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 is the next smallest edo to have a diatonic perfect fifth after 12edo, and is quite popular for that reason. The perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of 12edo's, lending itself to a diatonic scale with more constrasting large and small steps, so it can be seen as a tuning that emphasizes the hardness of Pythagorean tuning rather than mellowing it out as in meantone. It completely misses harmonic 5, with 5/4 and 6/5 both being about halfway between its steps, but it approximates harmonics 7, 11, 13, and 23 acceptably, with a sharp tuning for all of them. It can thus be treated as a temperament of the 2.3.25.7.11.13.23 subgroup or any of its subsets, where it is quite accurate for its size.
A notable comma it tempers out is 64/63, which equates the harmonic seventh 7/4 with the pythagorean minor seventh 16/9, while its patent val does not temper out 81/80. This makes 17edo by default a superpythagorean system rather than a meantone one, being very close to 1/7-comma superpyth. Other commas it tempers out can be found in the #Commas section, each of which has its own effect on the structure of 17edo. If one wants to approximate JI with prime 5, then 17edo would not be the best option, and it would be better to use other systems like 19edo, 22edo, 27edo, or 31edo instead. That said, the 17c val (written using wart notation) does temper out 81/80 (while improving consistency as shown below in #Approximation to JI), while still tempering out 64/63, thus placing it on the meantone spectrum with the dominant extension.
As a means of extending harmony
The diatonic major triad, which is 0–6–10 steps, is quite dissonant compared to 4:5:6, as the major third is over 37 cents sharp from the traditional 5/4, and is instead closer to 9/7 or 14/11. Instead, a different construction based on the 2.3.7 subgroup follows naturally from its support of superpyth, and may be preferred. Such chords include the tetrads 6:7:8:9 and its utonal inverse, realized in 17edo as 0–4–7–10 and 0–3–6–10, respectively, in addition to the sus2-4 chord, realized as 0–3–7–10. Possible chromatic alterations include but are not limited to an approximation of 12:13:16:18, 0–2–7–10, and an approximation of 8:9:11:12, 0–3–8–10. It is important to note that the chromatic semitone in 17edo is 2 steps, rather than 1 step as in 12edo or 19edo. Similarly, the fourth-spanning triad 6:7:8 and its inverse can be used, with their wide voicing realized in 17edo as 0–14–27 and 0–13–27, respectively. Extensions of these chords include 0–12–14–27, representing 8:13:14:24, and 0–13–15–27, representing 7:12:13:21.
Since the intervals of the 2.3.7-subgroup cluster around 5edo, a pentatonic system of interval classification may be preferred over the heptatonic one, with 7/6 becoming a major interval and 8/7~9/8 becoming a minor one.
Of course, scales generated by the perfect fifth are not the only scales 17edo contains. Another type of scale is neutral third scales, which are generated by half a fifth (5\17), and take the mos patterns 4L 3s (mosh) and 7L 3s (dicoid). Other notable scales include that of bleu and glacier (generated by 2\17), and skwares (generated by 6\17). Non-mos scales also exist; a more complete list can be found in the #Scales section.
Because the 5th harmonic is not well approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.
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) | |
| Harmonic | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 | 41 | 43 | 45 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.8 | +11.8 | +29.2 | -15.6 | +17.3 | -14.0 | +31.0 | +10.5 | -5.5 | -17.4 | -25.5 | -30.2 |
| Relative (%) | +5.4 | +16.7 | +41.4 | -22.1 | +24.5 | -19.8 | +43.9 | +14.8 | -7.8 | -24.7 | -36.2 | -42.8 | |
| Steps (reduced) |
79 (11) |
81 (13) |
83 (15) |
84 (16) |
86 (1) |
87 (2) |
89 (4) |
90 (5) |
91 (6) |
92 (7) |
93 (8) |
94 (9) | |
Subsets and supersets
17edo is the seventh prime edo, following 13edo and coming before 19edo. It does not contain any nontrivial subset edos, though it contains 17ed4 and 17ed8. 17ed8, built by taking every third step of 17edo, is a system where all odd harmonics up to the 21st are mapped exactly as in 17edo, except for the 11th. Beyond that, the 27th, 31st, 35th, and 39th harmonics are likewise mapped identically.
