17edo

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17edo
Prime factorization 17 (prime)
Step size 70.588¢
Fifth 10\17 = 705.88¢
Major 2nd 3\17 = 212¢
Minor 2nd 1\17 = 71¢
Augmented 1sn 2\17 = 141¢

17 tone equal temperament, or 17-EDO, divides the octave in 17 equal steps, each 70.588 cents in size. It is the seventh prime EDO, following 13edo and coming before 19edo.

Theory

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19 prime 23
Error absolute (¢) 0.0 +3.9 -33.4 +19.4 +13.4 +6.5 -34.3 -15.2 +7.0
relative (%) 0 +6 -47 +27 +19 +9 -49 -21 +10
nearest edomapping 17 10 5 14 8 12 1 4 9
fifthspan 0 +1 -8 -2 -6 +8 -5 -3 +6

17-EDO can plausibly be treated as a 2.3.25.7.11.13.23 subgroup temperament, for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because the 3, 7, 11, and 13 are all sharp, it adapts well to octave shrinking; 27edt (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is 44ed6.

As a no-fives system, it is best used with timbres in which harmonic multiples of 5 are attenuated or absent. Also, the standard major chord (4:5:6) cannot be used since it includes the fifth harmonic.

Instead, the tonic chords of 17-EDO could be considered to be the tetrad 6:7:8:9 and its utonal inversion, the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the mu chord of Steely Dan fame). These are realized in 17-EDO as 0-4-7-10 and 0-3-6-10, respectively. Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add the 0-3-7-10 (which is a sus4 with added second, or sus2 with added fourth). These three chords comprise the three ways to divide the 17-EDO perfect fifth into two whole tones and one subminor third. Chromatic alterations of them also exist, for example, the 0-3-7-10 chord may be altered to 0-2-7-10 (which approximates 12:13:16:18) or 0-3-8-10 (which approximates 8:9:11:12). The 0-3-8-10 chord is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 0-3-6-10.

Intervals

Edo steps Cents Names of Intervals, extended

pythagorean note names

Ups and Downs Notation Approximate Ratios* Temperament(s) generated
0 0.00 Unison C unison P1 C 1/1
1 70.59 Super Unison/Minor Second Db
(B#) 		up 1sn, minor 2nd ^1, m2 ^C, Db 25/24, 26/25, 33/32, 24/23
2 141.18 Augmented Unison/Neutral Second C# aug 1sn, mid 2nd A1, ~2 C#, vD 13/12, 12/11, 14/13, 25/23 Bleu
3 211.76 Major Second/Sub Third D major 2nd M2 D 9/8, 8/7, 28/25, 25/22, 26/23 Machine
4 282.35 Minor Third/Super Second Eb minor 3rd m3 Eb 13/11, 7/6 Huxley/Lovecraft
5 352.94 Augmented Second/Neutral

Third/Diminished Fourth

D#
(Fb) 		mid 3rd ~3 vE 11/9, 16/13, 28/23 Maqamic/Hemif
6 423.53 Major Third/Sub Fourth E major 3rd M3 E 32/25, 9/7, 14/11, 33/26, 23/18 Skwares
7 494.12 Perfect Fourth F perfect 4th P4 F 4/3 Supra
8 564.71 Super Fourth/Diminshed Fifth Gb
(E#) 		mid 4th,

diminished 5th

~4,

d5

^F, Gb 11/8, 18/13, 32/23 Progress
9 635.29 Augmented Fourth/Sub Fifth F# augmented 4th,

mid 5th

A4, ~5 F#, vG 16/11, 13/9, 23/16 Progress
10 705.88 Perfect Fifth G perfect 5th P5 G 3/2 Supra
11 776.47 Super Fifth/Minor Sixth Ab minor 6th m6 Ab 25/16, 14/9, 11/7, 52/33, 36/23 Skwares
12 847.06 Augmented Fifth/Neutral

Sixth/Diminished Seventh

G# mid 6th ~6 vA 13/8, 18/11, 23/14 Maqamic/hemif
13 917.65 Major Sixth/Sub Seventh A major 6th M6 A 17/10, 22/13,12/7 Huxley
14 988.24 Minor Seventh/Super Sixth Bb minor 7th m7 Bb 16/9, 7/4, 25/14, 44/25, 23/13 Machine
15 1058.82 Augmented Sixth/Neutral

Seventh/Diminished Octave

A# (Cb) mid 7th ~7 vB 11/6, 24/13, 13/7, 46/25 Bleu
16 1129.41 Major Seventh/Sub Octave B major 7th M7 B 25/13, 48/25, 64/33, 23/12
17 1200.00 Perfect Octave C octave P8 C 2/1

* Ratios based on treating 17edo as a 2.3.7.11.13.23.25 subgroup

In 17edo, ups and downs can respectively be substituted with half-sharps and half-flats, since sharps and flats each span two edo steps. Using half-sharps and half-flats may be preferable for compatibility with the ups-and-downs notation in 34edo, in which an up or down respectively constitute a quarter-sharp or quarter-flat.

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

quality color monzo format examples
minor zo {a, b, 0, 1} 7/6, 7/4
" fourthward wa {a, b}, b < -1 32/27, 16/9
mid ilo {a, b, 0, 0, 1} 11/9, 11/6
" lu {a, b, 0, 0, -1} 12/11, 18/11
major fifthward wa {a, b}, b > 1 9/8, 27/16
" ru {a, b, 0, -1} 9/7, 12/7

Chord Names

All 17edo chords can be named using ups and downs. Here are the zo, ilo and ru triads:

color of the 3rd JI chord notes as edosteps notes of C chord written name spoken name
zo 6:7:9 0-4-10 C Eb G Cm C minor
ilo 18:22:27 0-5-10 C vE G C~ C mid
ru 14:18:21 0-6-10 C E G C C major or C

Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).

