9edt: Difference between revisions

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 {{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]].
+{{ED 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.
+== 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{{c}} flat. However, the [[5/1]] is 39{{c}} 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{{c}}, 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]].
+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.
+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]].
+=== Harmonics ===
+{{Harmonics in equal|9|3|1|}}
+{{Harmonics in equal|9|3|1|intervals=prime}}
 {| class="wikitable"
+|-
 ! rowspan="2" | Steps
 ! colspan="2" | Size
-! rowspan="2" | Comparable intervals
+! rowspan="2" | Comparable intervals (¢)
 |-
-!(in [[cent|&cent;]])
+! Cents
-!in hekts
+! [[Hekt]]s
 |-
 ! colspan="3" | 0
@@ Line 19: / 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]]
 |}
+== Music ==
+* [https://www.youtube.com/watch?v=sEQP1AtjPrA Far Away From Them / Spazzystackers] by [[Mandrake]]
 [[Category:Macrotonal]]
-[[Category:Edt]]