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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]].
{{Infobox ET}}
{{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 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]].  
 
=== Harmonics ===
{{Harmonics in equal|9|3|1|}}
{{Harmonics in equal|9|3|1|intervals=prime}}


{| class="wikitable"
{| class="wikitable"
|-
! rowspan="2" | Steps
! rowspan="2" | Steps
! colspan="2" | Size
! colspan="2" | Size
! rowspan="2" | Comparable intervals
! rowspan="2" | Comparable intervals (¢)
|-
|-
!(in [[cent|¢]])
! Cents
!in hekts
! [[Hekt]]s
|-
|-
| colspan="3" | 0
! colspan="3" | 0
| [[1/1]]
| [[1/1]]
|-
|-
| 1
| 1
| 211.328  
| 211.328  
|144.444
| 144.444
| [[9/8]] (204)
| [[9/8]] (204)
|-
|-
| 2
| 2
| 422.657  
| 422.657  
|288.889
| 288.889
| [[9/7]] (435)
| [[9/7]] (435)
|-
|-
| 3
| 3
| 633.985  
| 633.985  
|433.333
| 433.333
| [[13/9]] (637)
| [[13/9]] (637)
|-
|-
| 4
| 4
| 845.313
| 845.313
|577.778
| 577.778
| [[13/8]] (841), [[5/3]] (884), [[8/5]] (814)
| [[13/8]] (841), [[5/3]] (884), [[8/5]] (814)
|-
|-
| 5
| 5
| 1056.642
| 1056.642
|722.222
| 722.222
| [[9/5]] (1018), [[11/6]] (1049)
| [[9/5]] (1018), [[11/6]] (1049)
|-
|-
| 6
| 6
| 1267.970
| 1267.970
|866.667
| 866.667
| [[27/13]] (1265)
| [[27/13]] (1265)
|-
|-
| 7
| 7
| 1479.298
| 1479.298
|1011.111
| 1011.111
| [[7/3]] (1467)
| [[7/3]] (1467)
|-
|-
| 8
| 8
| 1690.627
| 1690.627
|1155.556
| 1155.556
| [[8/3]] (1698)
| [[8/3]] (1698)
|-
|-
| 9
| 9
| 1901.955
| 1901.955
|1300
| 1300
| [[3/1]]
| [[3/1]]
|}
|}
== Music ==
* [https://www.youtube.com/watch?v=sEQP1AtjPrA Far Away From Them / Spazzystackers] by [[Mandrake]]


[[Category:Macrotonal]]
[[Category:Macrotonal]]
[[Category:Edt]]

Latest revision as of 15:31, 31 July 2025

← 8edt 9edt 10edt →
Prime factorization 32
Step size 211.328 ¢ 
Octave 6\9edt (1267.97 ¢) (→ 2\3edt)
Consistency limit 3
Distinct consistency limit 3

9 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 9edt or 9ed3), is a nonoctave tuning system that divides the interval of 3/1 into 9 equal parts of about 211 ¢ each. Each step represents a frequency ratio of 31/9, or the 9th root of 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.

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

9edt is the third 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.

Harmonics

Approximation of harmonics in 9edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +68.0 +0.0 -75.4 -39.0 +68.0 +12.4 -7.4 +0.0 +28.9 +75.2 -75.4
Relative (%) +32.2 +0.0 -35.7 -18.5 +32.2 +5.9 -3.5 +0.0 +13.7 +35.6 -35.7
Steps
(reduced) 		6
(6) 		9
(0) 		11
(2) 		13
(4) 		15
(6) 		16
(7) 		17
(8) 		18
(0) 		19
(1) 		20
(2) 		20
(2)
Approximation of prime harmonics in 9edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +68.0 +0.0 -39.0 +12.4 +75.2 -2.6 -44.4 -25.6 +66.3 +87.6 -27.8
Relative (%) +32.2 +0.0 -18.5 +5.9 +35.6 -1.2 -21.0 -12.1 +31.4 +41.5 -13.2
Steps
(reduced) 		6
(6) 		9
(0) 		13
(4) 		16
(7) 		20
(2) 		21
(3) 		23
(5) 		24
(6) 		26
(8) 		28
(1) 		28
(1)
Steps Size Comparable intervals (¢)
Cents Hekts
0 1/1
1 211.328 144.444 9/8 (204)
2 422.657 288.889 9/7 (435)
3 633.985 433.333 13/9 (637)
4 845.313 577.778 13/8 (841), 5/3 (884), 8/5 (814)
5 1056.642 722.222 9/5 (1018), 11/6 (1049)
6 1267.970 866.667 27/13 (1265)
7 1479.298 1011.111 7/3 (1467)
8 1690.627 1155.556 8/3 (1698)
9 1901.955 1300 3/1

Music

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