34edo, which doubles 17edo, provides a great correction to harmonics 5 and 17; while 68edo, which quadruples it, provides additionally the primes 7, 19, and 31.
Intervals
| # | Cents | Approximate ratios[note 1] | Circle-of-fifths notation[note 2] | Ups and downs notation (EUs: vvA1 and ^d2) |
SKULO notation (U = 1) | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.0 | 1/1 | Unison | D | unison | P1 | D | unison | P1 | D |
| 1 | 70.6 | 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 |
| 2 | 141.2 | 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 |
| 3 | 211.8 | 8/7, 9/8, 17/15, 25/22, 26/23 | Major 2nd | E | major 2nd | M2 | E | major 2nd | M2 | E |
| 4 | 282.4 | 7/6, 13/11, 20/17 | Minor 3rd | F | minor 3rd | m3 | F | minor 3rd | m3 | F |
| 5 | 352.9 | 11/9, 27/22, 16/13, 39/32 | Diminished 4th (Neutral 3rd) |
Gb (Ft) |
mid 3rd | ~3 | ^F | neutral 3rd | N3 | UF, uF# |
| 6 | 423.5 | 9/7, 14/11, 23/18, 32/25, 51/40 | Major 3rd (Semidiminished 4th) |
F# (Gd) |
major 3rd | M3 | F# | major 3rd | M3 | F# |
| 7 | 494.1 | 4/3, 21/16, 85/64 | Perfect 4th | G | perfect 4th | P4 | G | perfect 4th | P4 | G |
| 8 | 564.7 | 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 |
| 9 | 635.3 | 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 |
| 10 | 705.9 | 3/2, 32/21, 128/85 | Perfect 5th | A | perfect 5th | P5 | A | perfect 5th | P5 | A |
| 11 | 776.5 | 11/7, 14/9, 25/16, 36/23, 80/51 | Minor 6th (Semiaugmented 5th) |
Bb (At) |
minor 6th | m6 | Bb | minor 6th | m6 | Bb |
| 12 | 847.1 | 13/8, 18/11, 44/27, 64/39 | Augmented 5th (Neutral 6th) |
A# (Bd) |
mid 6th | ~6 | vB | neutral 6th | N6 | UBb, uB |
| 13 | 917.6 | 12/7, 17/10, 22/13 | Major 6th | B | major 6th | M6 | B | major 6th | M6 | B |
| 14 | 988.2 | 7/4, 16/9, 23/13, 30/17, 44/25 | Minor 7th | C | minor 7th | m7 | C | minor 7th | m7 | C |
| 15 | 1058.8 | 11/6, 13/7, 24/13, 46/25 | Diminished 8ve (Neutral 7th) |
Db (Ct) |
mid 7th | ~7 | ^C | neutral 7th | N7 | UC, uC# |
| 16 | 1129.4 | 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 |
| 17 | 1200.0 | 2/1 | Octave | D | octave | P8 | D | octave | P8 | D |
- ↑ Based on treating 17edo as a 2.3.25.7.11.13.85.23 subgroup temperament; other approaches are also possible.
- ↑ 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
Ups and downs notation
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp. The gamut runs D, ^D/Eb, D#/vE, E, F etc.
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Sharp symbol | 
|
 |
 |
|
 |
|
| Flat symbol |  |
 |
|
 |
|
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 notation
This notation uses the same sagittal sequence as edos 24, 31, and 38, and is a subset of the notation for 34edo.
Evo and Revo flavors
Alternative Evo flavor
Evo-SZ flavor
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to the Stein-Zimmerman notation.
