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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
| 36/35, 33/32, 31/30
| 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
| minor third
| Db
| Eb
|- style="background: #eee"
| minor third
|-
| 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
| 64/45, 10/7, 17/12
| 44/31, 64/45
| 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 left-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 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 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|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.
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.


As 3 semitones make a period-reduced octave, and it alludes to the 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:
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 ===
=== Tristone ===
Line 428: Line 543:
[[RMS temperament measures|RMS]] error: 2.228679 cents
[[RMS temperament measures|RMS]] error: 2.228679 cents


[[Vals]]: 9ed5/3, 16ed5/3, 25ed5/3
[[Optimal ET sequence]]: 9ed5/3, 16ed5/3, 25ed5/3


==== Tridistone ====
==== Tridistone ====
[[Subgroup]]: 5/3.20/9.10/3.1000/189
[[Subgroup]]: 5/3.20/9.10/3.1000/189


[[Comma]] list: 2048/2025, 225/224
[[Comma]] list: 2048/2025, 225/224, 64/63, 50/49


[[POL2]] generator: ~5/4 = 389.6140
[[POL2]] generator: ~5/4 = 389.6140
Line 443: Line 558:
[[RMS temperament measures|RMS]] error: 8.489179 cents
[[RMS temperament measures|RMS]] error: 8.489179 cents


[[Vals]]: 9ed5/3, 16ed5/3
[[Optimal ET sequence]]: 9ed5/3, 16ed5/3  


=== Metatristone ===
=== Metatristone ===
[[Subgroup]]: 5/3.20/9.15/6
[[Subgroup]]: 5/3.20/9.5/2


[[Comma]] list: 2048/2025
[[Comma]] list: 2048/2025
Line 458: Line 573:
[[RMS temperament measures|RMS]] error: 2.021819 cents
[[RMS temperament measures|RMS]] error: 2.021819 cents


[[Vals]]: 9ed5/3, 16ed5/3, 25ed5/3
[[Optimal ET sequence]]: 9ed5/3, 16ed5/3, 25ed5/3  


==== Metatridistone ====
==== Metatridistone ====
[[Subgroup]]: 5/3.20/9.15/6.250/63
[[Subgroup]]: 5/3.20/9.5/2.250/63


[[Comma]] list: 2048/2025, 225/224
[[Comma]] list: 2048/2025, 225/224, 64/63, 50/49


[[POL2]] generator: ~5/4 = 390.5430
[[POL2]] generator: ~5/4 = 390.5430
Line 473: Line 588:
[[RMS temperament measures|RMS]] error: 7.910273 cents
[[RMS temperament measures|RMS]] error: 7.910273 cents


[[Vals]]: 9ed5/3, 16ed5/3
[[Optimal ET sequence]]: 9ed5/3, 16ed5/3  
[[Category:EdVI]]
 
'''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"