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{{Infobox ET}}
{{Infobox ET}}
'''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 is vaguely equivalent to [[22edo]].
'''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 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).
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). It can also be notated using the fifth-generated [[Blackcomb]] temperament.
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
! Other interpretations
! 2L 5s<5/3> mos-interval
! 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
! 2L 5s<5/3> notation
! 1L 4s<5/3> ([[Blackcomb]]) notation
|-
|-
| '''0'''
| '''0'''
| '''0.0000'''
| '''0.0000'''
| '''1'''
| '''1/1'''
|
| '''unison'''
| '''unison'''
| '''E'''
| '''unison'''
| '''unison'''
| '''E'''
| '''C'''
| '''C'''
| '''unison'''
|-
|-
| 1
| 1
| 55.2724
| 55.2724
| 36/35, 33/32, 31/30
| 31/30, 32/31, 33/32
| 36/35
| aug unison
| aug unison
| quartertone
| E#
| E#
| aug unison
| C#
| C#
| quartertone
|-
|-
| 2
| 2
| 110.5448
| 110.5448
| 16/15, (21/20)
| 16/15, 33/31
| 21/20
| min mos2nd
| min mos2nd
| minor second
| Fb
| Fb
| double-aug unison, dim second
| Cx, Dbb
| Cx, Dbb
| minor second
|-
|-
| 3
| 3
| 165.8173
| 165.8173
| 11/10
| 11/10
|
| maj mos2nd
| maj mos2nd
| neutral second
| F
| F
| minor second
| Db
| Db
| neutral second
|-
|-
| 4
| 4
| 221.0897
| 221.0897
| 25/22
| 8/7, 17/15
| 8/7, 17/15
| min mos3rd
| min mos3rd
| F#/Gb
| major second
| major second
| F#/Gb
| D
| D
| major second
|-
|-
| 5
| 5
| 276.3621
| 276.3621
| 75/64, 7/6, 20/17
| 75/64, 88/75
| 7/6, 20/17
| maj mos3rd
| maj mos3rd
| subminor third
| G
| G
| aug second
| D#
| D#
| subminor third
|-
|-
| 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
| G#/Ab
| Eb
| Eb
| minor third
|-
|-
| 7
| 7
| ''386.9069''
| ''386.9069''
| ''5/4''
| ''5/4''
|
| ''perf mos4th''
| ''perf mos4th''
| A
| major third
| major third
| A
| E
| E
| major third
|-
|-
| 8
| 8
| 442.1794
| 442.1794
| 31/24, 40/31
| 9/7, 35/27, 22/17
| 9/7, 35/27, 22/17
| aug mos4th
| aug mos4th
| A#/Bb
| aug third
| E#
| supermajor third
| supermajor third
| A#/Bb
|-
|-
| 9
| 9
| ''497.4517''
| ''497.4517''
| ''4/3''
| ''4/3''
|
| ''perf mos5th''
| ''perf mos5th''
| B
| dim fourth
| Fb
| just fourth
| just fourth
| B
| E#, Fb
|-
|-
| 10
| 10
| 552.7242
| 552.7242
| 25/18, 11/8, 18/13
| 11/8, 62/45
| 25/18, 18/13
| aug mos5th
| aug mos5th
| wide fourth
| B#
| B#
| perfect fourth
| F
| F
| wide fourth
|-
|-
| 11
| 11
| 607.9966
| 607.9966
| 64/45, 10/7, 17/12
| 44/31, 64/45
| 10/7, 17/12
| min mos6th
| min mos6th
| large tritone
| Cb
| Cb
| aug fourth
| F#
| F#
| large tritone
|-
|-
| 12
| 12
| 663.2690
| 663.2690
| 72/49, 22/15
| 22/15
| 72/49
| maj mos6th
| maj mos6th
| narrow fifth
| C
| C
| dim fifth
| Gb
| Gb
| narrow fifth
|-
|-
| 13
| 13
| 718.5415
| 718.5415
| 3/2, 50/33
| 50/33
| 3/2
| min mos7th
| min mos7th
| acute fifth
| C#/Db
| C#/Db
| perfect fifth
| G
| G
| acute fifth
|-
|-
| 14
| 14
| 773.8129
| 773.8129
| 25/16
| 25/16
|
| maj mos7th
| maj mos7th
| subminor sixth
| D
| D
| aug fifth
| G#
| G#
| subminor sixth
|-
|-
| 15
| 15
| 829.0863
| 829.0863
| 50/31
| 8/5, 13/8
| 8/5, 13/8
| dim mos8ave
| dim mos8ave
| minor sixth
| D#/Eb
| D#/Eb
| dim sixth
| Cb
| Cb
| minor sixth
|-
|-
| '''16'''
| '''16'''
| '''884.3587'''
| '''884.3587'''
| '''5/3'''
| '''5/3'''
|
| '''mosoctave'''
| '''mosoctave'''
| '''major sixth'''
| '''E'''
| '''E'''
| '''perfect sixth'''
| '''C'''
| '''C'''
| '''major sixth'''
|-
|-
| 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
| E#
|
|-
|-
| 18
| 18
| 994.9035
| 994.9035
| 16/9, (7/4)
| 16/9, 55/31
| 7/4
| min mos9th
| min mos9th
| Fb
| double-aug sixth, dim seventh
| Cx, Dbb
| minor seventh
| minor seventh
| Fb
|
|-
|-
| 19
| 19
| 1050.1760
| 1050.1760
| 11/6
| 11/6
|
| maj mos9th
| maj mos9th
| F
| minor seventh
| Db
| neutral seventh
| neutral seventh
| F
|
|-
|-
| 20
| 20
| 1105.4484
| 1105.4484
| 176/93, 125/66, 256/135
| 40/21, (27/14), 17/9
| 40/21, (27/14), 17/9
| min mos10th
| min mos10th
| F#/Gb
| major seventh
| D
| major seventh
| major seventh
| F#/Gb
|
|-
|-
| 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
| G
|
|-
|-
| 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
| G#/Ab
|
|}
|}


