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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, (21/20)
| 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
| 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
| 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, (7/4)
| 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
| 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
| 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 422: 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 429: 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 [[14/9]] 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:
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:
Line 450: 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 ====
Line 465: 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 480: 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 ====
Line 495: 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]]

Latest revision as of 19:20, 1 August 2025

← 15ed5/3 16ed5/3 17ed5/3 →
Prime factorization 24
Step size 55.2724 ¢ 
Octave 22\16ed5/3 (1215.99 ¢) (→ 11\8ed5/3)
Twelfth 34\16ed5/3 (1879.26 ¢) (→ 17\8ed5/3)
Consistency limit 2
Distinct consistency limit 2

16ed5/3 is the equal division of the just major sixth into sixteen parts of 55.2724 cents each, corresponding to 21.7106edo. It is very closely related to the 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 just fifth and 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

Approximation of harmonics in 16ed5/3
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +16.0 -22.7 -23.3 -22.7 -6.7 +2.8 -7.3 +9.9 -6.7 -5.9 +9.3
Relative (%) +28.9 -41.1 -42.1 -41.1 -12.1 +5.1 -13.2 +17.9 -12.1 -10.6 +16.8
Steps
(reduced) 		22
(6) 		34
(2) 		43
(11) 		50
(2) 		56
(8) 		61
(13) 		65
(1) 		69
(5) 		72
(8) 		75
(11) 		78
(14)

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 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.

Degree Cents 5/3.4/3.11/6.31/18 subgroup 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
0 0.0000 1/1 unison E unison C unison
1 55.2724 31/30, 32/31, 33/32 36/35 aug unison E# aug unison C# quartertone
2 110.5448 16/15, 33/31 21/20 min mos2nd Fb double-aug unison, dim second Cx, Dbb minor second
3 165.8173 11/10 maj mos2nd F minor second Db neutral second
4 221.0897 25/22 8/7, 17/15 min mos3rd F#/Gb major second D major second
5 276.3621 75/64, 88/75 7/6, 20/17 maj mos3rd G aug second D# subminor third
6 331.6345 40/33, 75/62 6/5, 17/14 dim mos4th G#/Ab minor third Eb minor third
7 386.9069 5/4 perf mos4th A major third E major third
8 442.1794 31/24, 40/31 9/7, 35/27, 22/17 aug mos4th A#/Bb aug third E# supermajor third
9 497.4517 4/3 perf mos5th B dim fourth Fb just fourth
10 552.7242 11/8, 62/45 25/18, 18/13 aug mos5th B# perfect fourth F wide fourth
11 607.9966 44/31, 64/45 10/7, 17/12 min mos6th Cb aug fourth F# large tritone
12 663.2690 22/15 72/49 maj mos6th C dim fifth Gb narrow fifth
13 718.5415 50/33 3/2 min mos7th C#/Db perfect fifth G acute fifth
14 773.8129 25/16 maj mos7th D aug fifth G# subminor sixth
15 829.0863 50/31 8/5, 13/8 dim mos8ave D#/Eb dim sixth Cb minor sixth
16 884.3587 5/3 mosoctave E perfect sixth C major sixth
17 939.6311 31/18, 55/32 12/7, 19/11 aug mos8ave E# aug sixth C# supermajor sixth
18 994.9035 16/9, 55/31 7/4 min mos9th Fb double-aug sixth, dim seventh Cx, Dbb minor seventh
19 1050.1760 11/6 maj mos9th F minor seventh Db neutral seventh
20 1105.4484 176/93, 125/66, 256/135 40/21, (27/14), 17/9 min mos10th F#/Gb major seventh D major seventh
21 1160.7208 88/45, 125/64 35/18, 43/22 maj mos10th G aug seventh D# narrow octave
22 1215.9932 200/99, 121/60, 125/62 2/1 dim mos11th G#/Ab minor octave Eb octave

These intervals are close to a few other related scales:

22edo 7ed5/4 23ed18\17 16ed5/3 9ed4/3 (Noleta) 43ed4 34edt 21edo
1 54.54545 55.188 55.2429 55.2724 55.338 55.8140 55.9399 57.1429
2 109.0909 110.375 110.4859 110.5448 110.677 111.6729 111.8797 114.2857
3 163.6364 165.563 165.7288 165.8173 166.015 167.4419 167.8196 171.4286
4 218.1818 220.751 220.9718 221.0897 221.353 223.2558 223.7594 228.5714
5 272.7273 275.938 276.2147 276.3621 276.692 279.0698 279.6993 285.7143
6 327.2727 331.126 331.4576 331.6345 332.030 334.8837 335.6391 342.8571
7 381.8182 386.314 386.7006 386.9069 387.368 390.6977 391.5790 400
8 436.3636 441.501 441.9435 442.1794 442.707 446.5116 447.5188 457.1429
9 490.9091 496.689 497.1865 497.4517 498.045 502.3256 503.4587 514.2857
10 545.5455 551.877 552.4294 552.7242 553.383 558.1395 559.3985 571.4286
11 600 607.064 607.6723 607.9966 608.722 613.9535 615.3384 628.5714
12 654.5455 662.252 662.9153 663.269 664.060 669.7674 671.2782 685.7143
13 709.0909 717.440 718.1582 718.5415 719.398 725.5814 727.2181 742.8571
14 763.6364 772.627 773.4011 773.8129 774.737 781.3954 783.1579 800
15 818.1818 827.815 828.6441 829.0863 830.075 837.7209 839.0978 857.1429
16 872.7273 883.003 883.8870 884.3587 885.413 893.0233 895.0376 914.2857

MOS Scales

16edVI supports the same MOS scales as 16edo, as such it contains the following scales:

Periods
per octave 		Generator Pattern
1 1\16 1L Ns
1 3\16 1L 4s, 5L 1s
1 5\16 3L 4s, 3L 7s
1 7\16 2L 5s, 7L 2s
2 1\16 2L 8s, 2L 10s, 2L 12s
2 3\16 4L 2s, 6L 4s
4 1\16 4L 4s, 4L 8s

For the 2L 5s scale, the genchain is this:

F# C# G# D# A# E# B# F C G D A E B Fb Cb Gb Db Ab Eb Db Fbb Cbb Gbb Dbb
A2 A6 A3 A7 A4 A1 A5 M2 M6 M3 M7 P4 P1 P5 m2 m6 m3 m7 d4 d1 d5 d2 d6 d3 d7

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, 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

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 ((16/15)2=(8/7)), and the diaschisma - 2048/2025 ((16/15)3=(6/5)).

The diaschisma can also be tempered by taking 5 generators to mean a 3/2 ((4/3)5=(3/2)·(5/3)2), while the marvel comma can also be tempered with a stack of 3 generators, making a 10/7 ((4/3)3=(10/7)·(5/3)).

The tempered marvel comma also means that the two large tritones (pental and 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, I propose the name tristone for the basic 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 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 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 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 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).

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