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'''16ed5/3''' (or less accurately '''16edVI''') is the [[EdVI|equal division of the just major sixth]] into sixteen parts of 55.2724 [[cent|cents]] each, corresponding to 21.7106 [[edo]]. It is very closely related to the [[Escapade family|escapade temperament]].
{{Infobox ET}}
'''16ed5/3''' is the [[Ed5/3|equal division of the just major sixth]] into sixteen parts of 55.2724 [[cent]]s each, corresponding to 21.7106[[edo]]. It is very closely related to the [[Escapade family|escapade temperament]]. It is vaguely equivalent to [[22edo]].


It very accurately approximates a number of low complexity just intervals, such as: 4/3 (<1¢), 5/4 (<1¢), 11/8 (<2¢), 11/10 (<1¢), 16/15 (<2¢), and 25/16 (<2¢). It also approximates the just fifth and octave to within 20¢, making it a flexible non-octave scale.  
It very accurately approximates a number of low complexity just intervals, such as: [[4/3]] (<1¢), [[5/4]] (<1¢), [[11/8]] (<2¢), [[11/10]] (<1¢), [[16/15]] (<2¢), and [[25/16]] (<2¢). It also approximates the [[3/2|just fifth]] and [[2/1|octave]] to within 17¢, making it a flexible non-octave scale. Notably, having a period of [[5/3]], the diatonic minor third ([[6/5]]) is the period-reduced diatonic octave. This means both are approximated identically (16¢ sharp).
 
