17ed4: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''17ed4''' is the [[Ed4|equal division of the double octave]] into 17 parts of 141.18 [[cent|cents]] each, corresponding to 8.5edo or every second step of [[17edo]].
{{ED intro}} It corresponds to 8.5edo or every second step of [[17edo]].
 
==Theory==
==Theory==
17ed4 is the smallest ED4 to contain a diatonic fifth, in this cas [[17edo]]'s sharp fifth, and it can be used to generate heptatonic (3L 4s) and decatonic (7L 3s) MOS scales with a period of [[4/1]]. The decatonic scale is the more usable of these two scales, corresponding to an octave-repeating pentatonic scale in terms of step sizes, while the heptatonic scale has too large step sizes, corresponding to an octave-repeating tritonic or tetratonic scale in terms of step sizes.
17ed4 is the smallest ED4 to contain a diatonic fifth, in this case [[17edo]]'s sharp fifth, and it can be used to generate heptatonic (3L 4s<4/1>) and decatonic ([[7L 3s (4/1-equivalent)|7L 3s<4/1>]]) MOS scales with a period of [[4/1]]. The decatonic scale is the more usable of these two scales, corresponding to an octave-repeating pentatonic scale in terms of step sizes, while the heptatonic scale has too large step sizes, corresponding to an octave-repeating tritonic or tetratonic scale in terms of step sizes.
 
==Intervals==
==Intervals==
{|class="wikitable"
{|class="wikitable"
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!Approximate ratios
!Approximate ratios
![[17edo]] notation
![[17edo]] notation
![[7L 3s (4/1-equivalent)|7L 3s<4/1>]] notation (J = 1/1)
![[7L 3s (4/1-equivalent)|Diaquadic]] notation (J = 1/1)
|-
|-
|0
|0
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|[[13/12]], [[12/11]], [[14/13]], [[25/23]]
|[[13/12]], [[12/11]], [[14/13]], [[25/23]]
|C#
|C#
|J#, Kbb
|J#, Kb
|-
|-
|2
|2
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|[[13/11]], [[7/6]]
|[[13/11]], [[7/6]]
|Eb
|Eb
|Jx, Kb
|K
|-
|-
|3
|3
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|[[32/25]], [[9/7]], [[14/11]], [[33/26]], [[23/18]]
|[[32/25]], [[9/7]], [[14/11]], [[33/26]], [[23/18]]
|E
|E
|K
|K#, Lb
|-
|-
|4
|4
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|[[11/8]], [[18/13]], [[32/23]]
|[[11/8]], [[18/13]], [[32/23]]
|^F
|^F
|K#, Lb
|L
|-
|-
|5
|5
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|[[3/2]], [[32/21]]
|[[3/2]], [[32/21]]
|G
|G
|L
|M
|-
|-
|6
|6
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|[[13/8]], [[18/11]], [[23/14]]
|[[13/8]], [[18/11]], [[23/14]]
|G#, vA
|G#, vA
|M
|M#, Nb
|-
|-
|7
|7
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|[[16/9]], [[7/4]], [[25/14]], [[44/25]], [[23/13]]
|[[16/9]], [[7/4]], [[25/14]], [[44/25]], [[23/13]]
|Bb
|Bb
|M#, Nb
|N
|-
|-
|8
|8
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|[[25/13]], [[48/25]], [[27/14]], [[64/33]], [[23/12]]
|[[25/13]], [[48/25]], [[27/14]], [[64/33]], [[23/12]]
|B
|B
|N
|N#, Ob
|-
|-
|9
|9
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|[[15/7]]
|[[15/7]]
|^C
|^C
|N#, Ob
|O
|-
|-
|10
|10
|1411.80
|1411.80
|[[16/7]]
|[[9/4]], [[16/7]]
|D
|D
|O
|O#, Pb
|-
|-
|11
|11
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|[[8/3]]
|[[8/3]]
|F
|F
|P#, Qb
|Q
|-
|-
|13
|13
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|[[3/1]]
|[[3/1]]
|F#
|F#
|Q
|Q#, Rb
|-
|-
|14
|14
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|[[16/5]]
|[[16/5]]
|^G, Ab
|^G, Ab
|Q#, Rb
|R
|-
|-
|15
|15
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|[[10/3]]
|[[10/3]]
|A
|A
|R
|R#, Sb
|-
|-
|16
|16
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|[[11/3]]
|[[11/3]]
|vB
|vB
|R#, Jb
|S
|-
|-
|17
|17
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|J
|J
|}
|}
== Harmonics ==
{{Harmonics in equal
| steps = 17
| num = 4
| denom = 1
}}
{{Harmonics in equal
| steps = 17
| num = 4
| denom = 1
| start = 12
| collapsed = 1
}}
[[Category:Macrotonal]]
[[Category:Macrotonal]]
[[Category:Nonoctave]]
[[Category:Nonoctave]]
[[Category:Ed4]]
{{todo|expand}}

