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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.
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 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 (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"
! rowspan="2" | Steps
! rowspan="2" | Steps
! colspan="2" | Nelindic 6L 1s
! colspan="2" | Nelindic 6L 1s
! colspan="2" |Tetrarchy 3L 4s
! colspan="2" |Tetrarchy 3L 4s
! rowspan="2" |Cents
! rowspan="2" |Cents
! ~ Ratios
! rowspan="2" |~ Ratios
|-
|-
!Note
!Note
Line 20: Line 34:
!Note
!Note
!Interval name
!Interval name
!
|-
|-
| 0
| 0
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| aug unison
| aug unison
|Abb
|Abb
|dim mos2nd
|dim 1-mosstep
|88.89
|88.89
| 21/20
| 21/20
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| 2
| 2
| Bbb
| Bbb
| ddim mos2nd
| ddim 1-mosstep
|G#
|G#
|aug unison
|aug unison
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| 3
| 3
| Bb
| Bb
| dim mos2nd
| dim 1-mosstep
|Ab
|Ab
|min mos2nd
|min 1-mosstep
|266.67
|266.67
| 7/6
| 7/6
|-
| '''4'''
| '''4'''
| '''B'''
| '''B'''
| '''perf mos2nd'''
| '''perf 1-mosstep'''
|Bbb
|Bbb
|ddim mos3rd
|ddim 2-mosstep
|'''355.56'''
|'''355.56'''
| '''16/13'''
| '''16/13'''
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| 5
| 5
| B#
| B#
| aug mos2nd
| aug 1-mosstep
|A
|A
|maj mos2nd
|maj 1-mosstep
|444.44
|444.44
| 9/7, 13/10
| 9/7, 13/10
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| 6
| 6
| Cbb
| Cbb
| dim mos3rd
| dim 2-mosstep
|Bb
|Bb
|dim mos3rd
|dim 2-mosstep
|533.33
|533.33
| 27/20, 19/14
| 27/20, 19/14
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| 7
| 7
| Cb
| Cb
| min mos3rd
| min 2-mosstep
|A#
|A#
|aug mos2nd
|aug 2-mosstep
|622.22
|622.22
| 10/7, 13/9
| 10/7, 13/9
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| '''8'''
| '''8'''
| C
| C
| maj mos3rd
| maj 2-mosstep
|'''B'''
|'''B'''
|'''perf mos3rd'''
|'''perf 2-mosstep'''
|'''711.11'''
|'''711.11'''
| '''3/2'''
| '''3/2'''
Line 95: Line 109:
| 9
| 9
| C#
| C#
| aug mos3rd
| aug 2-mosstep
|Cbb
|Cbb
|dim mos4th
|dim 3-mosstep
|800.00
|800.00
| 8/5
| 8/5
Line 103: Line 117:
| 10
| 10
| Dbb
| Dbb
| dim mos4th
| dim 3-mosstep
|B#
|B#
|aug mos3rd
|aug 2-mosstep
|888.89
|888.89
| 5/3
| 5/3
Line 111: Line 125:
| 11
| 11
| Db
| Db
| min mos4th
| min 3-mosstep
|Cb
|Cb
|min mos4th
|min 3-mosstep
|977.78
|977.78
| 7/4
| 7/4
Line 119: Line 133:
| 12
| 12
| D
| D
| maj mos4th
| maj 3-mosstep
|Dbb
|Dbb
|ddim mos5th
|ddim 4-mosstep
|1066.67
|1066.67
| 13/7
| 13/7
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| 13
| 13
| D#
| D#
| aug mos4th
| aug 3-mosstep
|C
|C
|maj mos4th
|maj 3-mosstep
|1155.56
|1155.56
| 39/20, 35/18
| 39/20, 35/18
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| 14
| 14
| Ebb
| Ebb
| dim mos5th
| dim 4-mosstep
|Db
|Db
|min mos5th
|min 4-mosstep
|1244.44
|1244.44
| 80/39, 72/35
| 80/39, 72/35
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| 15
| 15
| Eb
| Eb
| min mos5th
| min 4-mosstep
|C#
|C#
|aug mos4th
|aug 3-mosstep
|1333.33
|1333.33
| 28/13
| 28/13
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| 16
| 16
| E
| E
| maj mos5th
| maj 4-mosstep
|D
|D
|maj mos5th
|maj 4-mosstep
|1422.22
|1422.22
| 16/7
| 16/7
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| 17
| 17
| E#
| E#
| aug mos5th
| aug 4-mosstep
|Eb
|Eb
|dim mos6th
|dim 5-mosstep
|1511.11
|1511.11
| 12/5
| 12/5
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| 18
| 18
| Fbb
| Fbb
| dim mos6th
| dim 5-mosstep
|D#
|D#
|aug mos5th
|aug 4-mosstep
|1600.00
|1600.00
| 5/2
| 5/2
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| '''19'''
| '''19'''
| Fb
| Fb
| min mos6th
| min 5-mosstep
|'''E'''
|'''E'''
|'''perf mos6th'''
|'''perf 5-mosstep'''
|'''1688.89'''
|'''1688.89'''
| '''8/3'''
| '''8/3'''
Line 183: Line 197:
| 20
| 20
| F
| F
| maj mos6th
| maj 5-mosstep
|Fbb
|Fbb
|dim mos7th
|dim 6-mosstep
|1777.78
|1777.78
| 14/5, 36/13
| 14/5, 36/13
Line 191: Line 205:
| 21
| 21
| F#
| F#
| aug mos6th
| aug 5-mosstep
|E#
|E#
|aug mos6th
|aug 5-mosstep
|1866.67
|1866.67
| 80/27, 38/13
| 80/27, 38/13
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| 22
| 22
| Gb
| Gb
| dim mos7th
| dim 6-mosstep
|Fb
|Fb
|min mos7th
|min 6-mosstep
|1955.56
|1955.56
| 28/9, 40/13
| 28/9, 40/13
|-
| '''23'''
| '''23'''
| '''G'''
| '''G'''
| '''perf mos7th'''
| '''perf 6-mosstep'''
|Gbb
|Gbb
|ddim tetratave
|ddim tetratave
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| 24
| 24
| G#
| G#
| aug mos7th
| aug 6-mosstep
|F
|F
|maj mos7th
|maj 6-mosstep
|2133.33
|2133.33
| 24/7
| 24/7
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| dim tetratave
| dim tetratave
|F#
|F#
|aug mos7th
|aug 6-mosstep
|2311.11
|2311.11
| 80/21
| 80/21
Line 364: Line 379:


== Temperaments ==
== 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.
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 ===
Tetrarchy is a [[Domain basis #Canonical form|noncanonical form]] of quarchy.


* [[Subgroup]]: 4.3/2.9/7
* [[Subgroup]]: 4.3/2.9/7
* [[Comma list]]: [[64/63]]
* [[Comma list]]: [[64/63]]
* [[Mapping]]: [⟨1 0 -1], ⟨0 1 4]]
* {{Mapping|legend=1|1 0 -1|0 1 4}}
* [[Support|Supporting]] ETs: [[17ed4|17]], 27
* [[Support|Supporting]] ETs: [[17ed4|17]], 27
* [[POTE tuning]]: ~[[3/2]] = 709.3213
* [[POTE tuning]]: ~[[3/2]] = 709.3213
Line 377: Line 393:
The Nelindic temperament is described in it's own article on [[Nelinda]].
The Nelindic temperament is described in it's own article on [[Nelinda]].
[[Category:Nonoctave]]
[[Category:Nonoctave]]
[[Category:Edonoi]]
[[Category:Ed4]]
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