8edf
← 7edf | 8edf | 9edf → |
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) |