17edo

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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), as well as the second edo containing a diatonic fifth, not counting degenerate cases 5edo and 7edo. 17edo has somewhat sharp diatonic major thirds and provides no good approximation of 5/4, which encourages many composers to use nonfunctional and modal harmony instead of standard functional harmony, although standard functional harmony can still justify itself via listeners' general familiarity with 12edo, whose major third is only 1/3 of an edostep off of 17edo's, compared to 5/4 being a half-edostep away. 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. These can be written in Unicode-supported text as t/d, or with ^/v; occasionally ‡ is used for a semisharp.

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	B♭, At	25/24
2	141.2	Neutral second	A♯, C♭ Bd	13/12
3	211.8	(Far)major second	B, Cd	9/8
4	282.4	(Far)minor third	C, Bt	33/28, 32/27
5	352.9	Neutral third	B♯, D♭, 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	E♭, Dt	25/18, 18/13
9	635.3	Semidiminished/neutral fifth	D♯, F♭, Ed	36/25, 13/9
10	705.9	Perfect fifth	E, Fd	3/2
11	776.5	(Far)minor sixth	F, Et	11/7
12	847.1	Neutral sixth	E♯, G♭, F	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	A♭, 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).

Scales

Some scales:

• MOS diatonic (5L2s, 3331331)
• Mosh (3L4s, 2323232)
• Tetrachordal mosh¹ or lissotonic² (3223322): modes of the ascending mode of Maqam Rast
• Trachytonic² or screamapillar³ (3313322)

¹ Inthar's name

² CellularAutomaton's names

³ Xen Wiki's name