27ed4

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27ed4 is an equal tuning that divides the 4/1 ratio (double-octave, tetratave, fifteenth) into steps of 888/9 cents.

It serves as a good first approximation to nelindic temperament, and is in many respects a "3n+1 cousin" of 5-limit 12et (even though it takes every other step of the dissimilar 27et), with relatively high error but low complexity, similar step size, and even sharing a common comma (128/125). Note the latter means that 27ed4 divides 4/1 into three approximate 8/5's, just as 12ed2 divides 2/1 into three 5/4's, and thus it has a 5/2 equally sharp of rational as the 5/4 in 12ed2. Its 7 and 13 approximations are a bit sharp themselves, and overall it lends itself well to IoE compression: the TE tuning gives one of 2395.819236 cents.

Intervals

The following table of intervals uses the 7-note (6L 1s) MOS scale of nelindic for the naturals, using a simple A-G notation and standard sharps/flats for the chroma. Extended to the 13-note (7L 6s) scale, these would include all of the sharps except for F#. Due to the L/s ratio of 3:1, in the 13-note case, most former diminished intervals become minor, most former minor intervals become augmented, and most former augmented intervals become major.

Steps Note Interval name Cents ~ Ratios
0 A unison 0.00 1/1
1 A# aug unison 88.89 21/20
2 Bbb ddim mos2nd 177.78 10/9
3 Bb dim mos2nd 266.67 7/6
4 B perf mos2nd 355.56 16/13
5 B# aug mos2nd 444.44 13/10, 9/7
6 Cbb dim mos3rd 533.33 27/20, 19/14
7 Cb min mos3rd 622.22 10/7, 13/9
8 C maj mos3rd 711.11 3/2
9 C# aug mos3rd 800.00 8/5
10 Dbb dim mos4th 888.89 5/3
11 Db min mos4th 977.78 7/4
12 D maj mos4th 1066.67 13/7
13 D# aug mos4th 1155.56 39/20, 35/18
14 Ebb dim mos5th 1244.44 80/39, 72/35
15 Eb min mos5th 1333.33 28/13
16 E maj mos5th 1422.22 16/7
17 E# aug mos5th 1511.11 12/5
18 Fbb dim mos6th 1600.00 5/2
19 Fb min mos6th 1688.89 8/3
20 F maj mos6th 1777.78 14/5, 36/13
21 F# aug mos6th 1866.67 80/27, 38/13
22 Gb dim mos7th 1955.56 40/13, 28/9
23 G perf mos7th 2044.44 13/4
24 G# aug mos7th 2133.33 24/7
25 Abb ddim tetratave 2222.22 18/5
26 Ab dim tetratave 2311.11 80/21
27 A tetratave 2400.00 4/1

The genchain for the nelindic scale is as follows:

Abb Bbb Cbb Dbb Ebb Fbb Gb Ab Bb Cb Db Eb Fb G A B C D E F G# A# B# C# D# E# F#
dd1 dd2 d3 d4 d5 d6 d7 d1 d2 m3 m4 m5 m6 P7 P1 P2 M3 M4 M5 M6 A7 A1 A2 A3 A4 A5 A6