== Harmonics ==
{{Harmonics in equal|16|5|3}}


== Intervals ==
== Intervals ==
16ed5/3 can be notated using steps 7 (~5/4) and 9 (~4/3) as generators, as these are accurate to within 0.6¢. The resulting scale is a heptatonic 2L 5s (similar to the octave repeating antidiatonic).
16ed5/3 can be notated using steps 7 (~5/4) and 9 (~4/3) as generators, as these are accurate to within 0.6¢. The resulting scale is a heptatonic 2L 5s (similar to the octave repeating antidiatonic). It can also be notated using the fifth-generated [[Blackcomb]] temperament as discussed in [[#Temperaments]], which lines up quite nicely with diatonic notation, aside from the "minor second" being in neutral second range and "perfect fourth" being in superfourth range.
{| class="wikitable"
{| class="wikitable center-all right-2"
!Degree
! Degree
!Cents
! Cents
!Approximate intervals
! 5/3.4/3.11/6.31/18 subgroup interval
!Mos-interval
! Other interpretations
!Diatonic interval
! 2L 5s<5/3> mos-interval
!Notation
! 2L 5s<5/3> notation
! 1L 4s<5/3> ([[Blackcomb]][5]) interval
! 1L 4s<5/3> ([[Blackcomb]][5]) notation
! Diatonic interval
|-
|-
|'''0'''
| '''0'''
|'''0'''
| '''0.0000'''
|'''1'''
| '''1/1'''
|'''unison'''
|
|'''unison'''
| '''unison'''
|'''A'''
| '''E'''
| '''unison'''
| '''C'''
| '''unison'''
|-
|-
|1
| 1
|55.2724
| 55.2724
|31/30, 33/32
| 31/30, 32/31, 33/32
|aug unison
| 36/35
|quatertone
| aug unison
|A#
| E#
| aug unison
| C#
| quartertone
|-
|-
|2
| 2
|110.5448
| 110.5448
|16/15
| 16/15, 33/31
|min mos2nd
| 21/20
|minor second
| min mos2nd
|Bb
| Fb
| double-aug unison, dim second
| Cx, Dbb
| minor second
|-
|-
|3
| 3
|165.8173
| 165.8173
|11/10
| 11/10
|maj mos2nd
|
|neutral second
| maj mos2nd
|B
| F
| minor second
| Db
| neutral second
|-
|-
|4
| 4
|221.0897
| 221.0897
|8/7, 17/15
| 25/22
|min mos3rd
| 8/7, 17/15
|major second
| min mos3rd
|Cb
| F#/Gb
| major second
| D
| major second
|-
|-
|5
| 5
|276.3621
| 276.3621
|7/6, 20/17, 75/64
| 75/64, 88/75
|maj mos3rd
| 7/6, 20/17
|subminor third
| maj mos3rd
|C
| G
| aug second
| D#
| subminor third
|-
|-
|6
| 6
|331.6345
| 331.6345
|6/5, 17/14
| 40/33, 75/62
|dim mos4th
| 6/5, 17/14
|minor third
| dim mos4th
|Db
| G#/Ab
| minor third
| Eb
| minor third
|-
|-
|''7''
| 7
|''386.9069''
| ''386.9069''
|''5/4''
| ''5/4''
|''perf mos4th''
|
|major third
| ''perf mos4th''
|D
| A
| major third
| E
| major third
|-
|-
|8
| 8
|442.1794
| 442.1794
|9/7, 22/17
| 31/24, 40/31
|aug mos4th
| 9/7, 35/27, 22/17
|supermajor third
| aug mos4th
|D#
| A#/Bb
| aug third
| E#
| supermajor third
|-
|-
|''9''
| 9
|''497.4517''
| ''497.4517''
|4/3
| ''4/3''
|''perf mos5th''
|
|just fourth
| ''perf mos5th''
|E
| B
| dim fourth
| Fb
| just fourth
|-
|-
|10
| 10
|552.7242
| 552.7242
|11/8
| 11/8, 62/45
|aug mos5th
| 25/18, 18/13
|wide fourth
| aug mos5th
|E#
| B#
| perfect fourth
| F
| wide fourth
|-
|-
|11
| 11
|607.9966
| 607.9966
|10/7
| 44/31, 64/45
|min mos6th
| 10/7, 17/12
|large tritone
| min mos6th
|Fb
| Cb
| aug fourth
| F#
| large tritone
|-
|-
|12
| 12
|663.269
| 663.2690
|22/15, 72/49
| 22/15
|maj mos6th
| 72/49
|narrow fifth
| maj mos6th
|F
| C
| dim fifth
| Gb
| narrow fifth
|-
|-
|13
| 13
|718.54145
| 718.5415
|3/2, 50/33
| 50/33
|min mos7th
| 3/2
|acute fifth
| min mos7th
|F#
| C#/Db
| perfect fifth
| G
| acute fifth
|-
|-
|14
| 14
|773.8129
| 773.8129
|25/16
| 25/16
|maj mos7th
|
|subminor sixth
| maj mos7th
|G
| D
| aug fifth
| G#
| subminor sixth
|-
|-
|15
| 15
|829.0863
| 829.0863
|8/5, 13/8
| 50/31
|dim mos8ave
| 8/5, 13/8
|minor sixth
| dim mos8ave
|G#
| D#/Eb
| dim sixth
| Cb
| minor sixth
|-
|-
|'''16'''
| '''16'''
|'''884.3587'''
| '''884.3587'''
|'''5/3'''
| '''5/3'''
|'''mosoctave'''
|
|'''major sixth'''
| '''mosoctave'''
|'''A'''
| '''E'''
| '''perfect sixth'''
| '''C'''
| '''major sixth'''
|-
|-
|17
| 17
|939.6311
| 939.6311
|12/7, 19/11
| 31/18, 55/32
|aug mos8ave
| 12/7, 19/11
|supermajor sixth
| aug mos8ave
|A#
| E#
| aug sixth
| C#
| supermajor sixth
|-
|-
|18
| 18
|994.9035
| 994.9035
|16/9
| 16/9, 55/31
|min mos9th
| 7/4
|minor seventh
| min mos9th
|Bb
| Fb
| double-aug sixth, dim seventh
| Cx, Dbb
| minor seventh
|-
|-
|19
| 19
|1050.1760
| 1050.1760
|11/6
| 11/6
|maj mos9th
|
|neutral seventh
| maj mos9th
|B
| F
| minor seventh
| Db
| neutral seventh
|-
|-
|20
| 20
|1105.4484
| 1105.4484
|17/9, 40/21
| 176/93, 125/66, 256/135
|min mos10th
| 40/21, (27/14), 17/9
|major seventh
| min mos10th
|Cb
| F#/Gb
| major seventh
| D
| major seventh
|-
|-
|21
| 21
|1160.7208
| 1160.7208
|35/18, 43/22
| 88/45, 125/64
|maj mos10th
| 35/18, 43/22
|narrow octave
| maj mos10th
|C
| G
| aug seventh
| D#
| narrow octave
|-
|-
|22
| 22
|1215.9932
| 1215.9932
|2/1
| 200/99, 121/60, 125/62
|dim mos11th
| 2/1
|octave
| dim mos11th
|C#
| G#/Ab
| minor octave
| Eb
| octave
|}
|}
These intervals are close to a few other related non-octave scales:
 
