293edo: Difference between revisions

Line 8:

293 edo does not approximate prime harmonics well all the way into the 41st, unless 30-relative cent errors are considered "well", in which case it equally represents all of them. The first harmonic that it approximates within 1 standard deviation of one step is 43rd, which is 10 cents flat compared to the just intonated interval.

When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], ~~and~~ [[25/19]]. [[21/16]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmoincs have +36 and +37 error respectively, which together cancels out to 1.

When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], [[25/19]], and respectively their octave inversions. [[21/16]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmoincs have +36 and +37 error respectively, which together cancels out to 1.

One step of 293edo is at the edge of human pitch perception of 3.5 cents. When combined with low harmonicity, this opens 293edo to a wide range of interpretations.

Line 106:

|511.9454

|[[43/32]]

|-

|130

|Vengeance superfourth

|532.4232

|[[34/25]]

|-

| 163

Line 111:

Line 116:

| 667.5768

|[[25/17]]

|-

|168

|43rd subharmonic

|688.0546

|64/43

|-

|171

Line 126:

Line 136:

| 782.2526

|[[11/7]]

|-

|216

|Major sixth

|884.6416

|[[5/3]]

|-

|236

@@ Line 8: / Line 8: @@
 edo does not approximate prime harmonics well all the way into the 41st, unless 30-relative cent errors are considered "well", in which case it equally represents all of them. The first harmonic that it approximates within 1 standard deviation of one step is 43rd, which is 10 cents flat compared to the just intonated interval.
-When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], and [[25/19]]. [[21/16]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmoincs have +36 and +37 error respectively, which together cancels out to 1.
+When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], [[25/19]], and respectively their octave inversions. [[21/16]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmoincs have +36 and +37 error respectively, which together cancels out to 1.
 One step of 293edo is at the edge of human pitch perception of 3.5 cents. When combined with low harmonicity, this opens 293edo to a wide range of interpretations.
@@ Line 106: / Line 106: @@
 |511.9454
 |[[43/32]]
+|-
+|130
+|Vengeance superfourth
+|532.4232
+|[[34/25]]
 |-
 | 163
@@ Line 111: / Line 116: @@
 | 667.5768
 |[[25/17]]
+|-
+|168
+|43rd subharmonic
+|688.0546
+|64/43
 |-
 |171
@@ Line 126: / Line 136: @@
 | 782.2526
 |[[11/7]]
+|-
+|216
+|Major sixth
+|884.6416
+|[[5/3]]
 |-
 |236