17edo
← 16edo | 17edo | 18edo → |
(semiconvergent)

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
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:
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.
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 |
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
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 |
- ↑ 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
- List of 17edo rank two temperaments by badness
- List of edo-distinct 17c rank two temperaments
- List of edo-distinct 17et rank two temperaments
- List of edo-distinct 17et no-fives rank two temperaments
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] |
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
- George Secor’s well temperament of this tuning
- Nicolai's 17-note Well Temperament
- Flora's 17-note well temperament
Introductory materials
- SeventeenTheory, an introduction to 17edo theory, through the eyes of the SeventeenTonePianoProject.
- The 17-tone Puzzle by George Secor, another introduction into 17edo theory.
- 17edo tetrachords
- Proyect 17-Perú [forbidden]
Music
- See also: [[
- Category:17edo tracks]]
Scores
- Prelude (PDF)
- Charles Loli 17edo[dead link] music for guitar heptadecatonic (2001) and armony inductive microtonally (1993)
- Heptadecatonic Peruvian[clarification needed]
Sound files
- Compositions from the Seventeen Tone Piano Project
- seventeen-tone piano project phase I
- Seventeen-tone piano project phase II
- Seventeen-tone piano project phase III
- Helas, pitié in 17edo with stretched octaves
- sing a blue (composed 2008, recorded 2010). This and the other pieces below by Andrew for cümbüş, steel tubes & voice.
- stringfinger it everybean (composed 2008, recorded 2010)
- cat feet belly (composed 2008, recorded 2010)
- 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
- Waltz
- Lost & Found Things #2 (studio) (live)
- Balladei (live), in 17edo, 29edo and 12edo (CD).
- Diamond Doll (Xen-Pop)
- Arm's Length Spotify, YouTube Link, Bandcamp Link
- A Calamitous Simultaneity in 17edo and 22edo
- First Impressions[dead link]
- I Insist[dead link]
- Etude no1 for 2 Pianos in 17 Equal Temperament[dead link]
- Etude no2 for 2 Pianos in 17 Equal Temperament[dead link]
- Demanding Two Faces (xen-pop)
- Where were you at the Apocalypse? (xen-pop)
- Spidermilk (prog album)
- Cretaceous Cosmos
- Rive
- An Other
- Xotla's Microtonal Funk & Blues Vol. 1 (whole album in 17edo)
- Field of mirrors
- Hidden Chamber
- Unrealities
- Rising Entities
- Sojourn
- Veering
- Sinking
- Recoil
- Restore
- Seethe
- Seeds of Moonlight (whole album in 17edo)
- Journey Lands
Instruments
External image: http://sphotos.ak.fbcdn.net/hphotos-ak-snc4/hs883.snc4/71639_167001659983806_100000219181856_601995_1526184_n.jpg [dead link]
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External image: http://sphotos.ak.fbcdn.net/hphotos-ak-ash2/hs382.ash2/66019_167001006650538_100000219181856_601987_48585_n.jpg [dead link]
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- 17 note per octave conversion from a "standard" Stratocaster copy - conversion by Brad Smith