27ed4: Difference between revisions
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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 mode numbering|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. | 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 mode numbering|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. | ||
{| class="wikitable center-all left-2 | {| class="wikitable center-all left-2 left-4" | ||
! rowspan="2" | Steps | ! rowspan="2" | Steps | ||
! colspan="2" | Nelindic 6L 1s | ! colspan="2" | Nelindic 6L 1s | ||
| Line 53: | Line 53: | ||
|266.67 | |266.67 | ||
| 7/6 | | 7/6 | ||
|- | |||
| '''4''' | | '''4''' | ||
| '''B''' | | '''B''' | ||
| Line 204: | Line 205: | ||
|1955.56 | |1955.56 | ||
| 28/9, 40/13 | | 28/9, 40/13 | ||
|- | |||
| '''23''' | | '''23''' | ||
| '''G''' | | '''G''' | ||
Revision as of 16:05, 20 July 2023
| ← 25ed4 | 27ed4 | 29ed4 → |
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.
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.
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.
| 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 mos2nd | 88.89 | 21/20 |
| 2 | Bbb | ddim mos2nd | G# | aug unison | 177.78 | 10/9 |
| 3 | Bb | dim mos2nd | Ab | min mos2nd | 266.67 | 7/6 |
| 4 | B | perf mos2nd | Bbb | ddim mos3rd | 355.56 | 16/13 |
| 5 | B# | aug mos2nd | A | maj mos2nd | 444.44 | 9/7, 13/10 |
| 6 | Cbb | dim mos3rd | Bb | dim mos3rd | 533.33 | 27/20, 19/14 |
| 7 | Cb | min mos3rd | A# | aug mos2nd | 622.22 | 10/7, 13/9 |
| 8 | C | maj mos3rd | B | perf mos3rd | 711.11 | 3/2 |
| 9 | C# | aug mos3rd | Cbb | dim mos4th | 800.00 | 8/5 |
| 10 | Dbb | dim mos4th | B# | aug mos3rd | 888.89 | 5/3 |
| 11 | Db | min mos4th | Cb | min mos4th | 977.78 | 7/4 |
| 12 | D | maj mos4th | Dbb | ddim mos5th | 1066.67 | 13/7 |
| 13 | D# | aug mos4th | C | maj mos4th | 1155.56 | 39/20, 35/18 |
| 14 | Ebb | dim mos5th | Db | min mos5th | 1244.44 | 80/39, 72/35 |
| 15 | Eb | min mos5th | C# | aug mos4th | 1333.33 | 28/13 |
| 16 | E | maj mos5th | D | maj mos5th | 1422.22 | 16/7 |
| 17 | E# | aug mos5th | Eb | dim mos6th | 1511.11 | 12/5 |
| 18 | Fbb | dim mos6th | D# | aug mos5th | 1600.00 | 5/2 |
| 19 | Fb | min mos6th | E | perf mos6th | 1688.89 | 8/3 |
| 20 | F | maj mos6th | Fbb | dim mos7th | 1777.78 | 14/5, 36/13 |
| 21 | F# | aug mos6th | E# | aug mos6th | 1866.67 | 80/27, 38/13 |
| 22 | Gb | dim mos7th | Fb | min mos7th | 1955.56 | 28/9, 40/13 |
| 23 | G | perf mos7th | Gbb | ddim tetratave | 2044.44 | 13/4 |
| 24 | G# | aug mos7th | F | maj mos7th | 2133.33 | 24/7 |
| 25 | Abb | ddim tetratave | Gb | dim tetratave | 2222.22 | 18/5 |
| 26 | Ab | dim tetratave | F# | aug mos7th | 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. 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
- 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.