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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|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 23: 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 167: Line 238:
| Abb
| Abb
| ddim tetratave
| ddim tetratave
| 2222.22
|Gb
|dim tetratave
|2222.22
| 18/5
| 18/5
|-
|-
Line 173: Line 246:
| Ab
| Ab
| dim tetratave
| dim tetratave
| 2311.11
|F#
|aug 6-mosstep
|2311.11
| 80/21
| 80/21
|-
|-
Line 179: 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 242: 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]]

Latest revision as of 19:21, 1 August 2025

← 25ed4 27ed4 29ed4 →
Prime factorization 33
Step size 88.8889 ¢ 
Octave 14\27ed4 (1244.44 ¢)
Twelfth 21\27ed4 (1866.67 ¢) (→ 7\9ed4)
Consistency limit 1
Distinct consistency limit 1

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.

Harmonics

Approximation of harmonics in 27ed4
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)
Approximation of harmonics in 27ed4
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.


The Nelindic temperament is described in it's own article on Nelinda.

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