "oneirotonic" scales similar to those in 13edo]] but with the 9/8 and 5/4 better in tune; this temperament is also represented by [[13edo]], [[18edo]] and [[44edo]].