Line 206: Line 258:
!23ed18\17
!23ed18\17
! 16ed5/3
! 16ed5/3
! [[Noleta|9ed4/3]]
! [[9ed4/3]] (Noleta)
! [[43ed4]]
! [[43ed4]]
! [[34edt]]
! [[34edt]]
Line 214: Line 266:
| 54.54545
| 54.54545
| 55.188
| 55.188
|55.2429
| 55.2429
| ''55.2724''
| ''55.2724''
| 55.338
| 55.338
Line 224: Line 276:
| 109.0909
| 109.0909
| 110.375
| 110.375
|110.4859
| 110.4859
| ''110.5448''
| ''110.5448''
| 110.677
| 110.677
Line 234: Line 286:
| 163.6364
| 163.6364
| 165.563
| 165.563
|165.7288
| 165.7288
| ''165.8173''
| ''165.8173''
| 166.015
| 166.015
Line 244: Line 296:
| 218.1818
| 218.1818
| 220.751
| 220.751
|220.9718
| 220.9718
| ''221.0897''
| ''221.0897''
| 221.353
| 221.353
Line 254: Line 306:
| 272.7273
| 272.7273
| 275.938
| 275.938
|276.2147
| 276.2147
| ''276.3621''
| ''276.3621''
| 276.692
| 276.692
Line 264: Line 316:
| 327.2727
| 327.2727
| 331.126
| 331.126
|331.4576
| 331.4576
| ''331.6345''
| ''331.6345''
| 332.030
| 332.030
Line 274: Line 326:
| 381.8182
| 381.8182
| 386.314
| 386.314
|386.7006
| 386.7006
| ''386.9069''
| ''386.9069''
| 387.368
| 387.368
Line 284: Line 336:
| 436.3636
| 436.3636
| 441.501
| 441.501
|441.9435
| 441.9435
| ''442.1794''
| ''442.1794''
| 442.707
| 442.707
Line 294: Line 346:
| 490.9091
| 490.9091
| 496.689
| 496.689
|497.1865
| 497.1865
| ''497.4517''
| ''497.4517''
| 498.045
| 498.045
Line 304: Line 356:
| 545.5455
| 545.5455
| 551.877
| 551.877
|552.4294
| 552.4294
| ''552.7242''
| ''552.7242''
| 553.383
| 553.383
Line 314: Line 366:
| 600
| 600
| 607.064
| 607.064
|607.6723
| 607.6723
| ''607.9966''
| ''607.9966''
| 608.722
| 608.722
Line 324: Line 376:
| 654.5455
| 654.5455
| 662.252
| 662.252
|662.9153
| 662.9153
| ''663.269''
| ''663.269''
| 664.060
| 664.060
Line 334: Line 386:
| 709.0909
| 709.0909
| 717.440
| 717.440
|718.1582
| 718.1582
| ''718.5415''
| ''718.5415''
| 719.398
| 719.398
Line 344: Line 396:
| 763.6364
| 763.6364
| 772.627
| 772.627
|773.4011
| 773.4011
| ''773.8129''
| ''773.8129''
| 774.737
| 774.737
Line 354: Line 406:
| 818.1818
| 818.1818
| 827.815
| 827.815
|828.6441
| 828.6441
| ''829.0863''
| ''829.0863''
| 830.075
| 830.075
Line 364: Line 416:
| 872.7273
| 872.7273
| 883.003
| 883.003
|883.8870
| 883.8870
| ''884.3587''
| ''884.3587''
| 885.413
| 885.413
Line 381: Line 433:
| 1
| 1
| 1\16
| 1\16
| 1L ns (pathological)
| 1L Ns
|-
|-
| 1
| 1
Line 429: Line 481:
| Ab
| Ab
| Eb
| Eb
| Db
| Bb
| Fbb
| Fbb
| Cbb
| Cbb
Line 463: Line 515:


== Commas ==
== Commas ==
Depending on your mapping, 16ed5/3 can be said to temper a number of commas, including the '''diaschisma''', the '''marvel comma''', '''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''' ([[245/243]]), as the half mosoctave is midway between [[9/7]] and [[35/27]]. This is analogous to the tritone in 2n edo systems. The '''keema''' ([[875/864]]) is tempered due to the septimal interpretation of the diatonic sevenths, and the '''Motwellsma''' ([[99/98]]) is tempered by two major mos3rds ([[7/6]]) resulting in an augmented mos5th ([[11/8]]).
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 470: 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.


Both of the 7-limit approaches also temper Archytas' comma ([[64/63]]) 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).
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.
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:
16ed5/3 primes can be mapped on the 31-limit to the val ⟨65 103 151 183 225 241 266 276 294 316 322], using every 3 steps of a shrinked [[65edo]] (-2.431¢ per octave). It differs from the patent val of 65edo in the mapping of prime 7 (val 65d).
 
As 3 semitones make a period-reduced octave, and it alludes to tritone tempering, [[User:Ayceman|Ayceman]] proposes 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:


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]]).
=== Tristone ===
=== Tristone ===
[[Subgroup]]: 5/3.20/9.10/3
[[Subgroup]]: 5/3.20/9.10/3
Line 492: Line 545:
[[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 ====
Line 507: Line 560:
[[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 522: Line 575:
[[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 ====
Line 537: Line 590:
[[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]]).
 
== See also ==
* [[Alpha, beta, and gamma family of equal divisions]]
 
[[Category:Nonoctave]]
[[Category:Nonoctave]]
[[Category:Edonoi]]
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