3159811edo: Difference between revisions

(15 intermediate revisions by 2 users not shown)

Line 1:

== Theory ==

3159811edo is [[consistent]] in the 65-odd-limit with a lower [[relative error]] than any previous equal temperaments in the 61-limit. It is the smallest edo which is purely consistent{{idio}} in the 63-odd-limit (i.e. does not exceed 25% relative error on the first 63 harmonics of the [[harmonic series]]).

Although its step size is far smaller than the human melodic [[just-noticeable difference]], 3159811edo is [[consistent]] in the 65-odd-limit with a lower [[relative error]] than any previous equal temperaments in the 61-limit. It is the smallest edo which is purely consistent{{idio}} in the 63-odd-limit (i.e. does not exceed 25% relative error on the first 63 harmonics of the [[harmonic series]]).

While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, like all other equal temperaments, the consistency of this tuning is finite, and the sequence of intervals may eventually start to deviate from their true JI counterparts.

=== Prime harmonics ===

Line 12:

Line 14:

== Scales ==

=== Harmonic scales ===

3159811edo accurately approximates mode 32 of the [[harmonic series]]. ~~All~~ interval pairs are distinguished.

As mentioned, 3159811edo accurately approximates [[32afdo|mode 32]] of the [[harmonic series]]. Additionally, unlike in [[10edo]]'s approximation of [[4afdo|mode 4]], [[87edo]]'s approximation of [[8afdo|mode 8]], or [[311edo]]'s approximation of [[16afdo|mode 16]], all interval pairs are distinguished.

{| class="wikitable center-all"

Line 38:

Line 40:

| 5/4

|-

! ~~… in~~ cents

! …in cents

| 0

| 53.273

| 104.955

| 155.14

| 155.140

| 203.91

| 203.910

| 251.344

| 297.513

Line 83:

Line 85:

| 3/2

|-

! ~~… in~~ cents

! …in cents

| 429.062

| 470.781

Line 126:

Line 128:

| 7/4

|-

! ~~… in~~ cents

! …in cents

| 737.652

| 772.627

| 806.91

| 806.910

| 840.528

| 873.505

Line 169:

Line 171:

| 2/1

|-

! ~~… in~~ cents

! …in cents

| 999.468

| 1029.577

Line 190:

Line 192:

|}

* The scale in adjacent steps is 140277, 136089, 132144, 128421, 124902, 121571, 118413, 115415, 112565, 109852, 107267, 104801, 102446, 100194, 98039, 95975, 93996, 92097, 90273, 88520, 86834, 85211, 83647, 82140, 80686, 79283, 77927, 76618, 75351, 74126, 72940, 71791.

The scale in adjacent steps is 140277, 136089, 132144, 128421, 124902, 121571, 118413, 115415, 112565, 109852, 107267, 104801, 102446, 100194, 98039, 95975, 93996, 92097, 90273, 88520, 86834, 85211, 83647, 82140, 80686, 79283, 77927, 76618, 75351, 74126, 72940, 71791.

@@ Line 1: / Line 1: @@
-{{Niche}}{{clear}}
+{{Mathematical interest}}{{clear}}
 {{Infobox ET|Consistency=65|Distinct consistency=65}}
 {{ED intro}}
 == Theory ==
-3159811edo is [[consistent]] in the 65-odd-limit with a lower [[relative error]] than any previous equal temperaments in the 61-limit. It is the smallest edo which is purely consistent{{idio}} in the 63-odd-limit (i.e. does not exceed 25% relative error on the first 63 harmonics of the [[harmonic series]]).
+Although its step size is far smaller than the human melodic [[just-noticeable difference]], 3159811edo is [[consistent]] in the 65-odd-limit with a lower [[relative error]] than any previous equal temperaments in the 61-limit. It is the smallest edo which is purely consistent{{idio}} in the 63-odd-limit (i.e. does not exceed 25% relative error on the first 63 harmonics of the [[harmonic series]]).
+While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, like all other equal temperaments, the consistency of this tuning is finite, and the sequence of intervals may eventually start to deviate from their true JI counterparts.
 === Prime harmonics ===
@@ Line 12: / Line 14: @@
 == Scales ==
 === Harmonic scales ===
-3159811edo accurately approximates mode 32 of the [[harmonic series]]. All interval pairs are distinguished.
+As mentioned, 3159811edo accurately approximates [[32afdo|mode 32]] of the [[harmonic series]]. Additionally, unlike in [[10edo]]'s approximation of [[4afdo|mode 4]], [[87edo]]'s approximation of [[8afdo|mode 8]], or [[311edo]]'s approximation of [[16afdo|mode 16]], all interval pairs are distinguished.
 {| class="wikitable center-all"
@@ Line 38: / Line 40: @@
 | 5/4
 |-
-! … in cents
+! …in cents
 | 0
 | 53.273
 | 104.955
-| 155.14
+| 155.140
-| 203.91
+| 203.910
 | 251.344
 | 297.513
@@ Line 83: / Line 85: @@
 | 3/2
 |-
-! … in cents
+! …in cents
 | 429.062
 | 470.781
@@ Line 126: / Line 128: @@
 | 7/4
 |-
-! … in cents
+! …in cents
 | 737.652
 | 772.627
-| 806.91
+| 806.910
 | 840.528
 | 873.505
@@ Line 169: / Line 171: @@
 | 2/1
 |-
-! … in cents
+! …in cents
 | 999.468
 | 1029.577
@@ Line 190: / Line 192: @@
 |}
-* The scale in adjacent steps is 140277, 136089, 132144, 128421, 124902, 121571, 118413, 115415, 112565, 109852, 107267, 104801, 102446, 100194, 98039, 95975, 93996, 92097, 90273, 88520, 86834, 85211, 83647, 82140, 80686, 79283, 77927, 76618, 75351, 74126, 72940, 71791.
+The scale in adjacent steps is 140277, 136089, 132144, 128421, 124902, 121571, 118413, 115415, 112565, 109852, 107267, 104801, 102446, 100194, 98039, 95975, 93996, 92097, 90273, 88520, 86834, 85211, 83647, 82140, 80686, 79283, 77927, 76618, 75351, 74126, 72940, 71791.