Latest revision as of 21:18, 31 July 2025

← 15ed4 17ed4 19ed4 →
Prime factorization 17 (prime)
Step size 141.176 ¢ 
Octave 9\17ed4 (1270.59 ¢)
Twelfth 13\17ed4 (1835.29 ¢)
Consistency limit 1
Distinct consistency limit 1

17 equal divisions of the 4th harmonic (abbreviated 17ed4) is a nonoctave tuning system that divides the interval of 4/1 into 17 equal parts of about 141 ¢ each. Each step represents a frequency ratio of 41/17, or the 17th root of 4. It corresponds to 8.5edo or every second step of 17edo.

Theory

17ed4 is the smallest ED4 to contain a diatonic fifth, in this case 17edo's sharp fifth, and it can be used to generate heptatonic (3L 4s<4/1>) and decatonic (7L 3s<4/1>) MOS scales with a period of 4/1. The decatonic scale is the more usable of these two scales, corresponding to an octave-repeating pentatonic scale in terms of step sizes, while the heptatonic scale has too large step sizes, corresponding to an octave-repeating tritonic or tetratonic scale in terms of step sizes.

Intervals

# Cents Approximate ratios 17edo notation Diaquadic notation (J = 1/1)
0 0.00 1/1 C J
1 141.18 13/12, 12/11, 14/13, 25/23 C# J#, Kb
2 282.36 13/11, 7/6 Eb K
3 423.54 32/25, 9/7, 14/11, 33/26, 23/18 E K#, Lb
4 564.72 11/8, 18/13, 32/23 ^F L
5 705.90 3/2, 32/21 G M
6 847.08 13/8, 18/11, 23/14 G#, vA M#, Nb
7 988.26 16/9, 7/4, 25/14, 44/25, 23/13 Bb N
8 1129.44 25/13, 48/25, 27/14, 64/33, 23/12 B N#, Ob
9 1270.62 15/7 ^C O
10 1411.80 9/4, 16/7 D O#, Pb
11 1552.98 12/5, 5/2 vE P
12 1694.16 8/3 F Q
13 1835.34 3/1 F# Q#, Rb
14 1976.52 16/5 ^G, Ab R
15 2117.70 10/3 A R#, Sb
16 2258.88 11/3 vB S
17 2400.00 4/1 C J

Harmonics

Approximation of harmonics in 17ed4
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +70.6 -66.7 +0.0 +37.2 +3.9 +19.4 +70.6 +7.9 -33.4 -57.2 -66.7
Relative (%) +50.0 -47.2 +0.0 +26.4 +2.8 +13.7 +50.0 +5.6 -23.6 -40.5 -47.2
Steps
(reduced) 		9
(9) 		13
(13) 		17
(0) 		20
(3) 		22
(5) 		24
(7) 		26
(9) 		27
(10) 		28
(11) 		29
(12) 		30
(13)
Approximation of harmonics in 17ed4
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -64.1 -51.2 -29.4 +0.0 +36.2 -62.7 -15.2 +37.2 -47.3 +13.4 -63.6
Relative (%) -45.4 -36.3 -20.9 +0.0 +25.7 -44.4 -10.7 +26.4 -33.5 +9.5 -45.0
Steps
(reduced) 		31
(14) 		32
(15) 		33
(16) 		34
(0) 		35
(1) 		35
(1) 		36
(2) 		37
(3) 		37
(3) 		38
(4) 		38
(4)
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