17edo: Difference between revisions

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17edo is the smallest edo that has a sharper-than-just diatonic fifth (705.8c, compared to just 3/2 = 702.0c), not counting the degenerate case 5edo. 17edo has discordant diatonic major thirds and provides no good approximation of 5/4, which to some extent forces one to use nonfunctional and modal harmony instead of standard functional harmony. It is also notable for having neutral intervals.
17edo is the smallest edo that has a sharper-than-just diatonic fifth (705.8c, compared to just 3/2 = 702.0c), not counting the degenerate case 5edo. 17edo has discordant diatonic major thirds and provides no good approximation of 5/4, which to some extent forces one to use nonfunctional and modal harmony instead of standard functional harmony. It is also notable for having neutral intervals.
Since 17edo's perfect eleventh is highly divisible (24 steps), you can do 2\17 x 12, 3\17 x 8, 4\17 x 6, 6\17 x 4, 8\17 x 3, or 12\17 x 2 movements to pump this "comma".


== Tuning theory ==
== Tuning theory ==


=== Intervals ===
=== Intervals ===
17edo can be notated with [[diatonic notation]] with #/b = 2\17. Since the sharp can be split in two, half-sharps and half-flats for 1\17 alterations can be used too.
{| class="wikitable"
{| class="wikitable"
!Edostep
!Edostep
!Cents
!Cents
!Name
!Name (Neutral diatonic / ADIN)
!Diatonic notation with half sharps/flats
!Notation
!JI interpretation (2.3.25.(11/7).13, 68edo subset)
|-
|-
|0
|0
Line 18: Line 22:
|Unison
|Unison
|A
|A
|1/1
|-
|-
|1
|1
|70.6
|70.6
|Minor second
|(Far)minor second
|Bb, At
|Bb, At
|25/24
|-
|-
|2
|2
Line 28: Line 34:
|Neutral second
|Neutral second
|A#, Bd
|A#, Bd
|13/12
|-
|-
|3
|3
|211.8
|211.8
|Major second
|(Far)major second
|B
|B
|9/8
|-
|-
|4
|4
|282.4
|282.4
|Minor third
|(Far)minor third
|C
|C
|33/28, 32/27
|-
|-
|5
|5
Line 43: Line 52:
|Neutral third
|Neutral third
|Db, Ct
|Db, Ct
|16/13
|-
|-
|6
|6
|423.5
|423.5
|Major third
|(Far)major third
|C#, Dd
|C#, Dd
|14/11
|-
|-
|7
|7
Line 53: Line 64:
|Perfect fourth
|Perfect fourth
|D
|D
|4/3
|-
|-
|8
|8
|564.7
|564.7
|Small tritone
|Semiaugmented/neutral fourth
|Eb, Dt
|Eb, Dt
|25/18, 18/13
|-
|-
|9
|9
|635.3
|635.3
|Large tritone
|Semidiminished/neutral fifth
|D#, Ed
|D#, Ed
|36/25, 13/9
|-
|-
|10
|10
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|Perfect fifth
|Perfect fifth
|E
|E
|3/2
|-
|-
|11
|11
|776.5
|776.5
|Minor sixth
|(Far)minor sixth
|F
|F
|11/7
|-
|-
|12
|12
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|Neutral sixth
|Neutral sixth
|Gb, Ft
|Gb, Ft
|13/8
|-
|-
|13
|13
|917.6
|917.6
|Major sixth
|(Far)major sixth
|F#, Gd
|F#, Gd
|56/33, 27/16
|-
|-
|14
|14
|988.2
|988.2
|Minor seventh
|(Far)minor seventh
|G
|G
|16/9
|-
|-
|15
|15
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|Neutral seventh
|Neutral seventh
|Ab, Gt
|Ab, Gt
|24/13
|-
|-
|16
|16
|1129.4
|1129.4
|Major seventh
|(Far)major seventh
|G#, Ad
|G#, Ad
|48/25
|-
|-
|17
|17
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|Octave
|Octave
|A
|A
|2/1
|}
|}
The major third is a farmajor third, interpreted as 14/11 either with 17edo's patent val or as a subset of 68edo; 17edo is at the upper edge of [[gentle tuning]]. While 17edo has reasonable approximations of 7 and 11, they are not shared with 68edo, although their ratio 14/11 is.
The edostep is a near-just 25/24, sometimes seen as the optimal size for a leading tone.


