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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]].
'''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]].


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 20¢, 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).


== 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).
{| class="wikitable"
{| class="wikitable center-all right-2"
!Degree
! Degree
!Cents
! Cents
!Approximate intervals
! Approximate intervals
!Mos-interval
! Mos-interval
!Diatonic interval
! Diatonic interval
!Notation
! Notation
|- style="background: #eee"
| '''0'''
| '''0.0000'''
| '''1'''
| '''unison'''
| '''unison'''
| '''A'''
|-
|-
|'''0'''
| 1
|'''0'''
| 55.2724
|'''1'''
| 31/30, 33/32
|'''unison'''
| aug unison
|'''unison'''
| quatertone
|'''A'''
| A#
|-
|-
|1
| 2
|55.2724
| 110.5448
|31/30, 33/32
| 16/15
|aug unison
| min mos2nd
|quatertone
| minor second
|A#
| Bb
|-
|-
|2
| 3
|110.5448
| 165.8173
|16/15
| 11/10
|min mos2nd
| maj mos2nd
|minor second
| neutral second
|Bb
| B
|-
|-
|3
| 4
|165.8173
| 221.0897
|11/10
| 8/7, 17/15
|maj mos2nd
| min mos3rd
|neutral second
| major second
|B
| Cb
|-
|-
|4
| 5
|221.0897
| 276.3621
|8/7, 17/15
| 75/64, 7/6, 20/17
|min mos3rd
| maj mos3rd
|major second
| subminor third
|Cb
| C
|-
|-
|5
| 6
|276.3621
| 331.6345
|7/6, 20/17, 75/64
| 6/5, 40/33, 17/14
|maj mos3rd
| dim mos4th
|subminor third
| minor third
|C
| Db
|- style="background: #eee"
| 7
| ''386.9069''
| ''5/4''
| ''perf mos4th''
| major third
| D
|-
|-
|6
| 8
|331.6345
| 442.1794
|6/5, 17/14
| 9/7, 22/17
|dim mos4th
| aug mos4th
|minor third
| supermajor third
|Db
| D#
|- style="background: #eee"
| 9
| ''497.4517''
| ''4/3''
| ''perf mos5th''
| just fourth
| E
|-
|-
|''7''
| 10
|''386.9069''
| 552.7242
|''5/4''
| 25/18, 11/8, 18/13
|''perf mos4th''
| aug mos5th
|major third
| wide fourth
|D
| E#
|-
|-
|8
| 11
|442.1794
| 607.9966
|9/7, 22/17
| 10/7, 17/12
|aug mos4th
| min mos6th
|supermajor third
| large tritone
|D#
| Fb
|-
|-
|''9''
| 12
|''497.4517''
| 663.2690
|''4/3''
| 72/49, 22/15
|''perf mos5th''
| maj mos6th
|just fourth
| narrow fifth
|E
| F
|-
|-
|10
| 13
|552.7242
| 718.5415
|11/8
| 3/2, 50/33
|aug mos5th
| min mos7th
|wide fourth
| acute fifth
|E#
| F#
|-
|-
|11
| 14
|607.9966
| 773.8129
|10/7, 17/12
| 25/16
|min mos6th
| maj mos7th
|large tritone
| subminor sixth
|Fb
| G
|-
|-
|12
| 15
|663.269
| 829.0863
|22/15, 72/49
| 8/5, 13/8
|maj mos6th
| dim mos8ave
|narrow fifth
| minor sixth
|F
| G#
|- style="background: #eee"
| '''16'''
| '''884.3587'''
| '''5/3'''
| '''mosoctave'''
| '''major sixth'''
| '''A'''
|-
|-
|13
| 17
|718.54145
| 939.6311
|3/2, 50/33
| 12/7, 19/11
|min mos7th
| aug mos8ave
|acute fifth
| supermajor sixth
|F#
| A#
|-
|-
|14
| 18
|773.8129
| 994.9035
|25/16
| 16/9
|maj mos7th
| min mos9th
|subminor sixth
| minor seventh
|G
| Bb
|-
|-
|15
| 19
|829.0863
| 1050.1760
|8/5, 13/8
| 11/6
|dim mos8ave
| maj mos9th
|minor sixth
| neutral seventh
|G#
| B
|-
|-
|'''16'''
| 20
|'''884.3587'''
| 1105.4484
|'''5/3'''
| 40/21, 17/9
|'''mosoctave'''
| min mos10th
|'''major sixth'''
| major seventh
|'''A'''
| Cb
|-
|-
|17
| 21
|939.6311
| 1160.7208
|12/7, 19/11
| 35/18, 43/22
|aug mos8ave
| maj mos10th
|supermajor sixth
| narrow octave
|A#
| C
|-
|-
|18
| 22
|994.9035
| 1215.9932
|16/9
| 2/1
|min mos9th
| dim mos11th
|minor seventh
| octave
|Bb
| C#
|-
|19
|1050.1760
|11/6
|maj mos9th
|neutral seventh
|B
|-
|20
|1105.4484
|17/9, 40/21
|min mos10th
|major seventh
|Cb
|-
|21
|1160.7208
|35/18, 43/22
|maj mos10th
|narrow octave
|C
|-
|22
|1215.9932
|2/1
|dim mos11th
|octave
|C#
|}
|}
These intervals are close to a few other related non-octave scales:
These intervals are close to a few other related non-octave scales:
{| class="wikitable"
{| class="wikitable left-all"
|+
|+
!
