17edo

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Prime factorization 17 (prime)
Step size 70.5882¢
Fifth 10\17 (705.882¢)
(semiconvergent)
Semitones (A1:m2) 2:1 (141.2¢ : 70.59¢)
Consistency limit 3
Distinct consistency limit 3
English Wikipedia has an article on:

17 equal divisions of the octave (17edo), or 17(-tone) equal temperament (17tet, 17et) when viewed from a regular temperament perspective, is the tuning system derived from dividing the octave in 17 equal steps, each around 70.6 cents in size.

Theory

17edo 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 triad is quite dissonant as the major third is closer to 9/7 than the traditional 5/4.

Instead, the tonic chords of 17edo could be considered to be the tetrad 6:7:8:9 and its utonal inversion (representing 14:16:18:21 as 64/63 is tempered out), 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 17edo 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 17edo 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.

Another construction of septimal chords involves 4:7:12 and its inversion 7:12:21. These triads span a twelfth, realized in 17edo as 0-14-27 and 0-13-27, respectively. To this we may add 0-12-14-27, representing 8:13:14:24, or 0-13-15-27, representing 7:12:13:21.

Odd harmonics

Approximation of odd harmonics in 17edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) +3.9 -33.4 +19.4 +7.9 +13.4 +6.5 -29.4 -34.4 -15.2 +23.3 +7.0
relative (%) +6 -47 +27 +11 +19 +9 -42 -49 -21 +33 +10
Steps
(reduced) 		27
(10) 		39
(5) 		48
(14) 		54
(3) 		59
(8) 		63
(12) 		66
(15) 		69
(1) 		72
(4) 		75
(7) 		77
(9)

Miscellaneous properties

17edo is the seventh prime edo, following 13edo and coming before 19edo.

Intervals

See also: 17edo solfege
Edosteps Cents Extended circle-of-fifths notation * Ups and Downs Notation 3L 4s notation Approximate Ratios†
0 0.00 Unison D unison P1 D unison D 1/1
1 70.59 Minor 2nd
(Semiaugmented 1sn) 		Eb
(D+) 		up unison,

minor 2nd

^1, m2 Eb augmented 1sn D# 22/21, 25/24, 26/25, 28/27, 33/32, 24/23
2 141.18 Augmented 1sn
(Neutral 2nd) 		D#
(Ed) 		augmented 1sn,

mid 2nd

A1, ~2 vE minor 2nd Eb 13/12, 12/11, 14/13, 25/23
3 211.76 Major 2nd E major 2nd M2 E major 2nd E 9/8, 8/7, 28/25, 25/22, 26/23
4 282.35 Minor 3rd F minor 3rd m3 F diminished 3rd Fb 13/11, 7/6
5 352.94 Diminished 4th
(Neutral 3rd) 		Gb
(F+) 		mid 3rd ~3 ^F perfect 3rd F 11/9, 16/13, 28/23
6 423.53 Major 3rd
(Semidiminished 4th) 		F#
(Gd) 		major 3rd M3 F# augmented 3rd F# 32/25, 9/7, 14/11, 33/26, 23/18
7 494.12 Perfect 4th G perfect 4th P4 G minor 4th G 4/3, 21/16
8 564.71 Diminshed 5th
(Semiaugmented 4th) 		Ab
(G+) 		mid 4th,
diminished 5th 		~4, d5 ^G, Ab major 4th G# 11/8, 18/13, 32/23
9 635.29 Augmented 4th
(Semidiminished 5th) 		G#
(Ad) 		augmented 4th,
mid 5th 		A4, ~5 G#, vA minor 5th Ab 16/11, 13/9, 23/16
10 705.88 Perfect 5th A perfect 5th P5 A major 5th A 3/2, 32/21
11 776.47 Minor 6th
(Semiaugmented 5th) 		Bb
(A+) 		minor 6th m6 Bb diminished 6th Bb 25/16, 14/9, 11/7, 52/33, 36/23
12 847.06 Augmented 5th
(Neutral 6th) 		A#
(Bd) 		mid 6th ~6 vB perfect 6th B 13/8, 18/11, 23/14
13 917.65 Major 6th B major 6th M6 B augmented 6th B# 17/10, 22/13, 12/7
14 988.24 Minor 7th C minor 7th m7 C minor 7th Cb 16/9, 7/4, 25/14, 44/25, 23/13
15 1058.82 Diminished 8ve
(Neutral 7th) 		Db
(C+) 		mid 7th ~7 ^C major 7th C 11/6, 24/13, 13/7, 46/25
16 1129.41 Major 7th
(Semidiminished 8ve) 		C#
(Dd) 		major 7th,

down 8ve

M7, v8 C# diminished 8ve Db 21/11, 25/13, 48/25, 27/14, 64/33, 23/12
17 1200.00 Octave D octave P8 D octave D 2/1

* Half-sharps and half-flats (denoted "+" and "d", respectively) can be used to alter the note by a single step, 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.

† Ratios based on treating 17edo as a 2.3.7.11.13.23.25 subgroup temperament.

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.

Notation

Sagittal

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

17edo Sagittal.png

JI approximation

15-odd-limit interval 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.

