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27ed4 is an equal tuning that divides the 4/1 ratio (double-octave, tetratave, fifteenth) into steps of 88+(8/9) 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"
 
!'''Degree'''
{| class="wikitable center-all left-2 left-4"
!'''Note'''
! rowspan="2" | Steps
!'''Interval name'''
! colspan="2" | Nelindic 6L 1s
!'''Cents'''
! colspan="2" |Tetrarchy 3L 4s
!'''~ Ratios'''
! rowspan="2" |Cents
! rowspan="2" |~ Ratios
|-
!Note
!Interval name
!Note
!Interval name
|-
|-
|'''0'''
| 0
|'''A'''
| A
|'''unison'''
| unison
|'''0'''
|G
|'''1/1'''
|unison
|0.00
| 1/1
|-
|-
|1
| 1
|A#
| A#
|aug unison
| aug unison
|Abb
|dim 1-mosstep
|88.89
|88.89
|21/20
| 21/20
|-
|-
|2
| 2
|Bbb
| Bbb
|ddim mos2nd
| ddim 1-mosstep
|G#
|aug unison
|177.78
|177.78
|10/9
| 10/9
|-
|-
|3
| 3
|Bb
| Bb
|dim mos2nd
| dim 1-mosstep
|Ab
|min 1-mosstep
|266.67
|266.67
|7/6
| 7/6
|-
|-
|''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
|A
|maj 1-mosstep
|444.44
|444.44
|13/10, 9/7
| 9/7, 13/10
|-
|-
|6
| 6
|Cbb
| Cbb
|dim mos3rd
| dim 2-mosstep
|Bb
|dim 2-mosstep
|533.33
|533.33
|27/20, 19/14
| 27/20, 19/14
|-
|-
|7
| 7
|Cb
| Cb
|min mos3rd
| min 2-mosstep
|A#
|aug 2-mosstep
|622.22
|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
|Cbb
|dim 3-mosstep
|800.00
|800.00
|8/5
| 8/5
|-
|-
|10
| 10
|Dbb
| Dbb
|dim mos4th
| dim 3-mosstep
|B#
|aug 2-mosstep
|888.89
|888.89
|5/3
| 5/3
|-
|-
|11
| 11
|Db
| Db
|min mos4th
| min 3-mosstep
|Cb
|min 3-mosstep
|977.78
|977.78
|7/4
| 7/4
|-
|-
|12
| 12
|D
| D
|maj mos4th
| maj 3-mosstep
|Dbb
|ddim 4-mosstep
|1066.67
|1066.67
|13/7
| 13/7
|-
|-
|13
| 13
|D#
| D#
|aug mos4th
| aug 3-mosstep
|C
|maj 3-mosstep
|1155.56
|1155.56
|39/20, 35/18
| 39/20, 35/18
|-
|-
|14
| 14
|Ebb
| Ebb
|dim mos5th
| dim 4-mosstep
|Db
|min 4-mosstep
|1244.44
|1244.44
|80/39, 72/35
| 80/39, 72/35
|-
|-
|15
| 15
|Eb
| Eb
|min mos5th
| min 4-mosstep
|C#
|aug 3-mosstep
|1333.33
|1333.33
|28/13
| 28/13
|-
|-
|16
| 16
|E
| E
|maj mos5th
| maj 4-mosstep
|D
|maj 4-mosstep
|1422.22
|1422.22
|16/7
| 16/7
|-
|-
|17
| 17
|E#
| E#
|aug mos5th
| aug 4-mosstep
|Eb
|dim 5-mosstep
|1511.11
|1511.11
|12/5
| 12/5
|-
|-
|18
| 18
|Fbb
| Fbb
|dim mos6th
| dim 5-mosstep
|D#
|aug 4-mosstep
|1600.00
|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
|Fbb
|dim 6-mosstep
|1777.78
|1777.78
|14/5, 36/13
| 14/5, 36/13
|-
|-
|21
| 21
|F#
| F#
|aug mos6th
| aug 5-mosstep
|E#
|aug 5-mosstep
|1866.67
|1866.67
|80/27, 38/13
| 80/27, 38/13
|-
|-
|22
| 22
|Gb
| Gb
|dim mos7th
| dim 6-mosstep
|Fb
|min 6-mosstep
|1955.56
|1955.56
|40/13, 28/9
| 28/9, 40/13
|-
|-
|''23''
| '''23'''
|''G''
| '''G'''
|''perf mos7th''
| '''perf 6-mosstep'''
|''2044.44''
|Gbb
|''13/4''
|ddim tetratave
|'''2044.44'''
| '''13/4'''
|-
|-
|24
| 24
|G#
| G#
|aug mos7th
| aug 6-mosstep
|F
|maj 6-mosstep
|2133.33
|2133.33
|24/7
| 24/7
|-
|-
|25
| 25
|Abb
| Abb
|ddim tetratave
| ddim tetratave
|Gb
|dim tetratave
|2222.22
|2222.22
|18/5
| 18/5
|-
|-
|26
| 26
|Ab
| Ab
|dim tetratave
| dim tetratave
|F#
|aug 6-mosstep
|2311.11
|2311.11
|80/21
| 80/21
|-
|-
|'''27'''
| 27
|'''A'''
| A
|'''tetratave'''
| tetratave
|'''2400'''
|G
|'''4/1'''
|tetratave
|2400.00
| 4/1
|}
|}
The [[Generator|genchain]] for the nelindic scale is as follows:
 
The genchain for the Nelindic scale is as follows:
 
{| class="wikitable"
| 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:
{| class="wikitable"
{| class="wikitable"
|Abb
| Gbb
|Bbb
| Bbb
|Cbb
| Dbb
|Dbb
| Fbb
|Ebb
| Abb
|Fbb
| Cbb
|Gb
| Eb
|Ab
| Gb
|Bb
| Bb
|Cb
| Db
|Db
| Fb
|Eb
| Ab
|Fb
| Cb
|G
| E
|A
| G
|B
| B
|C
| D
|D
| F
|E
| A
|F
| C
|G#
| E#
|A#
| G#
|B#
| B#
|C#
| D#
|D#
| F#
|E#
| A#
|F#
| C#
|-
|-
|dd1
| dd1
|dd2
| dd3
|d3
| d5
|d4
| d7
|d5
| d2
|d6
| d4
|d7
| d6
|d1
| d1
|d2
| d3
|m3
| m5
|m4
| m7
|m5
| m2
|m6
| m4
|P7
| P6
|P1
| P1
|P2
| P3
|M3
| M5
|M4
| M7
|M5
| M2
|M6
| M4
|A7
| A6
|A1
| A1
|A2
| A3
|A3
| A5
|A4
| A7
|A5
| A2
|A6
| 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"