9edt: Difference between revisions

Line 3:

== Theory ==

It has a decent seventh harmonic ([[7/1]]) which is 12.4¢ sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6¢ flat. However, the [[5/1]] is 39¢ flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth—or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845¢, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]]—making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].

It has a decent seventh harmonic ([[7/1]]) which is 12.4 cents sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6 cents flat. However, the [[5/1]] is 39 cents flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth — or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845 cents, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]] — making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].

Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[4L 5s (3/1-equivalent)|3/1-equivalent "lambda" scale]], of which 9edt offers the "equalized" interpretation of {{nowrap|L {{=}} s}}, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.

Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[~~4L_5s_~~(3/1-equivalent)|3/1-equivalent ~~4L 5s]], aka~~ lambda~~, [[moment of symmetry~~]] ~~scale~~, of which 9edt offers the "equalized" interpretation of L = s, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.

9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos zeta peak edt]].

=== Relation to edos ===

9edt is related to [[17edo]], by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to [[3/1]].

Line 19:

Line 17:

{| class="wikitable"

|-

! rowspan="2" | Steps

! colspan="2" | Size

! rowspan="2" | Comparable intervals (in [[cent|¢]])

|-

!(in [[cent|¢]])

! (in [[cent|¢]])

!in [[hekt]]s

! in [[hekt]]s

|-

! colspan="3" | 0

Line 31:

Line 30:

| 1

| 211.328

|144.444

| 144.444

| [[9/8]] (204)

|-

| 2

| 422.657

|288.889

| 288.889

| [[9/7]] (435)

|-

| 3

| 633.985

|433.333

| 433.333

| [[13/9]] (637)

|-

| 4

| 845.313

|577.778

| 577.778

| [[13/8]] (841), [[5/3]] (884), [[8/5]] (814)

|-

| 5

| 1056.642

|722.222

| 722.222

| [[9/5]] (1018), [[11/6]] (1049)

|-

| 6

| 1267.970

|866.667

| 866.667

| [[27/13]] (1265)

|-

| 7

| 1479.298

|1011.111

| 1011.111

| [[7/3]] (1467)

|-

| 8

| 1690.627

|1155.556

| 1155.556

| [[8/3]] (1698)

|-

| 9

| 1901.955

|1300

| 1300

| [[3/1]]

|}

Line 77:

Line 76:

== Music ==

* [https://www.youtube.com/watch?v=sEQP1AtjPrA Far Away From Them / Spazzystackers] by [[Mandrake]]

[[Category:Macrotonal]]

[[Category:Edt]]

@@ Line 3: / Line 3: @@
 == Theory ==
+It has a decent seventh harmonic ([[7/1]]) which is 12.4¢ sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6¢ flat. However, the [[5/1]] is 39¢ flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth—or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845¢, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]]—making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].
-It has a decent seventh harmonic ([[7/1]]) which is 12.4 cents sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6 cents flat. However, the [[5/1]] is 39 cents flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth — or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845 cents, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]] — making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].
+Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[4L 5s (3/1-equivalent)|3/1-equivalent "lambda" scale]], of which 9edt offers the "equalized" interpretation of {{nowrap|L {{=}} s}}, analogous to [[7edo]] in diatonic ([[5L&nbsp;2s]]) music.
-Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[4L_5s_(3/1-equivalent)|3/1-equivalent 4L 5s]], aka lambda, [[moment of symmetry]] scale, of which 9edt offers the "equalized" interpretation of L = s, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.
 edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos zeta peak edt]].
 === Relation to edos ===
 edt is related to [[17edo]], by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to [[3/1]].
@@ Line 19: / Line 17: @@
 {| class="wikitable"
+|-
 ! rowspan="2" | Steps
 ! colspan="2" | Size
 ! rowspan="2" | Comparable intervals (in [[cent|&cent;]])
 |-
-!(in [[cent|&cent;]])
+! (in [[cent|&cent;]])
-!in [[hekt]]s
+! in [[hekt]]s
 |-
 ! colspan="3" | 0
@@ Line 31: / Line 30: @@
 | 1
 | 211.328
-|144.444
+| 144.444
 | [[9/8]] (204)
 |-
 | 2
 | 422.657
-|288.889
+| 288.889
 | [[9/7]] (435)
 |-
 | 3
 | 633.985
-|433.333
+| 433.333
 | [[13/9]] (637)
 |-
 | 4
 | 845.313
-|577.778
+| 577.778
 | [[13/8]] (841), [[5/3]] (884), [[8/5]] (814)
 |-
 | 5
 | 1056.642
-|722.222
+| 722.222
 | [[9/5]] (1018), [[11/6]] (1049)
 |-
 | 6
 | 1267.970
-|866.667
+| 866.667
 | [[27/13]] (1265)
 |-
 | 7
 | 1479.298
-|1011.111
+| 1011.111
 | [[7/3]] (1467)
 |-
 | 8
 | 1690.627
-|1155.556
+| 1155.556
 | [[8/3]] (1698)
 |-
 | 9
 | 1901.955
-|1300
+| 1300
 | [[3/1]]
 |}
@@ Line 77: / Line 76: @@
 == Music ==
 * [https://www.youtube.com/watch?v=sEQP1AtjPrA Far Away From Them / Spazzystackers] by [[Mandrake]]
 [[Category:Macrotonal]]
 [[Category:Edt]]