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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.
{{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.


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.
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.
 
== Harmonics ==
{{Harmonics in equal
| steps = 27
| num = 4
| denom = 1
}}
{{Harmonics in equal
| steps = 27
| num = 4
| denom = 1
| start = 12
| collapsed = 1
}}


== Intervals ==
== 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.
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 (4/1-equivalent)|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.


{| class="wikitable center-all left-2 right-4"
{| class="wikitable center-all left-2 left-4"
! Steps
! rowspan="2" | Steps
! Note
! colspan="2" | Nelindic 6L 1s
! Interval name
! colspan="2" |Tetrarchy 3L 4s
! Cents
! rowspan="2" |Cents
! ~ Ratios
! rowspan="2" |~ Ratios
|-
!Note
!Interval name
!Note
!Interval name
|-
|-
| 0
| 0
| A
| A
| unison
| unison
| 0.00
|G
|unison
|0.00
| 1/1
| 1/1
|-
|-
Line 22: Line 46:
| A#
| A#
| aug unison
| aug unison
| 88.89
|Abb
|dim 1-mosstep
|88.89
| 21/20
| 21/20
|-
|-
| 2
| 2
| Bbb
| Bbb
| ddim mos2nd
| ddim 1-mosstep
| 177.78
|G#
|aug unison
|177.78
| 10/9
| 10/9
|-
|-
| 3
| 3
| Bb
| Bb
| dim mos2nd
| dim 1-mosstep
| 266.67
|Ab
|min 1-mosstep
|266.67
| 7/6
| 7/6
|- style="background: #eee"
|-
| 4
| '''4'''
| B
| '''B'''
| perf mos2nd
| '''perf 1-mosstep'''
| 355.56
|Bbb
| 16/13
|ddim 2-mosstep
|'''355.56'''
| '''16/13'''
|-
|-
| 5
| 5
| B#
| B#
| aug mos2nd
| aug 1-mosstep
| 444.44
|A
| 13/10, 9/7
|maj 1-mosstep
|444.44
| 9/7, 13/10
|-
|-
| 6
| 6
| Cbb
| Cbb
| dim mos3rd
| dim 2-mosstep
| 533.33
|Bb
|dim 2-mosstep
|533.33
| 27/20, 19/14
| 27/20, 19/14
|-
|-
| 7
| 7
| Cb
| Cb
| min mos3rd
| min 2-mosstep
| 622.22
|A#
|aug 2-mosstep
|622.22
| 10/7, 13/9
| 10/7, 13/9
|-
|-
| 8
| '''8'''
| C
| C
| maj mos3rd
| maj 2-mosstep
| 711.11
|'''B'''
| 3/2
|'''perf 2-mosstep'''
|'''711.11'''
| '''3/2'''
|-
|-
| 9
| 9
| C#
| C#
| aug mos3rd
| aug 2-mosstep
| 800.00
|Cbb
|dim 3-mosstep
|800.00
| 8/5
| 8/5
|-
|-
| 10
| 10
| Dbb
| Dbb
| dim mos4th
| dim 3-mosstep
| 888.89
|B#
|aug 2-mosstep
|888.89
| 5/3
| 5/3
|-
|-
| 11
| 11
| Db
| Db
| min mos4th
| min 3-mosstep
| 977.78
|Cb
|min 3-mosstep
|977.78
| 7/4
| 7/4
|-
|-
| 12
| 12
| D
| D
| maj mos4th
| maj 3-mosstep
| 1066.67
|Dbb
|ddim 4-mosstep
|1066.67
| 13/7
| 13/7
|-
|-
| 13
| 13
| D#
| D#
| aug mos4th
| aug 3-mosstep
| 1155.56
|C
|maj 3-mosstep
|1155.56
| 39/20, 35/18
| 39/20, 35/18
|-
|-
| 14
| 14
| Ebb
| Ebb
| dim mos5th
| dim 4-mosstep
| 1244.44
|Db
|min 4-mosstep
|1244.44
| 80/39, 72/35
| 80/39, 72/35
|-
|-
| 15
| 15
| Eb
| Eb
| min mos5th
| min 4-mosstep
| 1333.33
|C#
|aug 3-mosstep
|1333.33
| 28/13
| 28/13
|-
|-
| 16
| 16
| E
| E
| maj mos5th
| maj 4-mosstep
| 1422.22
|D
|maj 4-mosstep
|1422.22
| 16/7
| 16/7
|-
|-
| 17
| 17
| E#
| E#
| aug mos5th
| aug 4-mosstep
| 1511.11
|Eb
|dim 5-mosstep
|1511.11
| 12/5
| 12/5
|-
|-
| 18
| 18
| Fbb
| Fbb
| dim mos6th
| dim 5-mosstep
| 1600.00
|D#
|aug 4-mosstep
|1600.00
| 5/2
| 5/2
|-
|-
| 19
| '''19'''
| Fb
| Fb
| min mos6th
| min 5-mosstep
| 1688.89
|'''E'''
| 8/3
|'''perf 5-mosstep'''
|'''1688.89'''
| '''8/3'''
|-
|-
| 20
| 20
| F
| F
| maj mos6th
| maj 5-mosstep
| 1777.78
|Fbb
|dim 6-mosstep
|1777.78
| 14/5, 36/13
| 14/5, 36/13
|-
|-
| 21
| 21
| F#
| F#
| aug mos6th
| aug 5-mosstep
| 1866.67
|E#
|aug 5-mosstep
|1866.67
| 80/27, 38/13
| 80/27, 38/13
|-
|-
| 22
| 22
| Gb
| Gb
| dim mos7th
| dim 6-mosstep
| 1955.56
|Fb
| 40/13, 28/9
|min 6-mosstep
|- style="background: #eee"
|1955.56
| 23
| 28/9, 40/13
| G
|-
| perf mos7th
| '''23'''
| 2044.44
| '''G'''
| 13/4
| '''perf 6-mosstep'''
|Gbb
|ddim tetratave
|'''2044.44'''
| '''13/4'''
|-
|-
| 24
| 24
| G#
| G#
| aug mos7th
| aug 6-mosstep
| 2133.33
|F
|maj 6-mosstep
|2133.33
| 24/7
| 24/7
|-
|-
Line 166: Line 238:
| Abb
| Abb
| ddim tetratave
| ddim tetratave
| 2222.22
|Gb
|dim tetratave
|2222.22
| 18/5
| 18/5
|-
|-
Line 172: Line 246:
| Ab
| Ab
| dim tetratave
| dim tetratave
| 2311.11
|F#
|aug 6-mosstep
|2311.11
| 80/21
| 80/21
|-
|-
Line 178: Line 254:
| A
| A
| tetratave
| tetratave
| 2400.00
|G
|tetratave
|2400.00
| 4/1
| 4/1
|}
|}


The [[Generator|genchain]] for the nelindic scale is as follows:
The genchain for the Nelindic scale is as follows:


{| class="wikitable"
{| class="wikitable"
Line 241: Line 319:
| 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 ([http://x31eq.com/cgi-bin/rt.cgi?limit=4_3%2F2_9%2F7&ets=q17_q27&tuning=po&subgroup=on 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 [[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 ===
Tetrarchy is a [[Domain basis #Canonical form|noncanonical form]] of quarchy.
* [[Subgroup]]: 4.3/2.9/7
* [[Comma list]]: [[64/63]]
* {{Mapping|legend=1|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:Ed4]]
Retrieved from "https://en.xen.wiki/w/27ed4"