0-4-9 = C Eb vG = Cm(v5) = C minor down-five

0-5-9 = C vE vG = C~(v5) = C mid down-five

0-6-11 = C E ^G = C(^5) = C up-five

0-4-10-14 = C Eb G Bb = Cm7 = C minor seven

0-5-10-14 = C vE G Bb = C~,7 = C mid add seven

0-6-10-15 = C E G vB = C,~7 = C add mid-seven

0-5-10-15 = C vE G vB = C~7 = C mid-seven

For a more complete list, see Ups and Downs Notation - Chords and Chord Progressions.

Just approximation

Selected just intervals by error

15-odd-limit mappings

The following table shows how 15-odd-limit intervals are represented in 17edo (ordered by absolute error). Prime harmonics are in bold; inconsistent intervals are in italic.

Direct mapping (even if inconsistent)
Interval, complement Error (abs, ¢)
18/13, 13/9 1.324
13/12, 24/13 2.604
4/3, 3/2 3.927
11/9, 18/11 5.533
14/11, 11/7 6.021
16/13, 13/8 6.531
13/11, 22/13 6.857
9/8, 16/9 7.855
12/11, 11/6 9.461
9/7, 14/9 11.555
14/13, 13/7 12.878
11/8, 16/11 13.388
7/6, 12/7 15.482
7/5, 10/7 17.806
8/7, 7/4 19.409
15/14, 28/15 21.734
11/10, 20/11 23.828
15/11, 22/15 27.755
10/9, 9/5 29.361
16/15, 15/8 29.445
13/10, 20/13 30.685
6/5, 5/3 33.288
5/4, 8/5 33.373
15/13, 26/15 34.612
Patent val mapping
Interval, complement Error (abs, ¢)
18/13, 13/9 1.324
13/12, 24/13 2.604
4/3, 3/2 3.927
11/9, 18/11 5.533
14/11, 11/7 6.021
16/13, 13/8 6.531
13/11, 22/13 6.857
9/8, 16/9 7.855
12/11, 11/6 9.461
9/7, 14/9 11.555
14/13, 13/7 12.878
11/8, 16/11 13.388
7/6, 12/7 15.482
8/7, 7/4 19.409
16/15, 15/8 29.445
5/4, 8/5 33.373
15/13, 26/15 35.976
6/5, 5/3 37.300
13/10, 20/13 39.904
10/9, 9/5 41.227
15/11, 22/15 42.833
11/10, 20/11 46.760
15/14, 28/15 48.855
7/5, 10/7 52.782

Selected 13-limit intervals

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Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 17et.

3-limit 7-limit no-5 11-limit no-5 13-limit no-5
Octave stretch (¢) -1.24 -3.13 -3.31 -3.00
Error absolute (¢) 1.24 2.85 2.49 2.31
relative (%) 1.76 4.05 3.54 3.28
  • 17et has a lower relative error than any previous ETs in the no-5 11- and 13-limit. The next ET that does better in these subgroups is 41 and 207, respectively.

Scales

Tuning 17edo by ear

17edo is very close to a circle of seventeen 25/24 chromatic semitones: (25/24)^17 is only 1.43131 cents sharp of an octave. This means that if you can tune seventeen 25/24's accurately (by say, tuning 5/4 up, 3/2 down and 5/4 up, taking care to minimize the error at each step), you have a shot at approximating 17edo within melodic just noticeable difference.

Commas

17 EDO tempers out the following commas. (Note: This assumes val ⟨17 27 39 48 59 63], cent values ​​rounded to 5 digits.)

Prime
Limit 		Ratio[1] Monzo Cents Color name Name(s)
3 (18 digits) [27 -17⟩ 66.765 Sasawa 17-comma
5 25/24 [-3 -1 2⟩ 70.762 Yoyo Chromatic semitone, dicot comma
5 32805/32768 [-15 8 1⟩ 1.9537 Layo Schisma
7 525/512 [-9 1 2 1⟩ 43.408 Zoyoyo Avicennma, Avicennma's enharmonic diesis
7 64/63 [6 -2 0 -1⟩ 27.264 Ru Septimal comma, Archytas' comma, Leipziger Komma
7 245/243 [0 -5 1 2⟩ 14.191 Zozoyo Sensamagic
7 1728/1715 [6 3 -1 -3⟩ 13.074 Triru-agu Orwellisma, orwell comma
7 (12 digits) [-6 -8 2 5⟩ 1.1170 Quinzo-ayoyo Wizma
11 99/98 [-1 2 0 -2 1⟩ 17.576 Loruru Mothwellsma
11 896/891 [7 -4 0 1 -1⟩ 9.6880 Saluzo Pentacircle
11 243/242 [-1 5 0 0 -2⟩ 7.1391 Lulu Rastma
11 385/384 [-7 -1 1 1 1⟩ 4.5026 Lozoyo Keenanisma
13 1352/1331 [3 0 0 0 -3 2⟩ 27.101 Bithotrilu Lovecraft comma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Note that despite their relatively large size, the 17-comma, the avicennma and the chromatic semitone are all tempered out by the 13-limit patent val, as stated.

Temperaments

Rank-two temperaments

Well temperaments

Important MOSes include:

Introductory materials

Music

Scores

Sound files

Instruments

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