8edf

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← 7edf8edf9edf →
Prime factorization 23
Step size 87.7444¢
Octave 14\8edf (1228.42¢) (→7\4edf)
Semitones (A1:m2) 0:2 (0¢ : 175.5¢)
Consistency limit 2
Distinct consistency limit 2

8 equal divisions of the fifth (8edf, 8ed3/2) is the tuning system that divides the fifth into 8 steps of 87.7444 cents each, making it very nearly 88cET. It is related to the octacot temperament. 8edf corresponds to about 13.6761edo (similar to every third step of 41edo).

Intervals

8edf can be notated either using native uranian (sesquitave) notation, where the notation repeats every period (i.e. just diatonic fifth), or using double sesquitave (Annapolis) notation, where the notation repeats every two periods (i.e. major diatonic ninth). This interprets 8edf as 16ed9/4, resulting in a tuning for the Natural and Harmonic Minor modes of Annapolis[6L 4s]. It can also be notated using tetratonic 4edf-based notation.

The naturals result from a semiwolf generator (~7/6). For sesquitave notation, letters A-E can be used. For double sesquitave notation, Greek numerals 1-10 can be used (Α,Β,Γ,Δ,Ε,Ϛ/Ϝ,Ζ,Η,Θ,Ι).

Scale Cents Approximate intervals Uranian Diatonic Notation
degree value 7-limit 11-limit 19-limit interval equivalent Uranian Annapolis Tetratonic notation
0 0 1 unison A Α C
1 87.7444 21/20 22/21 20/19, 19/18 min mos2nd minor second A# Α# ^C, vD
2 175.48875 10/9 21/19 maj mos2nd major second B Β D
3 263.2331 7/6 perf mos3rd subminor third C Γ ^D, vE
4 350.9775 60/49, 49/40 11/9 aug mos3rd neutral third C# Γ# E
5 438.7219 9/7 perf mos4th supermajor third D Δ ^E, vF
6 526.46625 27/20 19/14 min mos5th wolf fourth D# Δ# F
7 614.2106 10/7 27/19 maj mos5th augmented fourth E Ε ^F, vC
8 701.955 3/2 sesquitave just fifth A Ϛ/Ϝ C
9 789.6994 63/40 11/7 30/19 min mos7th minor sixth A# Ϛ#/Ϝ# ^C, vD
10 877.44375 5/3 63/38 maj mos7th major sixth B Ζ D
11 965.1881 7/4 perf mos8th subminor seventh C Η ^D, vE
12 1052.9325 90/49, 35/18 11/6 aug mos8th neutral seventh C# Η# E
13 1140.6769 27/14 perf mos9th supermajor seventh D Θ ^E, vF
14 1228.42125 81/40 57/28 min mos10th acute octave D# Θ# F
15 1316.1656 15/7 81/38 maj mos10th minor ninth E Ι ^F, vC
16 1403.91 9/4 double sesquitave major ninth A Α C

Scale workshop link for a keyboard/MIDI playable 8EDF (with graphical uranian scale, A=220Hz)

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: 85.7143 cents (4\7/8 = 1\14) to 90 cents (3\5/8 = 3\40)

Fifth Cents Comments
4\7 85.714
27\47 86.170
23\40 86.250
42\73 86.301
19\33 86.36
53\92 86.413
34\59 86.441
49\85 86.471
15\26 86.5385
56\97 86.598
41\71 86.620
67\116 86.638
26\45 86.6 Flattone is in this region
63\109 86.697
37\64 86.719
48\83 86.747
11\19 86.842
51\88 86.9318
40\69 86.9565
69\119 86.975
29\50 87.000
76\131 87.023 Golden meantone (696.2145¢)
47\81 87.037
65\112 87.054
18\31 87.097 Meantone is in this region
61\105 87.143
43\74 87.162
68\117 87.1795
25\43 87.209
57\98 87.245
32\55 87.27
39\67 87.313
7\12 87.500
38\65 87.692
31\53 87.736 The fifth closest to a just 3/2 for EDOs less than 200
55\94 87.766 Garibaldi / Cassandra
24\41 87.805
65\111 87.837
41\70 87.857
58\99 87.87
17\29 87.931
61\104 87.981
44\75 88.000
71\121 88.0165 Golden neogothic (704.0956¢)
27\46 88.0435 Neogothic is in this region
64\109 88.073
37\63 88.095
47\80 88.125
10\17 88.235
43\73 88.356
33\56 88.392
56\95 88.421 The generator closest to a just 5/3 for EDOs less than 1600
23\39 88.4615
59\100 88.500
36\61 88.525
49\83 88.554
13\22 88.63 Archy is in this region
42\71 88.732
29\49 88.7755
45\76 88.816
16\27 88.8
35\59 88.931
19\32 89.0625
22\37 89.189
3\5 90.000

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.

Rank 2 temperaments

MOS scales and temperament listed by generator size and period:

Periods

per octave

Generator Scale pattern Temperaments
1 1\8 1L5s, 1L6s (pathological)
1 3\8 3L2s (uranian) Semiwolf
2 3\8 2L 2s (augmented-like)

Images

8EDF cheat sheet (sesquitave notation) - table only