17edo
← 16edo | 17edo | 18edo → |
(semiconvergent)
17 equal divisions of the octave (abbreviated 17edo or 17ed2), also called 17-tone equal temperament (17tet) or 17 equal temperament (17et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 17 equal parts of about 70.6 ¢ each. Each step represents a frequency ratio of 21/17, or the 17th root of 2.
Theory
17edo's perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of 12edo's, lending itself to an expressive diatonic scale. Meanwhile, it approximates harmonics 7, 11, 13, and 23 to reasonable degrees, despite completely missing harmonic 5. Thus it 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 these harmonics are all tempered 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.
Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.
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.
In the no-5 13-odd-limit, 17edo maintains the smallest relative error [clarification needed ] of any edo until 166edo.
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 (%) | +5.6 | -47.3 | +27.5 | +11.1 | +19.0 | +9.3 | -41.7 | -48.7 | -21.5 | +33.1 | +9.9 | |
Steps (reduced) |
27 (10) |
39 (5) |
48 (14) |
54 (3) |
59 (8) |
63 (12) |
66 (15) |
69 (1) |
72 (4) |
75 (7) |
77 (9) |
Subsets and supersets
17edo is the seventh prime edo, following 13edo and coming before 19edo. 34edo, which doubles it, provides a good correction to the harmonic 5.
Intervals
# | Cents | Approximate Ratios[note 1] | Circle-of-fifths Notation† | Ups and Downs Notation | SKULO notation (U = 1) | 3L 4s Notation | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0.00 | 1/1 | Unison | D | unison | P1 | D | unison | P1 | D | unison | D |
1 | 70.59 | 24/23, 25/24, 26/25, 27/26, 28/27 | Minor 2nd (Semiaugmented 1sn) |
Eb (Dt) |
up unison, minor 2nd |
^1, m2 | Eb | uber unison, minor 2nd |
U1, m2 | UD, Eb | augmented 1sn | D# |
2 | 141.18 | 12/11, 13/12, 14/13, 25/23 | Augmented 1sn (Neutral 2nd) |
D# (Ed) |
augmented 1sn, mid 2nd |
A1, ~2 | vE | neutral 2nd | N2 | UEb, uE | minor 2nd | Eb |
3 | 211.76 | 8/7, 9/8, 17/15, 25/22, 26/23 | Major 2nd | E | major 2nd | M2 | E | major 2nd | M2 | E | major 2nd | E |
4 | 282.35 | 7/6, 13/11, 20/17 | Minor 3rd | F | minor 3rd | m3 | F | minor 3rd | m3 | F | diminished 3rd | Fb |
5 | 352.94 | 11/9, 27/22, 16/13, 39/32 | Diminished 4th (Neutral 3rd) |
Gb (Ft) |
mid 3rd | ~3 | ^F | neutral 3rd | N3 | UF, uF# | perfect 3rd | F |
6 | 423.53 | 9/7, 14/11, 23/18, 32/25, 51/40 | Major 3rd (Semidiminished 4th) |
F# (Gd) |
major 3rd | M3 | F# | major 3rd | M3 | F# | augmented 3rd | F# |
7 | 494.12 | 4/3, 21/16, 85/64 | Perfect 4th | G | perfect 4th | P4 | G | perfect 4th | P4 | G | minor 4th | G |
8 | 564.71 | 11/8, 18/13, 25/18, 32/23 | Diminished 5th (Semiaugmented 4th) |
Ab (Gt) |
mid 4th, diminished 5th |
~4, d5 | ^G, Ab | uber 4th/ neutral 4th |
U4/N4 | UG | major 4th | G# |
9 | 635.29 | 13/9, 16/11, 23/16, 36/25 | Augmented 4th (Semidiminished 5th) |
G# (Ad) |
augmented 4th, mid 5th |
A4, ~5 | G#, vA | unter 5th/ neutral 5th |
u5/N5 | uA | minor 5th | Ab |
10 | 705.88 | 3/2, 32/21, 128/85 | Perfect 5th | A | perfect 5th | P5 | A | perfect 5th | P5 | A | major 5th | A |
11 | 776.47 | 11/7, 14/9, 25/16, 36/23, 80/51 | Minor 6th (Semiaugmented 5th) |
Bb (At) |