{| class="wikitable"
These intervals are close to a few other related scales:
|+
{| class="wikitable left-all"
!
!
!16ed16\22
! [[22edo]]
![[7ed5/4]]
! [[7ed5/4]]
!16ed5/3
!23ed18\17
![[Noleta|9ed4/3]]
! 16ed5/3
![[43ed4]]
! [[9ed4/3]] (Noleta)
!16ed16\21
! [[43ed4]]
! [[34edt]]
! [[21edo]]
|-
|-
|1
| 1
|54.54545
| 54.54545
|55.188
| 55.188
|55.2724
|55.2429
|55.338
| ''55.2724''
|55.81395
| 55.338
|57.1429
| 55.8140
| 55.9399
| 57.1429
|-
|-
|2
| 2
|109.0909
| 109.0909
|110.375
| 110.375
|110.5448
|110.4859
|110.677
| ''110.5448''
|111.6729
| 110.677
|114.2857
| 111.6729
| 111.8797
| 114.2857
|-
|-
|3
| 3
|163.6364
| 163.6364
|165.563
| 165.563
|165.8173
|165.7288
|166.015
| ''165.8173''
|167.4419
| 166.015
|171.4286
| 167.4419
| 167.8196
| 171.4286
|-
|-
|4
| 4
|218.1818
| 218.1818
|220.751
| 220.751
|221.0897
|220.9718
|221.353
| ''221.0897''
|223.2558
| 221.353
|228.5714
| 223.2558
| 223.7594
| 228.5714
|-
|-
|5
| 5
|272.7273
| 272.7273
|275.938
| 275.938
|276.3621
|276.2147
|276.692
| ''276.3621''
|279.0698
| 276.692
|285.7143
| 279.0698
| 279.6993
| 285.7143
|-
|-
|6
| 6
|327.2727
| 327.2727
|331.126
| 331.126
|331.6345
|331.4576
|332.030
| ''331.6345''
|334.8837
| 332.030
|342.8571
| 334.8837
| 335.6391
| 342.8571
|-
|-
|7
| 7
|381.8182
| 381.8182
|386.314
| 386.314
|386.9069
|386.7006
|387.368
| ''386.9069''
|390.6977
| 387.368
|400
| 390.6977
| 391.5790
| 400
|-
|-
|8
| 8
|436.3636
| 436.3636
|441.501
| 441.501
|442.1794
|441.9435
|442.707
| ''442.1794''
|446.5116
| 442.707
|457.1429
| 446.5116
| 447.5188
| 457.1429
|-
|-
|9
| 9
|490.9091
| 490.9091
|496.689
| 496.689
|497.4517
|497.1865
|498.045
| ''497.4517''
|502.3256
| 498.045
|514.2857
| 502.3256
| 503.4587
| 514.2857
|-
|-
|10
| 10
|545.54545
| 545.5455
|551.877
| 551.877
|552.7242
|552.4294
|553.383
| ''552.7242''
|558.1395
| 553.383
|571.4286
| 558.1395
| 559.3985
| 571.4286
|-
|-
|11
| 11
|600
| 600
|607.064
| 607.064
|607.9966
|607.6723
|608.722
| ''607.9966''
|613.9535
| 608.722
|628.5714
| 613.9535
| 615.3384
| 628.5714
|-
|-
|12
| 12
|654.54545
| 654.5455
|662.252
| 662.252
|663.269
|662.9153
|664.060
| ''663.269''
|669.7674
| 664.060
|685.7143
| 669.7674
| 671.2782
| 685.7143
|-
|-
|13
| 13
|709.0909
| 709.0909
|717.440
| 717.440
|718.54145
|718.1582
|719.398
| ''718.5415''
|725.5814
| 719.398
|742.8571
| 725.5814
| 727.2181
| 742.8571
|-
|-
|14
| 14
|763.6364
| 763.6364
|772.627
| 772.627
|773.8129
|773.4011
|774.737
| ''773.8129''
|781.39535
| 774.737
|800
| 781.3954
| 783.1579
| 800
|-
|-
|15
| 15
|818.1818
| 818.1818
|827.815
| 827.815
|829.0863
|828.6441
|830.075
| ''829.0863''
|837.7209
| 830.075
|857.1429
| 837.7209
| 839.0978
| 857.1429
|-
|-
|16
| 16
|872.7273
| 872.7273
|883.003
| 883.003
|884.3587
|883.8870
|885.413
| ''884.3587''
|893.0233
| 885.413
|914.2857
| 893.0233
| 895.0376
| 914.2857
|}
|}


