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{{Infobox ET}} | {{Infobox ET}} | ||
27ed4 is an equal tuning that divides the 4/1 ratio (double-octave, tetratave, fifteenth) into steps of 88<sup>8</sup>/<sub>9</sub> cents. | 27ed4 is an equal tuning that divides the [[4/1]] ratio (double-octave, tetratave, fifteenth) into steps of 88<sup>8</sup>/<sub>9</sub> cents. | ||
It serves as a good first approximation to [[Nelinda#Xenharmonic Systems for Nelinda| | It serves as a good first approximation to [[Nelinda#Xenharmonic Systems for Nelinda|Nelindic temperament]], and is in many respects a "3n+1 cousin" of 5-limit [[12edo|12et]] (even though it takes every other step of the dissimilar [[27edo|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 clan|Archytas temperament]] for the [[4/1|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 == | == Intervals == | ||
The following table of intervals uses the 7-note | 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 right-4" | {| class="wikitable center-all left-2 right-4" | ||
! Steps | ! rowspan="2" | Steps | ||
! | ! colspan="2" | Nelindic 6L 1s | ||
! | ! colspan="2" |Tetrarchy 3L 4s | ||
! Cents | ! rowspan="2" |Cents | ||
! ~ Ratios | ! ~ Ratios | ||
|- | |||
!Note | |||
!Interval name | |||
!Note | |||
!Interval name | |||
! | |||
|- | |- | ||
| 0 | | 0 | ||
| A | | A | ||
| unison | | unison | ||
| 0.00 | |G | ||
|unison | |||
|0.00 | |||
| 1/1 | | 1/1 | ||
|- | |- | ||
| Line 23: | Line 33: | ||
| A# | | A# | ||
| aug unison | | aug unison | ||
| 88.89 | |Abb | ||
|dim mos2nd | |||
|88.89 | |||
| 21/20 | | 21/20 | ||
|- | |- | ||
| Line 29: | Line 41: | ||
| Bbb | | Bbb | ||
| ddim mos2nd | | ddim mos2nd | ||
| 177.78 | |G# | ||
|aug unison | |||
|177.78 | |||
| 10/9 | | 10/9 | ||
|- | |- | ||
| Line 35: | Line 49: | ||
| Bb | | Bb | ||
| dim mos2nd | | dim mos2nd | ||
| 266.67 | |Ab | ||
|min mos2nd | |||
|266.67 | |||
| 7/6 | | 7/6 | ||
|- style="background: #eee" | |- style="background: #eee" | ||
| 4 | | '''4''' | ||
| B | | '''B''' | ||
| perf mos2nd | | '''perf mos2nd''' | ||
| 355.56 | |Bbb | ||
| 16/13 | |ddim mos3rd | ||
|'''355.56''' | |||
| '''16/13''' | |||
|- | |- | ||
| 5 | | 5 | ||
| B# | | B# | ||
| aug mos2nd | | aug mos2nd | ||
| 444.44 | |A | ||
| 13/10 | |maj mos2nd | ||
|444.44 | |||
| 9/7, 13/10 | |||
|- | |- | ||
| 6 | | 6 | ||
| Cbb | | Cbb | ||
| dim mos3rd | | dim mos3rd | ||
| 533.33 | |Bb | ||
|dim mos3rd | |||
|533.33 | |||
| 27/20, 19/14 | | 27/20, 19/14 | ||
|- | |- | ||
| Line 59: | Line 81: | ||
| Cb | | Cb | ||
| min mos3rd | | min mos3rd | ||
| 622.22 | |A# | ||
|aug mos2nd | |||
|622.22 | |||
| 10/7, 13/9 | | 10/7, 13/9 | ||
|- | |- | ||
| 8 | | '''8''' | ||
| C | | C | ||
| maj mos3rd | | maj mos3rd | ||
| 711.11 | |'''B''' | ||
| 3/2 | |'''perf mos3rd''' | ||
|'''711.11''' | |||
| '''3/2''' | |||
|- | |- | ||
| 9 | | 9 | ||
| C# | | C# | ||
| aug mos3rd | | aug mos3rd | ||
| 800.00 | |Cbb | ||
|dim mos4th | |||
|800.00 | |||
| 8/5 | | 8/5 | ||
|- | |- | ||
| Line 77: | Line 105: | ||
| Dbb | | Dbb | ||
| dim mos4th | | dim mos4th | ||
| 888.89 | |B# | ||
|aug mos3rd | |||
|888.89 | |||
| 5/3 | | 5/3 | ||
|- | |- | ||
| Line 83: | Line 113: | ||
| Db | | Db | ||
| min mos4th | | min mos4th | ||
| 977.78 | |Cb | ||
|min mos4th | |||
|977.78 | |||
| 7/4 | | 7/4 | ||
|- | |- | ||
| Line 89: | Line 121: | ||
| D | | D | ||
| maj mos4th | | maj mos4th | ||
| 1066.67 | |Dbb | ||
|ddim mos5th | |||
