9edf

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← 8edf9edf10edf →
Prime factorization 32
Step size 77.995¢
Octave 15\9edf (1169.93¢) (→5\3edf)
Semitones (A1:m2) 3:0 (234¢ : 0¢)
Consistency limit 2
Distinct consistency limit 2

9edf is the equal division of the just perfect fifth into 9 parts of 78 cents each, corresponding to 15.391524edo. It is nearly identical to Carlos Alpha.

Approximation of harmonics

Approximation of harmonics in 9edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Error absolute (¢) -30.1 -30.1 +17.8 +21.5 +17.8 -15.0 -12.2 +17.8 -8.6 -17.6 -12.2 +5.2 +32.9 -8.6 +35.7 +8.7 -12.2 -27.8
relative (%) -39 -39 +23 +28 +23 -19 -16 +23 -11 -23 -16 +7 +42 -11 +46 +11 -16 -36
Steps
(reduced) 		15
(6) 		24
(6) 		31
(4) 		36
(0) 		40
(4) 		43
(7) 		46
(1) 		49
(4) 		51
(6) 		53
(8) 		55
(1) 		57
(3) 		59
(5) 		60
(6) 		62
(8) 		63
(0) 		64
(1) 		65
(2)

Intervals

# Cents Approximate ratios Neptunian notation using 7\9edf
0 0.0 1/1 C
1 78.0 25/24, 21/20 C#
2 156.0 12/11, 11/10 Db
3 234.0 8/7 D
4 312.0 6/5 D#, Eb
5 390.0 5/4 E
6 468.0 21/16 E#, Fb
7 546.0 15/11, 11/8 F
8 624.0 10/7, 36/25 F#, Cb
9 702.0 3/2 C

Scale tree

EDF scales can be approximated in EDOs by subdividing diatonic fifths. If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger EDOs pop up in our continuum.

Generator range: 76.1905 cents (4\7/9 = 4\63) to 80 cents (3\5/9 = 1\15)

Fifth Cents Comments
4\7 76.1905
27\47 76.596
23\40 76.6
42\73 76.712
19\33 76.76
53\92 76.812
34\59 76.836
49\85 76.893
15\26 76.932
56\97 76.976
41\71 76.995
67\116 77.0115
26\45 77.037 Flattone is in this region
63\109 77.064
37\64 77.083
48\83 77.108
11\19 77.193
51\88 77.27
40\69 77.295
69\119 77.311
29\50 77.3
76\131 77.354 Golden meantone (696.2145¢)
47\81 77.366
65\112 77.381
18\31 77.419 Meantone is in this region
61\105 77.460
43\74 77.4775
68\117 77.493
25\43 77.519
57\98 77.551
32\55 77.57
39\67 77.612
7\12 77.7
38\65 77.949
31\53 77.987 The fifth closest to a just 3/2 for EDOs less than 200
55\94 78.014 Garibaldi / Cassandra
24\41 78.049
65\111 78.078
41\70 78.095
58\99 78.1145
17\29 78.161
61\104 78.205
44\75 78.2
71\121 78.237 Golden neogothic (704.0956¢)
27\46 78.261 Neogothic is in this region

The generator closest to a just 11/7 for EDOs less than 1800

64\109 78.2875
37\63 78.307
47\80 78.3
10\17 78.431
43\73 78.539
33\56 78.571
56\95 78.5965
23\39 78.6325
59\100 78.6
36\61 78.6885
49\83 78.715
13\22 78.78 Archy is in this region
42\71 78.873
29\49 78.912
45\76 78.947
16\27 79.012
35\59 79.096
19\32 79.16
22\37 79.279
3\5 80.000

Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.

Scales within 9edf

Tightrope

  • 77.995
  • 701.955
  • 779.950
  • 1403.905
  • 1481.905
  • 2105.865
  • 2183.860
  • 2807.820
  • 2885.815
  • 3509.775
  • 3587.770


Corrugated

  • 155.990
  • 701.955
  • 857.945
  • 1403.910
  • 1559.900
  • 2105.865
  • 2261.855
  • 2807.820
  • 2963.810
  • 3509.775
  • 3665.765
  • 4211.730
  • 4367.720
  • 4913.685
  • 5069.675
  • 5615.640
  • 5771.630
  • 6317.595
  • 6473.585
  • 7019.550
  • 7175.540
  • 7721.505
  • 7877.495
  • 8423.460


Snowcone

  • 233.985
  • 701.955
  • 935.940
  • 1403.910
  • 1637.895
  • 2105.865
  • 2339.850
  • 2807.820
  • 3041.805
  • 3509.775
  • 3743.760
  • 4211.730
  • 4445.715
  • 4913.685
  • 5147.670
  • 5615.640
  • 5849.625
  • 6317.595
  • 6551.580
  • 7019.550
  • 7253.535
  • 7721.505
  • 7955.490
  • 8423.460


Swan

  • 311.980
  • 701.955
  • 1013.935
  • 1403.910
  • 1715.890
  • 2105.865
  • 2417.845


Cloudscape

  • 389.975
  • 701.955
  • 1091.930
  • 1403.910
  • 1793.885
  • 2105.865
  • 2495.840
  • 2807.820
  • 3197.795
  • 3509.775
  • 3899.750
  • 4211.730
  • 4601.705
  • 4913.685
  • 5303.660
  • 5615.640
  • 6005.615


Pylon

  • 467.970
  • 701.955
  • 1169.925
  • 1403.910
  • 1871.880
  • 2105.865
  • 2573.835
  • 2807.820
  • 3275.790
  • 3509.775
  • 3977.745
  • 4211.730
  • 4679.700
  • 4913.685
  • 5381.655
  • 5615.640
  • 6083.610
  • 6317.595
  • 6785.565
  • 7019.550
  • 7487.520
  • 7721.505
  • 8189.475
  • 8423.460


Quest

  • 545.965
  • 701.955
  • 1247.920
  • 1403.910
  • 1949.875
  • 2105.865
  • 2651.830
  • 2807.820
  • 3353.785
  • 3509.775
  • 4055.740
  • 4211.730
  • 4757.695
  • 4913.685
  • 5459.650
  • 5615.640
  • 6161.605
  • 6317.595
  • 6863.560
  • 7019.550
  • 7565.515
  • 7721.505
  • 8267.470
  • 8423.460


Purgatory

  • 623.960
  • 701.955
  • 1325.915
  • 1403.910
  • 2027.870
  • 2105.865
  • 2729.825
  • 2807.820
  • 3431.780
  • 3509.775
  • 4133.735
  • 4211.730
  • 4835.690


Molten Pelog

  • 155.990
  • 311.980
  • 701.955
  • 857.945
  • 1247.920


Molten Slendro (3edf)

  • 233.985
  • 467.970
  • 701.955
  • 935.940
  • 1169.925

Music

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