9edt: Difference between revisions

Line 1:

The '''9 equal division of 3''', the [[tritave]], divides it into 9 equal steps of size 211.328 [[cent]]s each. It has a decent 7 and an excellent 13, but a 5 which is 39 cents flat; if octaves were added and it was a sixth, it would count as a [[neutral sixth]]. The corresponding interval for [[5/3]] is 845 cents, which is a neutral sixth between [[8/5]] and [[5/3]], which is really more of a [[13/8]], though this is allegedly a no-twos tuning. On the 3.7.13 [[subgroup]] it tempers out [[351/343]] and [[2197/2187]]. 9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos zeta peak edt]].

~~Following~~ [[~~4edt~~]], ~~this~~ is the ~~next "Lambda"~~ (~~BP related~~) ~~equal division~~ of ~~the tritave~~; ~~in a certain sense analogous~~ to [[~~7edo~~]] in ~~diatonic music~~.

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]].

This scale is also 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]].

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

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

9edt is also 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]].

{| class="wikitable"

! rowspan="2" | Steps

! colspan="2" | Size

! rowspan="2" | Comparable intervals

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

|-

!(in [[cent|¢]])

!in ~~hekts~~

!in [[hekt]]s

|-

! colspan="3" | 0

Line 63:

Line 67:

|}

== ~~Prime harmonics~~ ==

== Harmonics ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''9 equal division of 3''', the [[tritave]], divides it into 9 equal steps of size 211.328 [[cent]]s each. It has a decent 7 and an excellent 13, but a 5 which is 39 cents flat; if octaves were added and it was a sixth, it would count as a [[neutral sixth]]. The corresponding interval for [[5/3]] is 845 cents, which is a neutral sixth between [[8/5]] and [[5/3]], which is really more of a [[13/8]], though this is allegedly a no-twos tuning. On the 3.7.13 [[subgroup]] it tempers out [[351/343]] and [[2197/2187]]. 9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos zeta peak edt]].
+{{EDO intro}}
-Following [[4edt]], this is the next "Lambda" (BP related) equal division of the tritave; in a certain sense analogous to [[7edo]] in diatonic music.
+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]].
-This scale is also 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]].
+edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos zeta peak edt]].
+Following [[4edt]], this is the next edt that supports [[4L_5s_(3/1-equivalent)|lambda]] temperament. This property is virtually the same as supporting a 3/1-equivalent 4L 5s [[moment of symmetry]] scale, of which 9edt offers the "equalized" interpretation of L = s, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.
+edt is also 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]].
 {| class="wikitable"
 ! rowspan="2" | Steps
 ! colspan="2" | Size
-! rowspan="2" | Comparable intervals
+! rowspan="2" | Comparable intervals (in [[cent|&cent;]])
 |-
 !(in [[cent|&cent;]])
-!in hekts
+!in [[hekt]]s
 |-
 ! colspan="3" | 0
@@ Line 63: / Line 67: @@
 |}
-== Prime harmonics ==
+== Harmonics ==
+{{Harmonics in equal|9|3|1|}}
 {{Harmonics in equal|9|3|1|intervals=prime}}