!
!16ed16\22
! 16ed16\22
![[7ed5/4]]
! [[7ed5/4]]
!16ed5/3
! 16ed5/3
![[Noleta|9ed4/3]]
! [[Noleta|9ed4/3]]
![[43ed4]]
! [[43ed4]]
!16ed16\21
! 16ed16\21
|-
|-
|1
| 1
|54.54545
| 54.54545
|55.188
| 55.188
|55.2724
| 55.2724
|55.338
| 55.338
|55.81395
| 55.81395
|57.1429
| 57.1429
|-
|-
|2
| 2
|109.0909
| 109.0909
|110.375
| 110.375
|110.5448
| 110.5448
|110.677
| 110.677
|111.6729
| 111.6729
|114.2857
| 114.2857
|-
|-
|3
| 3
|163.6364
| 163.6364
|165.563
| 165.563
|165.8173
| 165.8173
|166.015
| 166.015
|167.4419
| 167.4419
|171.4286
| 171.4286
|-
|-
|4
| 4
|218.1818
| 218.1818
|220.751
| 220.751
|221.0897
| 221.0897
|221.353
| 221.353
|223.2558
| 223.2558
|228.5714
| 228.5714
|-
|-
|5
| 5
|272.7273
| 272.7273
|275.938
| 275.938
|276.3621
| 276.3621
|276.692
| 276.692
|279.0698
| 279.0698
|285.7143
| 285.7143
|-
|-
|6
| 6
|327.2727
| 327.2727
|331.126
| 331.126
|331.6345
| 331.6345
|332.030
| 332.030
|334.8837
| 334.8837
|342.8571
| 342.8571
|-
|-
|7
| 7
|381.8182
| 381.8182
|386.314
| 386.314
|386.9069
| 386.9069
|387.368
| 387.368
|390.6977
| 390.6977
|400
| 400
|-
|-
|8
| 8
|436.3636
| 436.3636
|441.501
| 441.501
|442.1794
| 442.1794
|442.707
| 442.707
|446.5116
| 446.5116
|457.1429
| 457.1429
|-
|-
|9
| 9
|490.9091
| 490.9091
|496.689
| 496.689
|497.4517
| 497.4517
|498.045
| 498.045
|502.3256
| 502.3256
|514.2857
| 514.2857
|-
|-
|10
| 10
|545.54545
| 545.54545
|551.877
| 551.877
|552.7242
| 552.7242
|553.383
| 553.383
|558.1395
| 558.1395
|571.4286
| 571.4286
|-
|-
|11
| 11
|600
| 600
|607.064
| 607.064
|607.9966
| 607.9966
|608.722
| 608.722
|613.9535
| 613.9535
|628.5714
| 628.5714
|-
|-
|12
| 12
|654.54545
| 654.54545
|662.252
| 662.252
|663.269
| 663.269
|664.060
| 664.060
|669.7674
| 669.7674
|685.7143
| 685.7143
|-
|-
|13
| 13
|709.0909
| 709.0909
|717.440
| 717.440
|718.54145
| 718.54145
|719.398
| 719.398
|725.5814
| 725.5814
|742.8571
| 742.8571
|-
|-
|14
| 14
|763.6364
| 763.6364
|772.627
| 772.627
|773.8129
| 773.8129
|774.737
| 774.737
|781.39535
| 781.39535
|800
| 800
|-
|-
|15
| 15
|818.1818
| 818.1818
|827.815
| 827.815
|829.0863
| 829.0863
|830.075
| 830.075
|837.7209
| 837.7209
|857.1429
| 857.1429
|-
|-
|16
| 16
|872.7273
| 872.7273
|883.003
| 883.003
|884.3587
| 884.3587
|885.413
| 885.413
|893.0233
| 893.0233
|914.2857
| 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|MOS scales]] as [[16edo]], as such it contains the following scales:
{| class="wikitable"
{| class="wikitable center-all right-3"
!Periods
! Periods
per octave
per octave
!Generator
! Generator
!Pattern
! Pattern
|-
|-
|1
| 1
|1\16
| 1\16
|1L ns (pathological)
| 1L ns (pathological)
|-
|-
|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#
| 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
|Bb
| Bb
|Fb
| Fb
|Cb
| Cb
|Gb
| Gb
|Db
| Db
|Ab
| Ab
|Eb
| Eb
|Bbb
| Bbb
|Fbb
| Fbb
|Cbb
| Cbb
|Gbb
| Gbb
|-
|-
|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
|}
|}
== 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 4 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>). The tempered marvel comma also means that the two large [[Tritone|tritones]] ([[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. [[User:Ayceman|I]] propose the name '''tristone''' for this temperament, as 3 semitones make a period-reduced octave, and it alludes to the tritone tempering.
[[Category:EdVI]]
[[Category:EdVI]]
[[Category:Nonoctave]]
[[Category:Nonoctave]]
[[Category:Edonoi]]
[[Category:Edonoi]]
Retrieved from "https://en.xen.wiki/w/16ed5/3"