minor 6th | m6 | Bb | minor 6th | m6 | Bb | diminished 6th | Bb |
12 | 847.06 | 13/8, 18/11, 44/27, 64/39 | Augmented 5th (Neutral 6th) |
A# (Bd) |
mid 6th | ~6 | vB | neutral 6th | N6 | UBb, uB | perfect 6th | B |
13 | 917.65 | 12/7, 17/10, 22/13 | Major 6th | B | major 6th | M6 | B | major 6th | M6 | B | augmented 6th | B# |
14 | 988.24 | 7/4, 16/9, 23/13, 30/17, 44/25 | Minor 7th | C | minor 7th | m7 | C | minor 7th | m7 | C | minor 7th | Cb |
15 | 1058.82 | 11/6, 13/7, 24/13, 46/25 | Diminished 8ve (Neutral 7th) |
Db (Ct) |
mid 7th | ~7 | ^C | neutral 7th | N7 | UC, uC# | major 7th | C |
16 | 1129.41 | 23/12, 25/13, 27/14, 48/25, 52/27 | Major 7th (Semidiminished 8ve) |
C# (Dd) |
major 7th, down 8ve |
M7, v8 | C# | major 7th, unter octave |
M7, u8 | C#, uD | diminished 8ve | Db |
17 | 1200.00 | 2/1 | Octave | D | octave | P8 | D | octave | P8 | D | octave | D |
† Half-sharps and half-flats (denoted "t" and "d", respectively) can be used to alter the note by a single step, since sharps and flats each span two edosteps. 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.
Interval quality and chord names in color notation
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 |
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
Quarter tone notation
Since a sharp raises by 2 steps, 17edo can be notated using quarter-tone accidentals.
Step Offset | −4 | −3 | −2 | −1 | 0 | +1 | +2 | +3 | +4 |
---|---|---|---|---|---|---|---|---|---|
Symbol |
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:
Approximation to JI
15-odd-limit interval mappings
The following tables show how 15-odd-limit intervals are represented in 17edo. Prime harmonics are in bold; inconsistent intervals are in italics.
Interval and 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 |
7/5, 10/7 | 17.806 | 25.2 |
7/4, 8/7 | 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 |
9/5, 10/9 | 29.361 | 41.6 |
15/8, 16/15 | 29.445 | 41.7 |
13/10, 20/13 | 30.685 | 43.5 |
5/3, 6/5 | 33.288 | 47.2 |
5/4, 8/5 | 33.373 | 47.3 |
15/13, 26/15 | 34.612 | 49.0 |
Interval and 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 equal temperaments doing better in these subgroups are 41 and 207, respectively.
Uniform maps
Min. size | Max. size | Wart notation | Map |
---|---|---|---|
16.5000 | 16.5636 | 17bccdddeeefff | ⟨17 26 38 46 57 61] |
16.5636 | 16.5810 | 17bccdeeefff | ⟨17 26 38 47 57 61] |
16.5810 | 16.6196 | 17bdeeefff | ⟨17 26 39 47 57 61] |
16.6196 | 16.6212 | 17bdeeef | ⟨17 26 39 47 57 62] |
16.6212 | 16.7196 | 17bdef | ⟨17 26 39 47 58 62] |
16.7196 | 16.8899 | 17def | ⟨17 27 39 47 58 62] |
16.8899 | 16.9103 | 17de | ⟨17 27 39 47 58 63] |
16.9103 | 16.9198 | 17d | ⟨17 27 39 47 59 63] |
16.9198 | 17.0117 | 17 | ⟨17 27 39 48 59 63] |
17.0117 | 17.1601 | 17c | ⟨17 27 40 48 59 63] |
17.1601 | 17.1994 | 17cff | ⟨17 27 40 48 59 64] |
17.1994 | 17.2760 | 17ceeff | ⟨17 27 40 48 60 64] |
17.2760 | 17.3506 | 17cddeeff | ⟨17 27 40 49 60 64] |
17.3506 | 17.4304 | 17bbcddeeff | ⟨17 28 40 49 60 64] |
17.4304 | 17.4424 | 17bbcddeeffff | ⟨17 28 40 49 60 65] |
17.4424 | 17.4884 | 17bbcccddeeffff | ⟨17 28 41 49 60 65] |
17.4884 | 17.5000 | 17bbcccddeeeeffff | ⟨17 28 41 49 61 65] |
Commas
17et tempers out the following commas. (Note: This assumes patent val ⟨17 27 39 48 59 63 69], cent values rounded to 5 digits.)
