17ed4: Difference between revisions

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'''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]].
'''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]].
==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 cas [[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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|[[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
Line 112: Line 112:
|[[11/3]]
|[[11/3]]
|vB
|vB
|R#, Jb
|S
|-
|-
|17
|17

Revision as of 01:01, 9 May 2023

← 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

17ed4 is the equal division of the double octave into 17 parts of 141.18 cents each, corresponding to 8.5edo or every second step of 17edo.

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<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 7L 3s<4/1> 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
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