Sagittal songbook diagram
From the appendix to The Sagittal Songbook by Jacob A. Barton, a diagram of how to notate 17edo in the Revo flavor of Sagittal:
3L 4s (mosh) notation
The notation of Neutral[7]. The generator is the perfect 3rd. Notes are denoted as sLsLsLs = DEFGABCD, and raising and lowering by a chroma (L − s), 1 edostep in this instance, is denoted by ♯ and ♭.
| # | Cents | Note | Name | Associated ratios |
|---|---|---|---|---|
| 0 | 0.0 | D | Perfect 1sn | 1/1 |
| 1 | 70.6 | D# | Augmented 1sn | 33/32 |
| 2 | 141.2 | Eb | Minor 2nd | 12/11 |
| 3 | 211.8 | E | Major 2nd | 9/8 |
| 4 | 282.4 | Fb | Diminished 3rd | 32/27 |
| 5 | 352.9 | F | Perfect 3rd | 11/9, 27/22 |
| 6 | 423.5 | F# | Augmented 3rd | 81/64 |
| 7 | 494.1 | G | Minor 4th | 4/3 |
| 8 | 564.7 | G# | Major 4th | 11/8 |
| 9 | 635.3 | Ab | Minor 5th | 16/11 |
| 10 | 705.9 | A | Major 5th | 3/2 |
| 11 | 776.5 | Bb | Diminished 6th | 128/81 |
| 12 | 847.1 | B | Perfect 6th | 18/11, 44/27 |
| 13 | 917.6 | B# | Augmented 6th | 27/16 |
| 14 | 988.2 | Cb | Minor 7th | 16/9 |
| 15 | 1058.8 | C | Major 7th | 11/6 |
| 16 | 1129.4 | Db | Diminished 8ve | 64/33 |
| 17 | 1200.0 | D | Perfect 8ve | 2/1 |
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 |
| 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 |
| 13/10, 20/13 | 30.685 | 43.5 |
| 5/3, 6/5 | 33.288 | 47.2 |
| 15/13, 26/15 | 34.612 | 49.0 |
| 5/4, 8/5 | 37.216 | 52.7 |
| 15/8, 16/15 | 41.143 | 58.3 |
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 |
|---|---|---|---|
| 16.7196 | 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] |
Commas
17et tempers out the following commas. (Note: This assumes patent val ⟨17 27 39 48 59 63 69 72 77], cent values rounded to 1/100 of a cent.)
| Prime limit |
Ratio[note 1] | Monzo | Cents | Color name | Name |
|---|---|---|---|---|---|
| 3 | (18 digits) | [27 -17⟩ | 66.76 | Sasawa | Gothic comma |
| 5 | 25/24 | [-3 -1 2⟩ | 70.76 | Yoyo | Dicot comma |
| 5 | 32805/32768 | [-15 8 1⟩ | 1.95 | Layo | Schisma |
| 7 | 64/63 | [6 -2 0 -1⟩ | 27.26 | Ru | Septimal comma |
| 7 | 525/512 | [-9 1 2 1⟩ | 43.41 | Lazoyoyo | Avicennma |
| 7 | 245/243 | [0 -5 1 2⟩ | 14.19 | Zozoyo | Sensamagic comma |
| 7 | 1728/1715 | [6 3 -1 -3⟩ | 13.07 | Triru-agu | Orwellisma |
| 7 | 17496/16807 | [3 7 0 -5⟩ | 69.56 | Quinru | Bleu comma |
| 7 | 19683/19208 | [-3 9 0 -4⟩ | 42.29 | Laquadru | Skwares comma |
| 7 | (12 digits) | [-6 -8 2 5⟩ | 1.12 | Quinzo-ayoyo | Wizma |
| 11 | 45/44 | [-2 2 1 0 -1⟩ | 38.91 | Luyo | Cake comma |
| 11 | 99/98 | [-1 2 0 -2 1⟩ | 17.58 | Loruru | Mothwellsma |
| 11 | 896/891 | [7 -4 0 1 -1⟩ | 9.69 | Saluzo | Pentacircle comma |
| 11 | 243/242 | [-1 5 0 0 -2⟩ | 7.14 | Lulu | Rastma, neutral thirds comma |
| 11 | 385/384 | [-7 -1 1 1 1⟩ | 4.50 | Lozoyo | Keenanisma |
| 13 | 40/39 | [3 -1 1 0 0 -1⟩ | 43.83 | Thuyo | Unintendo comma |
