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 17edo, 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

Notations

From the appendix to The Sagittal Songbook by Jacob A. Barton, a diagram of how to notate 17-EDO in the Revo flavor of Sagittal:

17edo Sagittal.png

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
13 364/363 [2 -1 0 1 -2 1 4.763 Tholuluzo Gentle 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:

  • diatonic (leapfrog/archy) 5L2s 3331331 (10\17, 1\1)
  • maqamic 3L4s 3232322 (5\17, 1\1)
  • maqamic 7L3s 2221221221 (5\17, 1\1)
  • squares 3L5s 1141414 (6\17, 1\1)
  • squares 3L8s 13113113 (6\17, 1\1)
  • Pathological squares 3L11s 11211121112 (6\17, 1\1)
  • lovecraft 4L5s 313131311 (4\17, 1\1)
  • Pathological 1L 13s 4 1 1 1 1 1 1 1 1 1 1 1 1 (1\17, 1\1)
  • Pathological 1L 14s 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\17, 1\1)
  • Pathological 2L 13s 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 (8\17, 1\1)
  • Pathological 1L 15s 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\17, 1\1)

Introductory materials

Music

Scores

Sound files

Instruments

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17P1050829r.JPG