== MOS Scales ==
== MOS Scales ==
16edVI supports the same [[MOS scale|MOS scales]] as [[16edo]], as such it contains the following scales:
16edVI supports the same [[MOS scale]]s as [[16edo]], as such it contains the following scales:
{| class="wikitable"
{| class="wikitable center-all left-3"
!Periods
! Periods <br> per octave
per octave
! Generator
!Generator
! Pattern
!Pattern
|-
|-
|1
| 1
|1\16
| 1\16
|1L ns (pathological)
| 1L Ns
|-
|-
|1
| 1
|3\16
| 3\16
|1L 4s, 5L 1s
| 1L 4s, 5L 1s
|-
|-
|1
| 1
|5\16
| 5\16
|3L 4s, 3L 7s
| 3L 4s, 3L 7s
|-
|-
|1
| 1
|7\16
| 7\16
|2L 5s, 7L 2s
| 2L 5s, 7L 2s
|-
|-
|2
| 2
|1\16
| 1\16
|2L 8s, 2L 10s, 2L 12s
| 2L 8s, 2L 10s, 2L 12s
|-
|-
|2
| 2
|3\16
| 3\16
|4L 2s, 6L 4s
| 4L 2s, 6L 4s
|-
|-
|4
| 4
|1\16
| 1\16
|4L 4s, 4L 8s
| 4L 4s, 4L 8s
|}
|}
For the 2L 5s scale, the genchain is this:
For the 2L 5s scale, the genchain is this:
{| class="wikitable"
{| class="wikitable center-all"
|B#
| F#
|F#
| C#
|C#
| G#
|G#
| D#
|D#
| A#
|A#
| E#
|E#
| B#
|B
| F
|F
| C
|C
| G
|G
| D
|D
| A
|A
| '''E'''
|E
| B
|Bb
| Fb
|Fb
| Cb
|Cb
| Gb
|Gb
| Db
|Db
| Ab
|Ab
| Eb
|Eb
| Db
|Bbb
| Fbb
|Fbb
| Cbb
|Cbb
| Gbb
|Gbb
| Dbb
|-
|-
|A2
| A2
|A6
| A6
|A3
| A3
|A7
| A7
|A5
| A4
|A1
| A1
|A4
| A5
|M2
| M2
|M6
| M6
|M3
| M3
|M7
| M7
|P5
| P4
|P1
| '''P1'''
|P4
| P5
|m2
| m2
|m6
| m6
|m3
| m3
|m7
| m7
|d5
| d4
|d1
| d1
|d4
| d5
|d2
| d2
|d6
| d6
|d3
| d3
|d7
| d7
|}
|}
[[Category:EdVI]]
 
== Commas ==
Depending on your mapping, 16ed5/3 can be said to temper a number of commas, including the [[diaschisma]], the [[marvel comma]], [[64/63|Archytas' comma]], and the [[jubilisma]], all discussed in the temperaments section. In addition, being an even division of the 5/3, it tempers the [[sensamagic comma]], as the half mosoctave is midway between [[9/7]] and [[35/27]]. This is analogous to the tritone in 2n edo systems. The [[keema]] is tempered due to the septimal interpretation of the diatonic sevenths, and the [[mothwellsma]] is tempered by two major mos3rds ([[7/6]]) resulting in an augmented mos5th ([[11/8]]).
 
== Temperaments ==
The 2L 5s scale is generated by a very accurate [[4/3]], such that two of them wind up on a near exact [[16/9]], which period-reduces to [[16/15]] (the minor mossecond). This interval taken 2 times is approximated by an [[8/7]], and taken 3 times is approximated by a [[6/5]] (or [[2/1]] in the next mosoctave). These 2 equivalencies result in two tempered commas: the marvel comma - [[225/224]] ((<sup>16</sup>/<sub>15</sub>)<sup>2</sup>=(<sup>8</sup>/<sub>7</sub>)), and the diaschisma - [[2048/2025]] ((<sup>16</sup>/<sub>15</sub>)<sup>3</sup>=(<sup>6</sup>/<sub>5</sub>)).
 