| 13 | 65/64 | [-6 0 1 0 0 1⟩ | 26.84 | Thoyo | Wilsorma |
| 13 | 78/77 | [1 1 0 -1 -1 1⟩ | 22.34 | Tholuru | Negustma |
| 13 | 144/143 | [4 2 0 0 -1 -1⟩ | 12.06 | Thulu | Grossma |
| 13 | 169/168 | [-3 -1 0 -1 0 2⟩ | 10.27 | Thothoru | Buzurgisma, dhanvantarisma |
| 13 | 352/351 | [5 -3 0 0 1 -1⟩ | 4.93 | Thulo | Major minthma |
| 13 | 364/363 | [2 -1 0 1 -2 1⟩ | 4.76 | Tholuluzo | Minor minthma |
| 13 | 512/507 | [9 -1 0 0 0 -2⟩ | 16.99 | Thuthu | Tridecimal neutral thirds comma |
| 13 | 1352/1331 | [3 0 0 0 -3 2⟩ | 27.10 | Bithotrilu | Lovecraft comma |
| 13 | 2197/2187 | [0 -7 0 0 0 3⟩ | 7.90 | Satritho | Threedie |
| 23 | 162/161 | [1 4 0 -1 0 0 0 0 -1⟩ | 10.72 | Twethuru | Minor kirnbergerisma |
| 23 | 208/207 | [4 -2 0 0 0 1 0 0 -1⟩ | 8.34 | Twethutho | Vicetone comma |
| 23 | 253/252 | [-2 -2 0 -1 1 0 0 0 1⟩ | 6.86 | Twetholoru | Middle neutravicema |
| 23 | 529/528 | [-4 -1 0 0 -1 0 0 0 2⟩ | 3.28 | Bitwetho-alu | Preziosisma |
| 23 | 736/729 | [5 -6 0 0 0 0 0 0 1⟩ | 16.54 | Satwetho | 23-limit Tenney/Cage comma (HEJI) |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.
Note that due to the inaccurate prime 5, the rather large commas 25/24, 525/512, 45/44, and 40/39 are all tempered out by 17edo's patent val.
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 | 3\17 | 211.76 | 26/23 | Shoal (trivial tuning) |
| 1 | 4\17 | 282.35 | 13/11 | Huxley / lovecraft / subklei (17c) |
| 1 | 5\17 | 352.94 | 11/9 | Suhajira / neutrominant (17c) / beatles (17c) / dichotic (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) |
Octave stretch or compression
17edo's approximations of harmonics 3, 7, 11, and 13 are all tempered sharp, so 17edo adapts well to slightly compressing the octave, if that is acceptable. 44ed6, 27edt and 56zpi are good demonstrations of this, where the octaves are flattened by about 1.5, 2.5 cents and 3 cents respectively.
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
- sLmLs: 2 5 3 5 2
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) (dedicated article: 17edo neutral scale)
- 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)
- lovecraft 4L 5s 3 1 3 1 3 1 3 1 1 (4\17, 1\1)
Well temperaments
Instruments
Fretted String Instruments
- 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
Keyboards
Lumatone mappings for 17edo are available.
The Striso Board can be tuned in many ways, but as it has 17 notes per octave and is organised in a circle of fifths based layout, it works particularly well with 17edo, letting you play far wider stretches of notes than a standard keyboard.
It is possible to rebuild some standard MIDI keyboards to have 17 note per octave by combining parts from multiple keyboards, as with the finished product shown in the following videos by Stephen Weigel and Chris Vaisvil:
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
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]