17edt

← 16edt

17edt

18edt →

Prime factorization	17 (prime)
Step size	111.88 ¢
Octave	11\17edt (1230.68 ¢)
Consistency limit	3
Distinct consistency limit	3

17EDT is equal division of the third harmonic into 17 parts of 111.880 cents each (corresponding to 10.726 EDO).

Properties

In the no-twos subgroup, 17EDT tempers out 245/243 and 16807/15625 in the 7-limit, 77/75 and 1331/1323 in the 11-limit, and 175/169 and 121/117 in the 13-limit. It supports the no-twos temperament tempering out 245/243 and 77/75, which in terms of tritave patent vals could be written b17&b21.

17EDT is the sixth zeta peak tritave division.

Approximation of prime harmonics in 17edt

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Error	Absolute (¢)	+30.7	+0.0	+10.7	-12.4	-11.8	+34.7	+17.8	+49.0	+53.8	-11.8	-15.4	+13.9	-51.9	-22.5	+47.3
	Relative (%)	+27.4	+0.0	+9.5	-11.1	-10.5	+31.0	+15.9	+43.8	+48.1	-10.6	-13.8	+12.4	-46.4	-20.1	+42.3
Steps (reduced)		11 (11)	17 (0)	25 (8)	30 (13)	37 (3)	40 (6)	44 (10)	46 (12)	49 (15)	52 (1)	53 (2)	56 (5)	57 (6)	58 (7)	60 (9)

Discussion

17EDT is closely related to 13EDT, the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13EDT and 17EDT have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13EDT is a calm 2:1, in 17EDT it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly in return for gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17EDT tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents), leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to 605/189-1.7 cents, which is also a 16/5 which is only .15 cents sharp (in addition to equaling 256).

Intervals

degree	note name	cents value	hekts	cents value octave reduced
0	C	0
1	Db = B#	111.9	76.5
2	Eb = C#	223.8	152.9
3	D	335.6	229.4
4	E	447.5	305.9
5	F = D#	559.4	382.35
6	Gb = E#	671.3	458.8
7	Hb = F#	783.2	535.3
8	G	895.1	611.8
9	H	1006.9	688.2
10	Jb = G#	1118.8	764.7
11	Ab = H#	1230.7	841.2	30.7
12	J	1342.6	917.65	142.6
13	A	1454.4	994.1	254.4
14	Bb = J#	1566.3	1070.6	366.3
15	Cb = A#	1678.2	1147.1	478.2
16	B	1790.1	1223.5	590.1
17	C	1902	1300	702
18		2013.8	1376.5	813.8
19		2125.7	1452.9	925.7
20		2237.6	1529.4	1037.6
21		2349.5	1605.9	1149.5
22		2461.35	1682.35	61.35
23		2573.2	1758.8	173.2
24		2685.1	1835.3	285.1
25		2797	1911.8	397
26		2908.9	1988.2	508.9
27		3020.75	2064.7	620.75
28		3132.6	2141.2	732.6
29		3244.5	2217.65	844.5
30		3356.4	2294.1	956.4
31		3468.3	2370.6	1068.3
32		3580.15	2447.1	1180.15
33		3692	2523.5	92
34		3803.9	2600	203.9

• Notes named so that C D E F G H J A B C = Lambda mode

It's a weird coincidence how the schemes of 17EDO and 17EDT diatonicism are so similar and how their approximations of 9/7 are off by such similar amounts in opposite directions (17EDO -11.6 cents and 17EDT +12.4 cents).

Regular temperament

17EDT is also be thought of as a generator of the vavoom temperament. As one degree of 17EDT is very close to 16/15, an unnoticeable comma [-68 18 17⟩ is tempered out in the vavoom temperament.

Vavoom (118&783)

5-limit
Comma: [-68 18 17⟩
Mapping: [⟨1 0 4], ⟨0 17 -18]]
POTE generator: ~16/15 = 111.876
Optimal ET sequence: 11, 32, 43, 75, 118, 429, 547, 665, 783, 901, 1684
Badness: 0.098376

Z function

Below is a plot of the no-twos Z function in the vicinity of 17EDT.

Music

Togenom

• "Circle is Square" from Xenharmonics, Vol. 5 (2024) – Spotify | Bandcamp | YouTube