The diaschisma can also be tempered by taking 5 generators to mean a [[3/2]] ((<sup>4</sup>/<sub>3</sub>)<sup>5</sup>=(<sup>3</sup>/<sub>2</sub>)·(<sup>5</sup>/<sub>3</sub>)<sup>2</sup>), while the marvel comma can also be tempered with a stack of 3 generators, making a [[10/7]] ((<sup>4</sup>/<sub>3</sub>)<sup>3</sup>=(<sup>10</sup>/<sub>7</sub>)·(<sup>5</sup>/<sub>3</sub>)).
 
The tempered marvel comma also means that the two large [[tritone]]s ([[64/45|pental]] and [[10/7|septimal]]) are addressed by the same scale step. The tempered diaschisma, on the other hand, means that both pental tritones are also addressed by the same scale step.
 
Both of the 7-limit approaches also temper Archytas' comma as a result of equating the [[16/9]] with [[7/4]], and the jubilisma ([[50/49]]) due to tritone equivalence. These are relatively large commas, given the step size (about half, and 7/11ths respectively).
 
This shows the close relationships with [[srutal]] and [[pajara]] octave temperaments. In 16ed5/3's case, there is a close equivalence to [[22edo]]'s pajara tuning.
 
As 3 semitones make a period-reduced octave, and it alludes to tritone tempering, [[User:Ayceman|I]] propose the name '''tristone''' for the basic [[Diaschismic family|diaschismic temperament]], based on the 16/15 to 6/5 relationship, as well as the following variants and extensions:
 
=== Tristone ===
[[Subgroup]]: 5/3.20/9.10/3
 
[[Comma]] list: 2048/2025
 
[[POL2]] generator: ~5/4 = 389.8224
 
[[Mapping]]: [⟨1 2 5], ⟨0 -1 -6]]
 
TE [[complexity]]: 1.988720
 
[[RMS temperament measures|RMS]] error: 2.228679 cents
 
[[Optimal ET sequence]]: 9ed5/3, 16ed5/3, 25ed5/3
 
==== Tridistone ====
[[Subgroup]]: 5/3.20/9.10/3.1000/189
 
[[Comma]] list: 2048/2025, 225/224, 64/63, 50/49
 
[[POL2]] generator: ~5/4 = 389.6140
 
[[Mapping]]: [⟨1 2 5 5], ⟨0 -1 -6 -4]]
 
TE [[complexity]]: 1.724923
 
[[RMS temperament measures|RMS]] error: 8.489179 cents
 
[[Optimal ET sequence]]: 9ed5/3, 16ed5/3
 
=== Metatristone ===
[[Subgroup]]: 5/3.20/9.5/2
 
[[Comma]] list: 2048/2025
 
[[POL2]] generator: ~5/4 = 390.5180
 
[[Mapping]]: [⟨1 2 4], ⟨0 -1 -5]]
 
TE [[complexity]]: 2.192193
 
[[RMS temperament measures|RMS]] error: 2.021819 cents
 
[[Optimal ET sequence]]: 9ed5/3, 16ed5/3, 25ed5/3
 
==== Metatridistone ====
[[Subgroup]]: 5/3.20/9.5/2.250/63
 
[[Comma]] list: 2048/2025, 225/224, 64/63, 50/49
 
[[POL2]] generator: ~5/4 = 390.5430
 
[[Mapping]]: [⟨1 2 4 4], ⟨0 -1 -5 -3]]
 
TE [[complexity]]: 1.895168
 
[[RMS temperament measures|RMS]] error: 7.910273 cents
 
[[Optimal ET sequence]]: 9ed5/3, 16ed5/3
 
'''16ed5/3''' also supports [[Blackcomb]] temperament which is built on [[5/4]] and [[3/2]] in a very similar way to octave-repeating [[meantone]] but is less accurate. Blackcomb tempers out the comma [[250/243]], the amount by which 3 [[3/2]]'s exceed [[5/4]] sixth-reduced, in the 5/3.2.3 subgroup (equal to the [[5-limit]]).
 
[[Category:Nonoctave]]
[[Category:Nonoctave]]
Retrieved from "https://en.xen.wiki/w/16ed5/3"