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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 ==
Line 9: Line 12:
! 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
! Diatonic interval
! 2L 5s<5/3> notation
! 2L 5s<5/3> notation
! 1L 4s<5/3> ([[Blackcomb]][5]) interval
! 1L 4s<5/3> ([[Blackcomb]][5]) interval
! 1L 4s<5/3> ([[Blackcomb]][5]) notation
! 1L 4s<5/3> ([[Blackcomb]][5]) notation
! Diatonic interval
|-
|-
| '''0'''
| '''0'''
| '''0.0000'''
| '''0.0000'''
| '''1'''
| '''1/1'''
| '''unison'''
|  
| '''unison'''
| '''unison'''
| '''E'''
| '''E'''
| '''unison'''
| '''unison'''
| '''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
| 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
| 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
| 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
| major second
| F#/Gb
| F#/Gb
| major second
| major second
| 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
| 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
| minor third
| G#/Ab
| G#/Ab
| minor third
| minor third
| Eb
| Eb
| minor third
|-
|-
| 7
| 7
| ''386.9069''
| ''386.9069''
| ''5/4''
| ''5/4''
|
| ''perf mos4th''
| ''perf mos4th''
| major third
| A
| A
| major third
| major third
| 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
| supermajor third
| A#/Bb
| A#/Bb
| aug third
| aug third
| E#
| E#
| supermajor third
|-
|-
| 9
| 9
| ''497.4517''
| ''497.4517''
| ''4/3''
| ''4/3''
|
| ''perf mos5th''
| ''perf mos5th''
| just fourth
| B
| B
| dim fourth
| dim fourth
| Fb
| Fb
| just fourth
|-
|-
| 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
| 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
| 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
| 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
| 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
| 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
| dim sixth
| Cb
| Cb
| minor sixth
|-
|-
| '''16'''
| '''16'''
| '''884.3587'''
| '''884.3587'''
| '''5/3'''
| '''5/3'''
|
| '''mosoctave'''
| '''mosoctave'''
| '''major sixth'''
| '''E'''
| '''E'''
| '''perfect sixth'''
| '''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
| supermajor sixth
| E#
| E#
| aug sixth
| aug sixth
| C#
| C#
| supermajor sixth
|-
|-
| 18
| 18
| 994.9035
| 994.9035
| 16/9, (7/4)
| 16/9, 55/31
| 7/4
| min mos9th
| min mos9th
| minor seventh
| Fb
| Fb
| double-aug sixth, dim seventh
| double-aug sixth, dim seventh
|Cx, Dbb
| Cx, Dbb
| minor seventh
|-
|-
| 19
| 19
| 1050.1760
| 1050.1760
| 11/6
| 11/6
|
| maj mos9th
| maj mos9th
| neutral seventh
| F
| F
| minor seventh
| minor seventh
|Db
| Db
| neutral seventh
|-
|-
| 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
| major seventh
| F#/Gb
| D
| major seventh
| major seventh
|D
|-
|-
| 21
| 21
| 1160.7208
| 1160.7208
| 88/45, 125/64
| 35/18, 43/22
| 35/18, 43/22
| maj mos10th
| maj mos10th
| narrow octave
| G
| G
| aug seventh
| aug seventh
|D#
| D#
| narrow octave
|-
|-
| 22
| 22
| 1215.9932
| 1215.9932
| 200/99, 121/60, 125/62
| 2/1
| 2/1
| dim mos11th
| dim mos11th
| octave
| G#/Ab
| G#/Ab
| minor octave
| minor octave
|Eb
| Eb
| octave
|}
|}


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


Line 287: Line 433:
| 1
| 1
| 1\16
| 1\16
| 1L ns (pathological)
| 1L Ns
|-
|-
| 1
| 1
Line 384: Line 530:
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:


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 405: Line 550:
[[Comma]] list: 2048/2025, 225/224, 64/63, 50/49
[[Comma]] list: 2048/2025, 225/224, 64/63, 50/49


[[POL2]] generator: ~5/4 = [[Tel:389.6140|389.6140]]
[[POL2]] generator: ~5/4 = 389.6140


[[Mapping]]: [⟨1 2 5 5], ⟨0 -1 -6 -4]]
[[Mapping]]: [⟨1 2 5 5], ⟨0 -1 -6 -4]]
Line 444: Line 589:


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