27ed4
← 25ed4 | 27ed4 | 29ed4 → |
27ed4 is an equal tuning that divides the 4/1 ratio (double-octave, tetratave, fifteenth) into steps of 88^{8}/_{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.
This tuning also lends itself to Tetrarchy temperament, effectively 7-limit Archytas temperament for the tetratave. In this case, the major mossecond (5 mossteps) represents 9/7 and the minor mossecond (3 mossteps), a very accurate 7/6. The generator is a sharp diatonic fifth (711.11¢), contextually a perfect mosthird (8 mossteps). The TE tuning gives a tetratave of 2393.9334 cents.
Harmonics
Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +44.4 | -35.3 | +0.0 | -30.8 | +9.2 | +9.0 | +44.4 | +18.3 | +13.7 | +26.5 | -35.3 |
Relative (%) | +50.0 | -39.7 | +0.0 | -34.6 | +10.3 | +10.1 | +50.0 | +20.6 | +15.4 | +29.8 | -39.7 | |
Steps (reduced) |
14 (14) |
21 (21) |
27 (0) |
31 (4) |
35 (8) |
38 (11) |
41 (14) |
43 (16) |
45 (18) |
47 (20) |
48 (21) |
Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +3.9 | -35.5 | +22.8 | +0.0 | -16.1 | -26.1 | -30.8 | -30.8 | -26.3 | -18.0 | -6.1 |
Relative (%) | +4.4 | -39.9 | +25.7 | +0.0 | -18.1 | -29.4 | -34.7 | -34.6 | -29.6 | -20.2 | -6.8 | |
Steps (reduced) |
50 (23) |
51 (24) |
53 (26) |
54 (0) |
55 (1) |
56 (2) |
57 (3) |
58 (4) |
59 (5) |
60 (6) |
61 (7) |
Intervals
The following table of intervals uses both the 7-note 6L 1s MOS scale of Nelindic for the naturals (simple A-G notation and standard sharps/flats for the chroma) and the 7-note 3L 4s scale (standard A-G notation using the typical genchain from mosh) for Tetrarchy. The 6L 1s scale can be 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. Similarly, the 3L 4s scale can be extended to a 7L 3s scale, by dividing the long intervals into sets of 3 and 2 mossteps. These extended scales will usually be melodically preferable over the 6-note and 7-note scales, which have extremely wide melodic spacing comparable to 3edo.
Steps | Nelindic 6L 1s | Tetrarchy 3L 4s | Cents | ~ Ratios | ||
---|---|---|---|---|---|---|
Note | Interval name | Note | Interval name | |||
0 | A | unison | G | unison | 0.00 | 1/1 |
1 | A# | aug unison | Abb | dim 1-mosstep | 88.89 | 21/20 |
2 | Bbb | ddim 1-mosstep | G# | aug unison | 177.78 | 10/9 |
3 | Bb | dim 1-mosstep | Ab | min 1-mosstep | 266.67 | 7/6 |
4 | B | perf 1-mosstep | Bbb | ddim 2-mosstep | 355.56 | 16/13 |
5 | B# | aug 1-mosstep | A | maj 1-mosstep | 444.44 | 9/7, 13/10 |
6 | Cbb | dim 2-mosstep | Bb | dim 2-mosstep | 533.33 | 27/20, 19/14 |
7 | Cb | min 2-mosstep | A# | aug 2-mosstep | 622.22 | 10/7, 13/9 |
8 | C | maj 2-mosstep | B | perf 2-mosstep | 711.11 | 3/2 |
9 | C# | aug 2-mosstep | Cbb | dim 3-mosstep | 800.00 | 8/5 |
10 | Dbb | dim 3-mosstep | B# | aug 2-mosstep | 888.89 | 5/3 |
11 | Db | min 3-mosstep | Cb | min 3-mosstep | 977.78 | 7/4 |
12 | D | maj 3-mosstep | Dbb | ddim 4-mosstep | 1066.67 | 13/7 |
13 | D# | aug 3-mosstep | C | maj 3-mosstep | 1155.56 | 39/20, 35/18 |
14 | Ebb | dim 4-mosstep | Db | min 4-mosstep | 1244.44 | 80/39, 72/35 |
15 | Eb | min 4-mosstep | C# | aug 3-mosstep | 1333.33 | 28/13 |
16 | E | maj 4-mosstep | D | maj 4-mosstep | 1422.22 | 16/7 |
17 | E# | aug 4-mosstep | Eb | dim 5-mosstep | 1511.11 | 12/5 |
18 | Fbb | dim 5-mosstep | D# | aug 4-mosstep | 1600.00 | 5/2 |
19 | Fb | min 5-mosstep | E | perf 5-mosstep | 1688.89 | 8/3 |
20 | F | maj 5-mosstep | Fbb | dim 6-mosstep | 1777.78 | 14/5, 36/13 |
21 | F# | aug 5-mosstep | E# | aug 5-mosstep | 1866.67 | 80/27, 38/13 |
22 | Gb | dim 6-mosstep | Fb | min 6-mosstep | 1955.56 | 28/9, 40/13 |
23 | G | perf 6-mosstep | Gbb | ddim tetratave | 2044.44 | 13/4 |
24 | G# | aug 6-mosstep | F | maj 6-mosstep | 2133.33 | 24/7 |
25 | Abb | ddim tetratave | Gb | dim tetratave | 2222.22 | 18/5 |
26 | Ab | dim tetratave | F# | aug 6-mosstep | 2311.11 | 80/21 |
27 | A | tetratave | G | 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 |
The genchain for the Tetrarchy scale is as follows:
Gbb | Bbb | Dbb | Fbb | Abb | Cbb | Eb | Gb | Bb | Db | Fb | Ab | Cb | E | G | B | D | F | A | C | E# | G# | B# | D# | F# | A# | C# |
dd1 | dd3 | d5 | d7 | d2 | d4 | d6 | d1 | d3 | m5 | m7 | m2 | m4 | P6 | P1 | P3 | M5 | M7 | M2 | M4 | A6 | A1 | A3 | A5 | A7 | A2 | A4 |
Temperaments
There rank-2 temperament interpretation of the 3L 4s is called Tetrarchy (regular temperament finder link). The name is derived from „tetratave Archytas”, as it's the double octave interpretation of 7-limit Archytas. This scale tempers Archytas' comma (64/63), as 3/2 stacked twice approximates 16/7, stacked thrice, it approximates 24/7, and stacked 4 times: 36/7, which is 9/7 above the tetratave.
Tetrarchy
Tetrarchy is a noncanonical form of quarchy.
- Subgroup: 4.3/2.9/7
- Comma list: 64/63
- Mapping: [⟨1 0 -1], ⟨0 1 4]]
- Supporting ETs: 17, 27
- POTE tuning: ~3/2 = 709.3213
The Nelindic temperament is described in it's own article on Nelinda.