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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 [[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).
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 center-all right-2"
{| 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
! 2L 5s<5/3> mos-interval
! 2L 5s<5/3> notation
! 1L 4s<5/3> ([[Blackcomb]][5]) interval
! 1L 4s<5/3> ([[Blackcomb]][5]) notation
! Diatonic interval
! Diatonic interval
! Notation
|-
|- style="background: #eee"
| '''0'''
| '''0'''
| '''0.0000'''
| '''0.0000'''
| '''1'''
| '''1/1'''
|
| '''unison'''
| '''E'''
| '''unison'''
| '''unison'''
| '''C'''
| '''unison'''
| '''unison'''
| '''A'''
|-
|-
| 1
| 1
| 55.2724
| 55.2724
| 31/30, 33/32
| 31/30, 32/31, 33/32
| 36/35
| aug unison
| E#
| aug unison
| aug unison
| quatertone
| C#
| A#
| quartertone
|-
|-
| 2
| 2
| 110.5448
| 110.5448
| 16/15
| 16/15, 33/31
| 21/20
| min mos2nd
| min mos2nd
| Fb
| double-aug unison, dim second
| Cx, Dbb
| minor second
| minor second
| Bb
|-
|-
| 3
| 3
| 165.8173
| 165.8173
| 11/10
| 11/10
|
| maj mos2nd
| maj mos2nd
| F
| minor second
| Db
| neutral second
| neutral second
| B
|-
|-
| 4
| 4
| 221.0897
| 221.0897
| 25/22
| 8/7, 17/15
| 8/7, 17/15
| min mos3rd
| min mos3rd
| F#/Gb
| major second
| D
| major second
| major second
| Cb
|-
|-
| 5
| 5
| 276.3621
| 276.3621
| 75/64, 7/6, 20/17
| 75/64, 88/75
| 7/6, 20/17
| maj mos3rd
| maj mos3rd
| G
| aug second
| D#
| subminor third
| subminor third
| C
|-
|-
| 6
| 6
| 331.6345
| 331.6345
| 6/5, 40/33, 17/14
| 40/33, 75/62
| 6/5, 17/14
| dim mos4th
| dim mos4th
| G#/Ab
| minor third
| Eb
| minor third
| minor third
| Db
|-
|- style="background: #eee"
| 7
| 7
| ''386.9069''
| ''386.9069''
| ''5/4''
| ''5/4''
|
| ''perf mos4th''
| ''perf mos4th''
| A
| major third
| E
| major third
| major third
| D
|-
|-
| 8
| 8
| 442.1794
| 442.1794
| 9/7, 22/17
| 31/24, 40/31
| 9/7, 35/27, 22/17
| aug mos4th
| aug mos4th
| A#/Bb
| aug third
| E#
| supermajor third
| supermajor third
| D#
|-
|- style="background: #eee"
| 9
| 9
| ''497.4517''
| ''497.4517''
| ''4/3''
| ''4/3''
|
| ''perf mos5th''
| ''perf mos5th''
| B
| dim fourth
| Fb
| just fourth
| just fourth
| E
|-
|-
| 10
| 10
| 552.7242
| 552.7242
| 25/18, 11/8, 18/13
| 11/8, 62/45
| 25/18, 18/13
| aug mos5th
| aug mos5th
| B#
| perfect fourth
| F
| wide fourth
| wide fourth
| E#
|-
|-
| 11
| 11
| 607.9966
| 607.9966
| 44/31, 64/45
| 10/7, 17/12
| 10/7, 17/12
| min mos6th
| min mos6th
| Cb
| aug fourth
| F#
| large tritone
| large tritone
| Fb
|-
|-
| 12
| 12
| 663.2690
| 663.2690
| 72/49, 22/15
| 22/15
| 72/49
| maj mos6th
| maj mos6th
| C
| dim fifth
| Gb
| narrow fifth
| narrow fifth
| F
|-
|-
| 13
| 13
| 718.5415
| 718.5415
| 3/2, 50/33
| 50/33
| 3/2
| min mos7th
| min mos7th
| C#/Db
| perfect fifth
| G
| acute fifth
| acute fifth
| F#
|-
|-
| 14
| 14
| 773.8129
| 773.8129
| 25/16
| 25/16
|
| maj mos7th
| maj mos7th
| D
