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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]]
! [[34edt]]
! [[34edt]]
! 16ed16\21
! [[21edo]]
|-
|-
| 1
| 1
| 54.54545
| 54.54545
| 55.188
| 55.188
| 55.2724
|55.2429
| ''55.2724''
| 55.338
| 55.338
| 55.8140
| 55.8140
Line 198: Line 276:
| 109.0909
| 109.0909
| 110.375
| 110.375
| 110.5448
|110.4859
| ''110.5448''
| 110.677
| 110.677
| 111.6729
| 111.6729
Line 207: Line 286:
| 163.6364
| 163.6364
| 165.563
| 165.563
| 165.8173
|165.7288
| ''165.8173''
| 166.015
| 166.015
| 167.4419
| 167.4419
Line 216: Line 296:
| 218.1818
| 218.1818
| 220.751
| 220.751
| 221.0897
|220.9718
| ''221.0897''
| 221.353
| 221.353
| 223.2558
| 223.2558
Line 225: Line 306:
| 272.7273
| 272.7273
| 275.938
| 275.938
| 276.3621
|276.2147
| ''276.3621''
| 276.692
| 276.692
| 279.0698
| 279.0698
Line 234: Line 316:
| 327.2727
| 327.2727
| 331.126
| 331.126
| 331.6345
|331.4576
| ''331.6345''
| 332.030
| 332.030
| 334.8837
| 334.8837
Line 243: Line 326:
| 381.8182
| 381.8182
| 386.314
| 386.314
| 386.9069
|386.7006
| ''386.9069''
| 387.368
| 387.368
| 390.6977
| 390.6977
Line 252: Line 336:
| 436.3636
| 436.3636
| 441.501
| 441.501
| 442.1794
|441.9435
| ''442.1794''
| 442.707
| 442.707
| 446.5116
| 446.5116
Line 261: Line 346:
| 490.9091
| 490.9091
| 496.689
| 496.689
| 497.4517
|497.1865
| ''497.4517''
| 498.045
| 498.045
| 502.3256
| 502.3256
Line 270: Line 356:
| 545.5455
| 545.5455
| 551.877
| 551.877
| 552.7242
|552.4294
| ''552.7242''
| 553.383
| 553.383
| 558.1395
| 558.1395
Line 279: Line 366:
| 600
| 600
| 607.064
| 607.064
| 607.9966
|607.6723
| ''607.9966''
| 608.722
| 608.722
| 613.9535
| 613.9535
Line 288: Line 376:
| 654.5455
| 654.5455
| 662.252
| 662.252
| 663.269
|662.9153
| ''663.269''
| 664.060
| 664.060
| 669.7674
| 669.7674
Line 297: Line 386:
| 709.0909
| 709.0909
| 717.440
| 717.440
| 718.5415
|718.1582
| ''718.5415''
| 719.398
| 719.398
| 725.5814
| 725.5814
Line 306: Line 396:
| 763.6364
| 763.6364
| 772.627
| 772.627
| 773.8129
|773.4011
| ''773.8129''
| 774.737
| 774.737
| 781.3954
| 781.3954
Line 315: Line 406:
| 818.1818
| 818.1818
| 827.815
| 827.815
| 829.0863
|828.6441
| ''829.0863''
| 830.075
| 830.075
| 837.7209
| 837.7209
Line 324: Line 416:
| 872.7273
| 872.7273
| 883.003
| 883.003
| 884.3587
|883.8870
| ''884.3587''
| 885.413
| 885.413
| 893.0233
| 893.0233
Line 340: Line 433:
| 1
| 1
| 1\16
| 1\16
| 1L ns (pathological)
| 1L Ns
|-
|-
| 1
| 1
Line 368: 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 375: Line 467:
| A#
| A#
| E#
| E#
| B
| B#
| F
| F
| C
| C
Line 381: Line 473:
| D
| D
| A
| A
| E
| '''E'''
| Bb
| B
| Fb
| Fb
| Cb
| Cb
Line 389: Line 481:
| Ab
| Ab
| Eb
| Eb
| Bbb
| Db
| Fbb
| Fbb
| Cbb
| Cbb
| Gbb
| Gbb
| Dbb
|-
|-
| A2
| A2
Line 406: Line 499:
| M7
| M7
| P4
| P4
| P1
| '''P1'''
| P5
| P5
| m2
| m2
Line 420: 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 ==
Line 426: Line 522:
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 443: 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 458: 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 ===
Line 473: 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]]: 5/3.20/9.5/2.250/63
[[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 488: 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"