# 17edo

(Redirected from 17-EDO)

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 by George Secor.

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 pions 7mus Names of Intervals ups and downs notation Approximate Ratios* Temperament(s)generated
Pure octave 25/24et Pure octave 25/24et Pure octave 25/24et
0 Unison unison P1 C 1/1
1 70.59 70.67 74.82 74.91 90.35 (5A.5A16) 90.46 (5A.7616) Super Unison/Minor Second minor 2nd m2 Db 25/24, 26/25, 33/32
2 141.18 141.345 149.65 149.83 180.71 (B4.B516) 180.92 (B4.EC16) Augmented Unison/Neutral Second mid 2nd ~2 Dv 13/12, 12/11, 14/13 Bleu
3 211.765 212.02 224.47 224.74 271.06 (10F.0F16) 271.38 (10F.6216) Major Second/Sub Third major 2nd M2 D 9/8, 25/22, 8/7, 28/25 Machine
4 282.35 282.69 299.29 299.65 361.41 (169.6916) 361.84 (169.D816) Minor Third/Super Second minor 3rd m3 Eb 13/11, 7/6 Huxley
5 352.94 353.36 374.12 374.56 451.765 (1C3.C416) 452.3 (1C4.4E16) Augmented Second/Neutral Third/

Diminished Fourth

mid 3rd ~3 Ev 11/9, 16/13 Maqamic/Hemif
6 423.53 424.035 448.94 449.48 542.12 (21E.1E16) 542.76 (21E.C416) Major Third/Sub Fourth major 3rd M3 E 32/25, 33/26, 9/7, 14/11 Skwares
7 494.12 494.71 523.765 524.39 632.47 (278.78816) 633.225 (279.3A16) Perfect Fourth perfect 4th P4 F 4/3 Supra
8 564.71 565.38 588.59 599.3 722.82 (2D2.D316) 723.69 (2D3.B16) Super Fourth/Diminshed Fifth up 4th,

diminished 5th

^4, d5 F^, Gb 11/8, 18/13 Progress
9 635.29 636.05 673.41 674.215 813.18 (32D.2D16) 814.15 (32E.2516) Augmented Fourth/Sub Fifth augmented 4th,

down 5th

A4, v5 F#, Gv 16/11, 13/9, 23/16 Progress
10 705.88 706.72 748.235 749.13 903.53 (387.87816) 904.61 (388.9B16) Perfect Fifth perfect 5th P5 G 3/2 Supra
11 776.47 777.4 823.06 824.04 993.88 (3E1.E216) 995.07 (3E3.1116) Super Fifth/Minor Sixth minor 6th m6 Ab 25/16, 14/9, 11/7 Skwares
12 847.06 848.07 897.88 898.95 1084.235 (43C.3C16) 1085.53 (43D.8716) Augmented Fifth/Neutral Sixth/

Diminished Seventh

mid 6th ~6 Av 13/8, 18/11 Maqamic/hemif
13 917.65 918.74 972.71 973.87 1174.59 (496.9716) 1175.99 (497.FD16) Major Sixth/Sub Seventh major 6th M6 A 17/10, 22/13,12/7 Huxley
14 988.235 989.41 1047.53 1048.78 1264.94 (4F0.F116) 1266.45 (4F2.7316) Minor Seventh/Super Sixth minor 7th m7 Bb 16/9, 7/4, 25/14 Machine
15 1058.82 1060.09 1122.35 1123.69 1355.29 (54B.4B16) 1356.91 (54C.E916) Augmented Sixth/Neutral Seventh/

Diminished Octave

mid 7th ~7 Bv 11/6, 24/13, 13/7 Bleu
16 1129.41 1130.76 1197.18 1198.6 1445.65 (5A5.A616) 1447.37 (5A7.5F16) Major Seventh/Sub Octave major 7th M7 B 25/13, 48/25, 64/33
17 1200 1201.43 1272 1273.52 1536 (60016) 1537.83 (601.D516) 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

# 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.

# Instruments

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