| aug fifth
| G#
| subminor sixth
| subminor sixth
| G
|-
|-
| 15
| 15
| 829.0863
| 829.0863
| 50/31
| 8/5, 13/8
| 8/5, 13/8
| dim mos8ave
| dim mos8ave
| D#/Eb
| dim sixth
| Cb
| minor sixth
| minor sixth
| G#
|-
|- style="background: #eee"
| '''16'''
| '''16'''
| '''884.3587'''
| '''884.3587'''
| '''5/3'''
| '''5/3'''
|
| '''mosoctave'''
| '''mosoctave'''
| '''E'''
| '''perfect sixth'''
| '''C'''
| '''major sixth'''
| '''major sixth'''
| '''A'''
|-
|-
| 17
| 17
| 939.6311
| 939.6311
| 31/18, 55/32
| 12/7, 19/11
| 12/7, 19/11
| aug mos8ave
| aug mos8ave
| E#
| aug sixth
| C#
| supermajor sixth
| supermajor sixth
| A#
|-
|-
| 18
| 18
| 994.9035
| 994.9035
| 16/9
| 16/9, 55/31
| 7/4
| min mos9th
| min mos9th
| Fb
| double-aug sixth, dim seventh
| Cx, Dbb
| minor seventh
| minor seventh
| Bb
|-
|-
| 19
| 19
| 1050.1760
| 1050.1760
| 11/6
| 11/6
|
| maj mos9th
| maj mos9th
| F
| minor seventh
| Db
| neutral seventh
| neutral seventh
| B
|-
|-
| 20
| 20
| 1105.4484
| 1105.4484
| 40/21, 17/9
| 176/93, 125/66, 256/135
| 40/21, (27/14), 17/9
| min mos10th
| min mos10th
| F#/Gb
| major seventh
| D
| major seventh
| major seventh
| Cb
|-
|-
| 21
| 21
| 1160.7208
| 1160.7208
| 88/45, 125/64
| 35/18, 43/22
| 35/18, 43/22
| maj mos10th
| maj mos10th
| G
| aug seventh
| D#
| narrow octave
| narrow octave
| C
|-
|-
| 22
| 22
| 1215.9932
| 1215.9932
| 200/99, 121/60, 125/62
| 2/1
| 2/1
| dim mos11th
| dim mos11th
| G#/Ab
| minor octave
| Eb
| octave
| octave
| C#
|}
|}


These intervals are close to a few other related non-octave scales:
These intervals are close to a few other related scales:
{| class="wikitable left-all"
{| class="wikitable left-all"
|+
!
!
! 16ed16\22
! [[22edo]]
! [[7ed5/4]]
! [[7ed5/4]]
!23ed18\17
! 16ed5/3
! 16ed5/3
! [[Noleta|9ed4/3]]
! [[9ed4/3]] (Noleta)
! [[43ed4]]
! [[43ed4]]
! 16ed16\21
! [[34edt]]
! [[21edo]]
|-
|-
| 1
| 1
| 54.54545
| 54.54545
| 55.188
| 55.188
| 55.2724
|55.2429
| ''55.2724''
| 55.338
| 55.338
| 55.81395
| 55.8140
| 55.9399
| 57.1429
| 57.1429
|-
|-
Line 197: Line 276:
| 109.0909
| 109.0909
| 110.375
| 110.375
| 110.5448
|110.4859
| ''110.5448''
| 110.677
| 110.677
| 111.6729
| 111.6729
| 111.8797
| 114.2857
| 114.2857
|-
|-
Line 205: Line 286:
| 163.6364
| 163.6364
| 165.563
| 165.563
| 165.8173
|165.7288
| ''165.8173''
| 166.015
| 166.015
| 167.4419
| 167.4419
| 167.8196
| 171.4286
| 171.4286
|-
|-
Line 213: Line 296:
| 218.1818
| 218.1818
| 220.751
| 220.751
| 221.0897
|220.9718
| ''221.0897''
| 221.353
| 221.353
| 223.2558
| 223.2558
| 223.7594
| 228.5714
| 228.5714
|-
|-
Line 221: Line 306:
| 272.7273
| 272.7273
| 275.938
| 275.938
| 276.3621
|276.2147
| ''276.3621''
| 276.692
| 276.692
| 279.0698
| 279.0698
| 279.6993
| 285.7143
| 285.7143
|-
|-
Line 229: Line 316:
| 327.2727
| 327.2727
| 331.126
| 331.126
| 331.6345
|331.4576
| ''331.6345''
| 332.030
| 332.030
| 334.8837
| 334.8837
| 335.6391
| 342.8571
| 342.8571
|-
|-
Line 237: Line 326:
| 381.8182
