17edo

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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

An introduction to 17-EDO theory, through the eyes of the SeventeenTonePianoProject: SeventeenTheory.

Another introduction into 17-EDO theory: The 17-tone Puzzle [dead link] by George Secor.

17edo Solfege. 17edo tetrachords. Proyect 17-Perú [forbidden]

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

Degree Cents Names of Intervals ups and downs notation Approximate Ratios* Temperament(s)generated
0 0 Unison unison P1 C 1/1
1 70.59 Super Unison/Minor Second minor 2nd m2 Db 25/24, 26/25, 33/32
2 141.18 Augmented Unison/Neutral Second mid 2nd ~2 Dv 13/12, 12/11, 14/13 Bleu
3 211.765 Major Second/Sub Third major 2nd M2 D 9/8, 25/22, 8/7, 28/25 Machine
4 282.35 Minor Third/Super Second minor 3rd m3 Eb 13/11, 7/6 Huxley
5 352.94 Augmented Second/Neutral Third/

Diminished Fourth

mid 3rd ~3 Ev 11/9, 16/13 Maqamic/Hemif
6 423.53 Major Third/Sub Fourth major 3rd M3 E 32/25, 33/26, 9/7, 14/11 Skwares
7 494.11 Perfect Fourth perfect 4th P4 F 4/3 Supra
8 564.71 Super FourthDiminished Fifth up 4th,

diminished 5th

^4, d5 F^, Gb 11/8, 18/13 Progress
9 635.29 Augmented Fourth/Sub Fifth augmented 4th,

down 5th

A4, v5 F#, Gv 16/11, 13/9, 23/16 Progress
10 705.88 Perfect Fifth perfect 5th P5 G 3/2 Supra
11 776.47 Super Fifth/Minor Sixth minor 6th m6 Ab 25/16, 14/9, 11/7 Skwares
12 847.06 Augmented Fifth/Neutral Sixth/

Diminished Seventh

mid 6th ~6 Av 13/8, 18/11 Maqamic/hemif
13 917.65 Major Sixth/Sub Seventh major 6th M6 A 17/10, 22/13,12/7 Huxley
14 988.235 Minor Seventh/Super Sixth minor 7th m7 Bb 16/9, 7/4, 25/14 Machine
15 1058.82 Augmented Sixth/Neutral Seventh/

Diminished Octave

mid 7th ~7 Bv 11/6, 24/13, 13/7 Bleu
16 1129.41 Major Seventh/Sub Octave major 7th M7 B 25/13, 48/25, 64/33
17 1200 Perfect Octave octave P8 C 2/1

*Ratios based on treating 17edo as a 2.3.7.11.13.25 subgroup

Combining ups and downs notation with Kite's 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 Ev G C~ C mid
ru 14:18:27 0-6-10 C E G C C major or C

0-4-9 = C Eb Gv = Cm(v5) = C minor down-five

0-5-9 = C Ev G = 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 Ev G Bb = C7(~3) = C seven mid-three

0-6-10-15 = C E G Bv = C,~7 = C mid seven

0-5-10-15 = C Ev G Bv = C.~7 = C dot mid seven

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 17edo (ordered by absolute error).

Best direct mapping, even if inconsistent

Interval, complement Error (abs., in cents)
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

Patent val mapping

Interval, complement Error (abs., in cents)
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

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17ed2-001.svg

Commas

17 EDO tempers out the following commas. (Note: This assumes val < 17 27 39 48 59 63 |, cent values ​​rounded to 5 digits.)

Comma Monzo Value (Cents) Name 1 Name 2 Name 3
| 27 -17 > 66.765 17-Comma
25/24 | -3 -1 2 > 70.762 Chromatic semitone Dicot comma
32805/32768 | -15 8 1 > 1.9537 Schisma
64/63 | 6 -2 0 -1 > 27.264 Septimal Comma Archytas' Comma Leipziger Komma
245/243 | 0 -5 1 2 > 14.191 Sensamagic
1728/1715 | 6 3 -1 -3 > 13.074 Orwellisma Orwell Comma
| -6 -8 2 5 > 1.1170 Wizma
99/98 | -1 2 0 -2 1 > 17.576 Mothwellsma
896/891 | 7 -4 0 1 -1 > 9.6880 Pentacircle
243/242 | -1 5 0 0 -2 > 7.1391 Rastma
385/384 | -7 -1 1 1 1 > 4.5026 Keenanisma
525/512 | -9 1 2 1 > 43.408 Avicennma Avicennma's Enharmonic Diesis

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.

Scales

Temperaments

Music

Scores

Sound files

Instruments

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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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17P1050829r.JPG