editing disabled

Deutsch - 日本語

17 tone equal temperament

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.
17edo Solfege. 17edo tetrachords. Proyect 17-Perú

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 famous mu major 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, 75/64, 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, 51/40
Skwares
7
494.11
Perfect Fourth
perfect 4th
P4
F
4/3
Supra
8
564.71
Super Fourth
Diminished 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, 52/33, 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, 128/75, 12/7
Huxley
14
988.235
Minor Seventh/
Super Sixth
minor 7th
m7
Bb
16/9, 44/25, 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


Chord Names


Ups and down notation can be used to name 17edo chords.

0-4-10 = C Eb G = Cm = C minor
0-5-10 = C Ev G = C~ = C mid
0-6-10 = C E G = C = C major

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

0-4-7-10 = C Eb F G = Cm(11) = C minor add eleven (approximates 6:7:8:9)
0-3-6-10 = C D E G = C(9) = C add nine (approximates 1/(6:7:8:9) = 1/1 - 9/8 - 9/7 - 3/2)
0-3-7-10 = C D F G = C4(9) = C four add nine
0-2-7-10 = C Dv F G = C4(v9) = C four add down-nine (approximates 12:13:16:18)
0-3-8-10 = C D F^ G = C.^4(9) = C up-four add nine (approximates 8:9:11:12)

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).
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

alt : Your browser has no SVG support.


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


List of 17edo rank two temperaments by badness
List of edo-distinct 17c rank two temperaments

Music

Compositions


Sound files


Instruments




17P1050829r.JPG