| 381.8182
| 386.314
| 386.314
| 386.9069
|386.7006
| ''386.9069''
| 387.368
| 387.368
| 390.6977
| 390.6977
| 391.5790
| 400
| 400
|-
|-
Line 245: Line 336:
| 436.3636
| 436.3636
| 441.501
| 441.501
| 442.1794
|441.9435
| ''442.1794''
| 442.707
| 442.707
| 446.5116
| 446.5116
| 447.5188
| 457.1429
| 457.1429
|-
|-
Line 253: Line 346:
| 490.9091
| 490.9091
| 496.689
| 496.689
| 497.4517
|497.1865
| ''497.4517''
| 498.045
| 498.045
| 502.3256
| 502.3256
| 503.4587
| 514.2857
| 514.2857
|-
|-
| 10
| 10
| 545.54545
| 545.5455
| 551.877
| 551.877
| 552.7242
|552.4294
| ''552.7242''
| 553.383
| 553.383
| 558.1395
| 558.1395
| 559.3985
| 571.4286
| 571.4286
|-
|-
Line 269: Line 366:
| 600
| 600
| 607.064
| 607.064
| 607.9966
|607.6723
| ''607.9966''
| 608.722
| 608.722
| 613.9535
| 613.9535
| 615.3384
| 628.5714
| 628.5714
|-
|-
| 12
| 12
| 654.54545
| 654.5455
| 662.252
| 662.252
| 663.269
|662.9153
| ''663.269''
| 664.060
| 664.060
| 669.7674
| 669.7674
| 671.2782
| 685.7143
| 685.7143
|-
|-
Line 285: Line 386:
| 709.0909
| 709.0909
| 717.440
| 717.440
| 718.54145
|718.1582
| ''718.5415''
| 719.398
| 719.398
| 725.5814
| 725.5814
| 727.2181
| 742.8571
| 742.8571
|-
|-
Line 293: Line 396:
| 763.6364
| 763.6364
| 772.627
| 772.627
| 773.8129
|773.4011
| ''773.8129''
| 774.737
| 774.737
| 781.39535
| 781.3954
| 783.1579
| 800
| 800
|-
|-
Line 301: Line 406:
| 818.1818
| 818.1818
| 827.815
| 827.815
| 829.0863
|828.6441
| ''829.0863''
| 830.075
| 830.075
| 837.7209
| 837.7209
| 839.0978
| 857.1429
| 857.1429
|-
|-
Line 309: Line 416:
| 872.7273
| 872.7273
| 883.003
| 883.003
| 884.3587
|883.8870
| ''884.3587''
| 885.413
| 885.413
| 893.0233
| 893.0233
| 895.0376
| 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]]s as [[16edo]], as such it contains the following scales:
{| class="wikitable center-all right-3"
{| class="wikitable center-all left-3"
! Periods
! Periods <br> per octave
per octave
! Generator
! Generator
! Pattern
! Pattern
Line 325: Line 433:
| 1
| 1
| 1\16
| 1\16
| 1L ns (pathological)
| 1L Ns
|-
|-
| 1
| 1
Line 353: Line 461:
For the 2L 5s scale, the genchain is this:
For the 2L 5s scale, the genchain is this:
{| class="wikitable center-all"
{| class="wikitable center-all"
| B#
| F#
| F#
| C#
| C#
Line 360: Line 467:
| A#
| A#
| E#
| E#
| B
| B#
| F
| F
| C
| C
Line 366: Line 473:
| D
| D
| A
| A
| E
| '''E'''
| Bb
| B
| Fb
| Fb
| Cb
| Cb
Line 374: Line 481:
| Ab
| Ab
| Eb
| Eb
| Bbb
| Db
| Fbb
| Fbb
| Cbb
| Cbb
| Gbb
| Gbb
| Dbb
|-
|-
| A2
| A2
Line 391: Line 499:
| M7
| M7
| P4
| P4
| P1
| '''P1'''
| P5
| P5
| m2
| m2
Line 405: Line 513:
| d7
| d7
|}
|}
== 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 ==
== 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.
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>)).
[[Category:EdVI]]
 
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]]
[[Category:Edonoi]]
Retrieved from "https://en.xen.wiki/w/16ed5/3"