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 17-EDO, 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 |
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.
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 |
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
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
- Otonal 17
- Blues Peruvian 17edo
- 17edo neutral scale
- Scorp
- Screamapillar
- Hydra
- MOS scales of 17edo (horograms)
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 |
- ↑ 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
- George Secor’s well temperament of this tuning
- Sam's 17-note Well Temperament
- Flora's 17-note well temperament
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)
- lovecraft 4L5s 313131311 (4\17, 1\1)
Introductory materials
- 17-edo example composition by Schrodingasdawg (File:17edo 1MC score.pdf)
- SeventeenTheory, an introduction to 17-EDO theory, through the eyes of the SeventeenTonePianoProject.
- The 17-tone Puzzle by George Secor, another introduction into 17-EDO theory.
- 17edo Solfege
- 17edo tetrachords
- Proyect 17-Perú [forbidden]
Music
Scores
- Prelude (PDF) by Daniel Wolf
- Heptadecatonic Drops by Georg Hajdu
- Klangmoraste by Georg Hajdu
- Charles Loli 17edo [forbidden] music for guitar heptadecatonic (2001) and armony inductive microtonally (1993)
- microtonalismo Heptadecatonic Peruvian
- Multiverse by Jacob Barton
- Balladei by Christopher Bailey
- Sarabande from the Locrian Suite by Inthar
Sound files
- Arm's Length by Diamond Doll (xen-pop)
- Spidermilk (prog album) by The Mercury Tree
- Demanding Two Faces (xen-pop) by Stephen Weigel
- Where were you at the Apocalypse? (xen-pop) by Stephen Weigel*
- Shanidar Cave is a piece in 17 edo that features an electric 17 edo 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. by Chris Vaisvil
- Puhlops and Laugua's Big Adventure by Aaron Krister Johnson
- Adagio for Margo by Aaron Krister Johnson
- Transformation by Herman Miller [dead link]
- Waltz by Christopher Bailey
- Lost & Found Things #2 (studio) (live) by Christopher Bailey
- Balladei (live) (in 17, 29 and 12) by Christopher Bailey , CD available here .
- Two-Part Invention in 17ET [dead link] by Aaron Andrew Hunt
- Overture to A Midsummer Night's Dream by Rick McGowan
- Fairy Lullaby from A Midsummer Night's Dream by Rick McGowan
- A Calamitous Simultaneity by Igliashon Jones (17edo and 22edo)
- First Impressions by Igliashon Jones
- I Insist by Igliashon Jones
- Etude no1 for 2 Pianos in 17 Equal Temperament [dead link] and Etude no2 for 2 Pianos in 17 Equal Temperament [dead link] by Jon Lyle Smith
- A number of compositions from seventeen-tone piano project phase I , seventeen-tone piano project phase II, seventeen-tone piano project phase III.
- sing a blue by Andrew Heathwaite (composed 2008, recorded 2010). This and the other pieces below by Andrew for cümbüş, steel tubes & voice.
- stringfinger it everybean by Andrew Heathwaite (composed 2008, recorded 2010).
- cat feet belly by Andrew Heathwaite (composed 2008, recorded 2010).
- 17 Tone Jam by Marmalade Man
- youtube videos tagged with 17edo
- No Love by Gregory Sanchez
- Micro7 by Ralph Jarzombek
- Helas, pitié by Alex Ness (in 17edo with stretched octaves)
- March of the Mushrooms by Jutomi
- by Chris Vaisvil:
- 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
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