|1066.67 | |||
| 13/7 | | 13/7 | ||
|- | |- | ||
| Line 95: | Line 129: | ||
| D# | | D# | ||
| aug mos4th | | aug mos4th | ||
| 1155.56 | |C | ||
|maj mos4th | |||
|1155.56 | |||
| 39/20, 35/18 | | 39/20, 35/18 | ||
|- | |- | ||
| Line 101: | Line 137: | ||
| Ebb | | Ebb | ||
| dim mos5th | | dim mos5th | ||
| 1244.44 | |Db | ||
|min mos5th | |||
|1244.44 | |||
| 80/39, 72/35 | | 80/39, 72/35 | ||
|- | |- | ||
| Line 107: | Line 145: | ||
| Eb | | Eb | ||
| min mos5th | | min mos5th | ||
| 1333.33 | |C# | ||
|aug mos4th | |||
|1333.33 | |||
| 28/13 | | 28/13 | ||
|- | |- | ||
| Line 113: | Line 153: | ||
| E | | E | ||
| maj mos5th | | maj mos5th | ||
| 1422.22 | |D | ||
|maj mos5th | |||
|1422.22 | |||
| 16/7 | | 16/7 | ||
|- | |- | ||
| Line 119: | Line 161: | ||
| E# | | E# | ||
| aug mos5th | | aug mos5th | ||
| 1511.11 | |Eb | ||
|dim mos6th | |||
|1511.11 | |||
| 12/5 | | 12/5 | ||
|- | |- | ||
| Line 125: | Line 169: | ||
| Fbb | | Fbb | ||
| dim mos6th | | dim mos6th | ||
| 1600.00 | |D# | ||
|aug mos5th | |||
|1600.00 | |||
| 5/2 | | 5/2 | ||
|- | |- | ||
| 19 | | '''19''' | ||
| Fb | | Fb | ||
| min mos6th | | min mos6th | ||
| 1688.89 | |'''E''' | ||
| 8/3 | |'''perf mos6th''' | ||
|'''1688.89''' | |||
| '''8/3''' | |||
|- | |- | ||
| 20 | | 20 | ||
| F | | F | ||
| maj mos6th | | maj mos6th | ||
| 1777.78 | |Fbb | ||
|dim mos7th | |||
|1777.78 | |||
| 14/5, 36/13 | | 14/5, 36/13 | ||
|- | |- | ||
| Line 143: | Line 193: | ||
| F# | | F# | ||
| aug mos6th | | aug mos6th | ||
| 1866.67 | |E# | ||
|aug mos6th | |||
|1866.67 | |||
| 80/27, 38/13 | | 80/27, 38/13 | ||
|- | |- | ||
| Line 149: | Line 201: | ||
| Gb | | Gb | ||
| dim mos7th | | dim mos7th | ||
| 1955.56 | |Fb | ||
| 40/13 | |min mos7th | ||
|1955.56 | |||
| 28/9, 40/13 | |||
|- style="background: #eee" | |- style="background: #eee" | ||
| 23 | | '''23''' | ||
| G | | '''G''' | ||
| perf mos7th | | '''perf mos7th''' | ||
| 2044.44 | |Gbb | ||
| 13/4 | |ddim tetratave | ||
|'''2044.44''' | |||
| '''13/4''' | |||
|- | |- | ||
| 24 | | 24 | ||
| G# | | G# | ||
| aug mos7th | | aug mos7th | ||
| 2133.33 | |F | ||
|maj mos7th | |||
|2133.33 | |||
| 24/7 | | 24/7 | ||
|- | |- | ||
| Line 167: | Line 225: | ||
| Abb | | Abb | ||
| ddim tetratave | | ddim tetratave | ||
| 2222.22 | |Gb | ||
|dim tetratave | |||
|2222.22 | |||
| 18/5 | | 18/5 | ||
|- | |- | ||
| Line 173: | Line 233: | ||
| Ab | | Ab | ||
| dim tetratave | | dim tetratave | ||
| 2311.11 | |F# | ||
|aug mos7th | |||
|2311.11 | |||
| 80/21 | | 80/21 | ||
|- | |- | ||
| Line 179: | Line 241: | ||
| A | | A | ||
| tetratave | | tetratave | ||
| 2400.00 | |G | ||
|tetratave | |||
|2400.00 | |||
| 4/1 | | 4/1 | ||
|} | |} | ||
The | The genchain for the Nelindic scale is as follows: | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 242: | Line 306: | ||
| A6 | | A6 | ||
|} | |} | ||
The genchain for the Tetrarchy scale is as follows: | |||
{| class="wikitable" | |||
| 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 [[64/63|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]] | |||
* [[Support|Supporting]] ETs: [[17ed4|17]], 27 | |||
* [[POTE tuning]]: ~[[3/2]] = 709.3213 | |||
The Nelindic temperament is described in it's own article on [[Nelinda]]. | |||
[[Category:Nonoctave]] | [[Category:Nonoctave]] | ||
[[Category:Edonoi]] | [[Category:Edonoi]] | ||
[[Category:Ed4]] | [[Category:Ed4]] | ||
Revision as of 16:00, 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.