=== Prime approximations ===
=== Prime approximations ===
{{Harmonics in ED|17|23}}
{{Harmonics in ED|17|23}}
=== Notation ===
Because a sharp is 2 steps in 17edo, it may be notated with neutral [[diatonic notation]] (semisharps and semiflats).
=== Tuning properties ===


== Scales ==
== Scales ==
Line 114: Line 144:
* Lissotonic (3223322)
* Lissotonic (3223322)
* Trachytonic (3313322)
* Trachytonic (3313322)
{{Navbox EDO}}


{{cat|Edos}}
{{cat|Edos}}

Latest revision as of 11:54, 8 June 2026

This page or section is a work in progress. It may lack sufficient justification, content, or organization, and is subject to future overhaul.

17edo, or 17 equal divisions of the octave, is the equal tuning featuring steps of (1200/17) ~= 70.6 cents, 17 of which stack to the octave 2/1.

17edo is the smallest edo that has a sharper-than-just diatonic fifth (705.8c, compared to just 3/2 = 702.0c), not counting the degenerate case 5edo. 17edo has discordant diatonic major thirds and provides no good approximation of 5/4, which to some extent forces one to use nonfunctional and modal harmony instead of standard functional harmony. It is also notable for having neutral intervals.

Since 17edo's perfect eleventh is highly divisible (24 steps), you can do 2\17 x 12, 3\17 x 8, 4\17 x 6, 6\17 x 4, 8\17 x 3, or 12\17 x 2 movements to pump this "comma".

Tuning theory

Intervals

17edo can be notated with diatonic notation with #/b = 2\17. Since the sharp can be split in two, half-sharps and half-flats for 1\17 alterations can be used too.

Edostep Cents Name (Neutral diatonic / ADIN) Notation JI interpretation (2.3.25.(11/7).13, 68edo subset)
0 0 Unison A 1/1
1 70.6 (Far)minor second Bb, At 25/24
2 141.2 Neutral second A#, Bd 13/12
3 211.8 (Far)major second B 9/8
4 282.4 (Far)minor third C 33/28, 32/27
5 352.9 Neutral third Db, Ct 16/13
6 423.5 (Far)major third C#, Dd 14/11
7 494.1 Perfect fourth D 4/3
8 564.7 Semiaugmented/neutral fourth Eb, Dt 25/18, 18/13
9 635.3 Semidiminished/neutral fifth D#, Ed 36/25, 13/9
10 705.9 Perfect fifth E 3/2
11 776.5 (Far)minor sixth F 11/7
12 847.1 Neutral sixth Gb, Ft 13/8
13 917.6 (Far)major sixth F#, Gd 56/33, 27/16
14 988.2 (Far)minor seventh G 16/9
15 1058.9 Neutral seventh Ab, Gt 24/13
16 1129.4 (Far)major seventh G#, Ad 48/25
17 1200 Octave A 2/1

The major third is a farmajor third, interpreted as 14/11 either with 17edo's patent val or as a subset of 68edo; 17edo is at the upper edge of gentle tuning. While 17edo has reasonable approximations of 7 and 11, they are not shared with 68edo, although their ratio 14/11 is.

The edostep is a near-just 25/24, sometimes seen as the optimal size for a leading tone.

Prime approximations

Approximation of prime harmonics in 17edo
Harmonic 2 3 5 7 11 13 17 19 23
Error Absolute (¢) 0.0 +3.9 -33.4 +19.4 +13.4 +6.5 -34.4 -15.2 +7.0
Relative (%) 0.0 +5.6 -47.3 +27.5 +19.0 +9.3 -48.7 -21.5 +9.9
Steps

(reduced)

17

(0)

27

(10)

39

(5)

48

(14)

59

(8)

63

(12)

69

(1)

72

(4)

77

(9)

Notation

Because a sharp is 2 steps in 17edo, it may be notated with neutral diatonic notation (semisharps and semiflats).

Tuning properties

Scales

Some scales:

  • MOS diatonic (5L2s, 3331331)
  • Mosh/"malacotonic" (3L4s, 2323232)
  • Lissotonic (3223322)
  • Trachytonic (3313322)


View • Talk • EditEqual temperaments
EDOs
Macrotonal 5 • 6 • 7 • 8 • 9 • 10 • 11
12-23 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23
24-35 24 • 25 • 26 • 27 • 29 • 31 • 32 • 34 • 35
36-47 36 • 37 • 39 • 40 • 41 • 43 • 44 • 45 • 46 • 47
48-59 48 • 50 • 51 • 53 • 54 • 56 • 57 • 58
60-71 60 • 63 • 64 • 65 • 67 • 68 • 70
72-83 72 • 77 • 80 • 81
84-95 84 • 87 • 89 • 90 • 93 • 94
Large EDOs 99 • 104 • 106 • 111 • 118 • 130 • 140 • 152 • 159 • 171 • 217 • 224 • 239 • 270 • 306 • 311 • 612 • 665
Nonoctave equal temperaments
Tritave 4 • 9 • 13 • 17 • 26 • 39
Fifth 8 • 9 • 11 • 20
Other
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