15-odd-limit intervals by direct approximation (even if inconsistent)
Interval, complement Error (abs, ¢) Error (rel, %)
18/13, 13/9 1.324 1.9
13/12, 24/13 2.604 3.7
4/3, 3/2 3.927 5.6
11/9, 18/11 5.533 7.8
14/11, 11/7 6.021 8.5
16/13, 13/8 6.531 9.3
13/11, 22/13 6.857 9.7
9/8, 16/9 7.855 11.1
12/11, 11/6 9.461 13.4
9/7, 14/9 11.555 16.4
14/13, 13/7 12.878 18.2
11/8, 16/11 13.388 19.0
7/6, 12/7 15.482 21.0
7/5, 10/7 17.806 25.2
8/7, 7/4 19.409 27.5
15/14, 28/15 21.734 30.8
11/10, 20/11 23.828 33.8
15/11, 22/15 27.755 39.3
10/9, 9/5 29.361 41.6
16/15, 15/8 29.445 41.7
13/10, 20/13 30.685 43.5
6/5, 5/3 33.288 47.2
5/4, 8/5 33.373 47.3
15/13, 26/15 34.612 49.0
15-odd-limit intervals by patent val mapping
Interval, complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/9, 18/13 1.324 1.9
13/12, 24/13 2.604 3.7
3/2, 4/3 3.927 5.6
11/9, 18/11 5.533 7.8
11/7, 14/11 6.021 8.5
13/8, 16/13 6.531 9.3
13/11, 22/13 6.857 9.7
9/8, 16/9 7.855 11.1
11/6, 12/11 9.461 13.4
9/7, 14/9 11.555 16.4
13/7, 14/13 12.878 18.2
11/8, 16/11 13.388 19.0
7/6, 12/7 15.482 21.9
7/4, 8/7 19.409 27.5
15/8, 16/15 29.445 41.7
5/4, 8/5 33.373 47.3
15/13, 26/15 35.976 51.0
5/3, 6/5 37.300 52.8
13/10, 20/13 39.904 56.5
9/5, 10/9 41.227 58.4
15/11, 22/15 42.833 60.7
11/10, 20/11 46.760 66.2
15/14, 28/15 48.855 69.2
7/5, 10/7 52.782 74.8

Selected 13-limit intervals

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Tuning 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.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [27 -17⟩ [⟨17 27]] -1.24 1.24 1.76
2.3.7 64/63, 17496/16807 [⟨17 27 48]] -3.13 2.85 4.05
2.3.7.11 64/63, 99/98, 243/242 [⟨17 27 48 59]] -3.31 2.49 3.54
2.3.7.11.13 64/63, 78/77, 99/98, 144/143 [⟨17 27 48 59 63]] -3.00 2.31 3.28

17et is lower in relative error than any previous equal temperaments in the no-5 11- and 13-limit. The next ETs doing better in these subgroups are 41 and 207, respectively.

Commas

17et 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 Lazoyoyo 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.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per Octave 		Generator Cents Associated
Ratio 		Temperament
1 2\17 141.18 13/12 Bleu / progression (17c)
1 3\17 211.76 8/7~9/8 Machine
1 4\17 282.35 13/11 Huxley[clarification needed]


Lovecraft

1 5\17 352.94 11/9 Neutral
Suhajira / maqamic (17c) / beatles (17c) / dicotic (17)
Hemif / mohamaq (17c) / salsa (17)
1 6\17 423.53 9/7 Skwares / squares (17c) / sentinel (17) / sidi (17)
1 7\17 494.12 4/3 Archy / supra / quasisuper (17c) / dominant (17c) / superpyth (17) / schism (17)
1 8\17 564.71 7/5 Lee / liese (17c) / pycnic (17)
Progress (17c)

Scales

MOS scales

Main article: MOS scales of 17edo
  • 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)

Well temperaments

Introductory materials

Music

See also: [[
Category:17edo tracks]]

Scores

Christopher Bailey (site)
Daniel Wolf
Georg Hajdu (site)
Inthar
Jacob Barton
Microtonalismo (site)


Sound files

YouTube playlist of 17edo pieces
YouTube videos tagged with 17edo
Compositions from the Seventeen Tone Piano Project
Aaron Andrew Hunt
Aaron Krister Johnson (site)
Alex Ness
Andrew Heathwaite
Beheld
benyamind
Chris Vaisvil
  • Shanidar Cave, a piece in 17edo that features an electric 17edo guitar and what is essentially an electric tanpura which ends up making this a sort of fusion of middle eastern and Indian music in a sense
  • On the Shores of the Dead Sea: blog | video
  • Only in Disneyland: blog | MP3 (guitar solo)
  • 17 Reasons I Hate the Blues: blog | MP3
  • Klingon Opera Overture: blog | MP3
  • Seventeen Selfless Notes: blog | MP3
  • 17et Jazz: blog | MP3 (60 x 60 winner)
  • 17 Pink Tuxedos: blog | MP3
  • Devil in the Deep Blue Sea: blog | MP3 (blues collaboration between The Two Regs (vocals / lyrics) and Norm Harris (percussion) and Chris Vaisvil (17 note per octave electric guitar and fretless bass))
  • Seventeen Years in the Sixties: blog | MP3
  • CT Scan: blog | MP3
  • Fish and a Grenade: blog | MP3 (parental advisory: language)
  • Seventeen Unsteady Hands: blog | video of performance
  • The Pond: blog | video
  • Graveyard: blog | MP3
  • For Brass and Voice Choirs in 17 edo: blog | MP3
  • And I Became One With My Pet Fungi: blog | MP3
  • Counterintuitive: blog | MP3 (guitar solo)
  • Flying Into O'Hare: blog | MP3
Christopher Bailey (site)
Diamond Doll (Xen-Pop)
Francium
Gregory Sanchez (site)
Herman Miller (site[dead link])
Igliashon Jones
Jon Lyle Smith
Jutomi
Marmalade Man
Nick, The NRG
Rick McGowan
Stephen Weigel (site)
The Mercury Tree (site)
Xotla

Instruments

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17P1050829r.JPG

See also

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