Prime limit |
Ratio[1] | Monzo | Cents | Color name | Name |
---|---|---|---|---|---|
3 | (18 digits) | [27 -17⟩ | 66.765 | Sasawa | 17-comma |
5 | 25/24 | [-3 -1 2⟩ | 70.762 | Yoyo | Dicot comma |
5 | 32805/32768 | [-15 8 1⟩ | 1.9537 | Layo | Schisma |
7 | 525/512 | [-9 1 2 1⟩ | 43.408 | Lazoyoyo | Avicennma |
7 | 64/63 | [6 -2 0 -1⟩ | 27.264 | Ru | Septimal comma |
7 | 245/243 | [0 -5 1 2⟩ | 14.191 | Zozoyo | Sensamagic comma |
7 | 1728/1715 | [6 3 -1 -3⟩ | 13.074 | Triru-agu | Orwellisma |
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 comma |
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 | Minor minthma |
17 | 136/135 | [3 -3 -1 0 0 0 1⟩ | 12.776 | Sogu 2nd | Diatisma |
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 8ve |
Generator | Cents | Associated ratio |
Temperament |
---|---|---|---|---|
1 | 2\17 | 141.18 | 13/12 | Bleu / progression (17c) / glacier |
1 | 3\17 | 211.76 | 8/7~9/8 | Machine |
1 | 4\17 | 282.35 | 13/11 | Huxley / lovecraft / subklei (17c) |
1 | 5\17 | 352.94 | 11/9 | Suhajira / neutrominant (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) Fiventeen |
1 | 8\17 | 564.71 | 7/5 | Lee / liese (17c) / pycnic (17) Progress (17c) |
Scales
- Antipental blues: 4 3 1 2 4 3
- Blues Peruvian: 4 3 1 1 1 4 3
- Hydra: 3 3 1 1 2 3 2 1 1
- Husayni Ascending: 2 2 3 3 2 2 3
- Otonal 17: 3 2 3 2 2 2 3
- Scorp: 3 2 3 1 3 2 3
- Screamapillar: 3 3 2 2 3 3 1
MOS scales
- diatonic (leapfrog/archy) 5L 2s 3 3 3 1 3 3 1 (10\17, 1\1)
- neutrominant 3L 4s 3 2 3 2 3 2 2 (5\17, 1\1)
- neutrominant 7L 3s 2 2 2 1 2 2 1 2 2 1 (5\17, 1\1)
- squares 3L 5s 1 1 4 1 4 1 4 (6\17, 1\1)
- squares 3L 8s 1 3 1 1 3 1 1 3 (6\17, 1\1)
- Pathological squares 3L 11s 1 1 2 1 1 1 2 1 1 1 2 (6\17, 1\1)
- lovecraft 4L 5s 3 1 3 1 3 1 3 1 1 (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
- 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
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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- 17 note per octave conversion from a "standard" Stratocaster copy - conversion by Brad Smith
- 17edo soprano Harmony ukulele with a 3D printed fretboard - conversion by Tristan Bay
See also
Notes
- ↑ Based on treating 17edo as a 2.3.25.7.11.13.85.23 subgroup temperament; other approaches are also possible.
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints