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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]].
'''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 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 20¢, making it a flexible non-octave scale.  
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).


== 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).
{| class="wikitable"
{| class="wikitable center-all right-2"
!Degree
! Degree
!Cents
! Cents
!Approximate intervals
! Approximate intervals
!Mos-interval
! Mos-interval
!Diatonic interval
! Diatonic interval
!Notation
! Notation
|- style="background: #eee"
| '''0'''
| '''0.0000'''
| '''1'''
| '''unison'''
| '''unison'''
| '''A'''
|-
|-
|'''0'''
| 1
|'''0'''
| 55.2724
|'''1'''
| 31/30, 33/32
|'''unison'''
| aug unison
|'''unison'''
| quatertone
|'''A'''
| A#
|-
|-
|1
| 2
|55.2724
| 110.5448
|31/30, 33/32
| 16/15
|aug unison
| min mos2nd
|quatertone
| minor second
|A#
| Bb
|-
|-
|2
| 3
|110.5448
| 165.8173
|16/15
| 11/10
|min mos2nd
| maj mos2nd
|minor second
| neutral second
|Bb
| B
|-
|-
|3
| 4
|165.8173
| 221.0897
|11/10
| 8/7, 17/15
|maj mos2nd
| min mos3rd
|neutral second
| major second
|B
| Cb
|-
|-
|4
| 5
|221.0897
| 276.3621
|8/7, 17/15
| 75/64, 7/6, 20/17
|min mos3rd
| maj mos3rd
|major second
| subminor third
|Cb
| C
|-
|-
|5
| 6
|276.3621
| 331.6345
|7/6, 20/17, 75/64
| 6/5, 40/33, 17/14
|maj mos3rd
| dim mos4th
|subminor third
| minor third
|C
| Db
|- style="background: #eee"
| 7
| ''386.9069''
| ''5/4''
| ''perf mos4th''
| major third
| D
|-
|-
|6
| 8
|331.6345
| 442.1794
|6/5, 17/14
| 9/7, 22/17
|dim mos4th
| aug mos4th
|minor third
| supermajor third
|Db
| D#
|- style="background: #eee"
| 9
| ''497.4517''
| ''4/3''
| ''perf mos5th''
| just fourth
| E
|-
|-
|''7''
| 10
|''386.9069''
| 552.7242
|''5/4''
| 25/18, 11/8, 18/13
|''perf mos4th''
| aug mos5th
|major third
| wide fourth
|D
| E#
|-
|-
|8
| 11
|442.1794
| 607.9966
|9/7, 22/17
| 10/7, 17/12
|aug mos4th
| min mos6th
|supermajor third
| large tritone
|D#
| Fb
|-
|-
|''9''
| 12
|''497.4517''
| 663.2690
|''4/3''
| 72/49, 22/15
|''perf mos5th''
| maj mos6th
|just fourth
| narrow fifth
|E
| F
|-
|-
|10
| 13
|552.7242
| 718.5415
|11/8
| 3/2, 50/33
|aug mos5th
| min mos7th
|wide fourth
| acute fifth
|E#
| F#
|-
|-
|11
| 14
|607.9966
| 773.8129
|10/7, 17/12
| 25/16
|min mos6th
| maj mos7th
|large tritone
| subminor sixth
|Fb
| G
|-
|-
|12
| 15
|663.269
| 829.0863
|22/15, 72/49
| 8/5, 13/8
|maj mos6th
| dim mos8ave
|narrow fifth
| minor sixth
|F
| G#
|- style="background: #eee"
| '''16'''
| '''884.3587'''
| '''5/3'''
| '''mosoctave'''
| '''major sixth'''
| '''A'''
|-
|-
|13
| 17
|718.54145
| 939.6311
|3/2, 50/33
| 12/7, 19/11
|min mos7th
| aug mos8ave
|acute fifth
| supermajor sixth
|F#
| A#
|-
|-
|14
| 18
|773.8129
| 994.9035
|25/16
| 16/9
|maj mos7th
| min mos9th
|subminor sixth
| minor seventh
|G
| Bb
|-
|-
|15
| 19
|829.0863
| 1050.1760
|8/5, 13/8
| 11/6
|dim mos8ave
| maj mos9th
|minor sixth
| neutral seventh
|G#
| B
|-
|-
|'''16'''
| 20
|'''884.3587'''
| 1105.4484
|'''5/3'''
| 40/21, 17/9
|'''mosoctave'''
| min mos10th
|'''major sixth'''
| major seventh
|'''A'''
| Cb
|-
|-
|17
| 21
|939.6311
| 1160.7208
|12/7, 19/11
| 35/18, 43/22
|aug mos8ave
| maj mos10th
|supermajor sixth
| narrow octave
|A#
| C
|-
|-
|18
| 22
|994.9035
| 1215.9932
|16/9
| 2/1
|min mos9th
| dim mos11th
|minor seventh
| octave
|Bb
| C#
|-
|19
|1050.1760
|11/6
|maj mos9th
|neutral seventh
|B
|-
|20
|1105.4484
|17/9, 40/21
|min mos10th
|major seventh
|Cb
|-
|21
|1160.7208
|35/18, 43/22
|maj mos10th
|narrow octave
|C
|-
|22
|1215.9932
|2/1
|dim mos11th
|octave
|C#
|}
|}
These intervals are close to a few other related non-octave scales:
These intervals are close to a few other related non-octave scales:
{| class="wikitable"
{| class="wikitable left-all"
|+
|+
!
!
!16ed16\22
! 16ed16\22
![[7ed5/4]]
! [[7ed5/4]]
!16ed5/3
! 16ed5/3
![[Noleta|9ed4/3]]
! [[Noleta|9ed4/3]]
![[43ed4]]
! [[43ed4]]
!16ed16\21
! 16ed16\21
|-
|-
|1
| 1
|54.54545
| 54.54545
|55.188
| 55.188
|55.2724
| 55.2724
|55.338
| 55.338
|55.81395
| 55.81395
|57.1429
| 57.1429
|-
|-
|2
| 2
|109.0909
| 109.0909
|110.375
| 110.375
|110.5448
| 110.5448
|110.677
| 110.677
|111.6729
| 111.6729
|114.2857
| 114.2857
|-
|-
|3
| 3
|163.6364
| 163.6364
|165.563
| 165.563
|165.8173
| 165.8173
|166.015
| 166.015
|167.4419
| 167.4419
|171.4286
| 171.4286
|-
|-
|4
| 4
|218.1818
| 218.1818
|220.751
| 220.751
|221.0897
| 221.0897
|221.353
| 221.353
|223.2558
| 223.2558
|228.5714
| 228.5714
|-
|-
|5
| 5
|272.7273
| 272.7273
|275.938
| 275.938
|276.3621
| 276.3621
|276.692
| 276.692
|279.0698
| 279.0698
|285.7143
| 285.7143
|-
|-
|6
| 6
|327.2727
| 327.2727
|331.126
| 331.126
|331.6345
| 331.6345
|332.030
| 332.030
|334.8837
| 334.8837
|342.8571
| 342.8571
|-
|-
|7
| 7
|381.8182
| 381.8182
|386.314
| 386.314
|386.9069
| 386.9069
|387.368
| 387.368
|390.6977
| 390.6977
|400
| 400
|-
|-
|8
| 8
|436.3636
| 436.3636
|441.501
| 441.501
|442.1794
| 442.1794
|442.707
| 442.707
|446.5116
| 446.5116
|457.1429
| 457.1429
|-
|-
|9
| 9
|490.9091
| 490.9091
|496.689
| 496.689
|497.4517
| 497.4517
|498.045
| 498.045
|502.3256
| 502.3256
|514.2857
| 514.2857
|-
|-
|10
| 10
|545.54545
| 545.54545
|551.877
| 551.877
|552.7242
| 552.7242
|553.383
| 553.383
|558.1395
| 558.1395
|571.4286
| 571.4286
|-
|-
|11
| 11
|600
| 600
|607.064
| 607.064
|607.9966
| 607.9966
|608.722
| 608.722
|613.9535
| 613.9535
|628.5714
| 628.5714
|-
|-
|12
| 12
|654.54545
| 654.54545
|662.252
| 662.252
|663.269
| 663.269
|664.060
| 664.060
|669.7674
| 669.7674
|685.7143
| 685.7143
|-
|-
|13
| 13
|709.0909
| 709.0909
|717.440
| 717.440
|718.54145
| 718.54145
|719.398
| 719.398
|725.5814
| 725.5814
|742.8571
| 742.8571
|-
|-
|14
| 14
|763.6364
| 763.6364
|772.627
| 772.627
|773.8129
| 773.8129
|774.737
| 774.737
|781.39535
| 781.39535
|800
| 800
|-
|-
|15
| 15
|818.1818
| 818.1818
|827.815
| 827.815
|829.0863
| 829.0863
|830.075
| 830.075
|837.7209
| 837.7209
|857.1429
| 857.1429
|-
|-
|16
| 16
|872.7273
| 872.7273
|883.003
| 883.003
|884.3587
| 884.3587
|885.413
| 885.413
|893.0233
| 893.0233
|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|MOS scales]] as [[16edo]], as such it contains the following scales:
{| class="wikitable"
{| class="wikitable center-all right-3"
!Periods
! Periods
per octave
per octave
!Generator
! Generator
!Pattern
! Pattern
|-
|-
|1
| 1
|1\16
| 1\16
|1L ns (pathological)
| 1L ns (pathological)
|-
|-
|1
| 1
|3\16
| 3\16
|1L 4s, 5L 1s
| 1L 4s, 5L 1s
|-
|-
|1
| 1
|5\16
| 5\16
|3L 4s, 3L 7s
| 3L 4s, 3L 7s
|-
|-
|1
| 1
|7\16
| 7\16
|2L 5s, 7L 2s
| 2L 5s, 7L 2s
|-
|-
|2
| 2
|1\16
| 1\16
|2L 8s, 2L 10s, 2L 12s
| 2L 8s, 2L 10s, 2L 12s
|-
|-
|2
| 2
|3\16
| 3\16
|4L 2s, 6L 4s
| 4L 2s, 6L 4s
|-
|-
|4
| 4
|1\16
| 1\16
|4L 4s, 4L 8s
| 4L 4s, 4L 8s
|}
|}
For the 2L 5s scale, the genchain is this:
For the 2L 5s scale, the genchain is this:
{| class="wikitable"
{| class="wikitable center-all"
|B#
| B#
|F#
| F#
|C#
| C#
|G#
| G#
|D#
| D#
|A#
| A#
|E#
| E#
|B
| B
|F
| F
|C
| C
|G
| G
|D
| D
|A
| A
|E
| E
|Bb
| Bb
|Fb
| Fb
|Cb
| Cb
|Gb
| Gb
|Db
| Db
|Ab
| Ab
|Eb
| Eb
|Bbb
| Bbb
|Fbb
| Fbb
|Cbb
| Cbb
|Gbb
| Gbb
|-
|-
|A2
| A2
|A6
| A6
|A3
| A3
|A7
| A7
|A5
| A4
|A1
| A1
|A4
| A5
|M2
| M2
|M6
| M6
|M3
| M3
|M7
| M7
|P5
| P4
|P1
| P1
|P4
| P5
|m2
| m2
|m6
| m6
|m3
| m3
|m7
| m7
|d5
| d4
|d1
| d1
|d4
| d5
|d2
| d2
|d6
| d6
|d3
| d3
|d7
| d7
|}
|}
== 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 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>). 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. [[User:Ayceman|I]] propose the name '''tristone''' for this temperament, as 3 semitones make a period-reduced octave, and it alludes to the tritone tempering.
[[Category:EdVI]]
[[Category:EdVI]]
[[Category:Nonoctave]]
[[Category:Nonoctave]]
[[Category:Edonoi]]
[[Category:Edonoi]]

Revision as of 16:09, 19 May 2021

16ed5/3 (or less accurately 16edVI) is the equal division of the just major sixth into sixteen parts of 55.2724 cents each, corresponding to 21.7106 edo. It is very closely related to the escapade temperament.

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 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).

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).

Degree Cents Approximate intervals Mos-interval Diatonic interval Notation
0 0.0000 1 unison unison A
1 55.2724 31/30, 33/32 aug unison quatertone A#
2 110.5448 16/15 min mos2nd minor second Bb
3 165.8173 11/10 maj mos2nd neutral second B
4 221.0897 8/7, 17/15 min mos3rd major second Cb
5 276.3621 75/64, 7/6, 20/17 maj mos3rd subminor third C
6 331.6345 6/5, 40/33, 17/14 dim mos4th minor third Db
7 386.9069 5/4 perf mos4th major third D
8 442.1794 9/7, 22/17 aug mos4th supermajor third D#
9 497.4517 4/3 perf mos5th just fourth E
10 552.7242 25/18, 11/8, 18/13 aug mos5th wide fourth E#
11 607.9966 10/7, 17/12 min mos6th large tritone Fb
12 663.2690 72/49, 22/15 maj mos6th narrow fifth F
13 718.5415 3/2, 50/33 min mos7th acute fifth F#
14 773.8129 25/16 maj mos7th subminor sixth G
15 829.0863 8/5, 13/8 dim mos8ave minor sixth G#
16 884.3587 5/3 mosoctave major sixth A
17 939.6311 12/7, 19/11 aug mos8ave supermajor sixth A#
18 994.9035 16/9 min mos9th minor seventh Bb
19 1050.1760 11/6 maj mos9th neutral seventh B
20 1105.4484 40/21, 17/9 min mos10th major seventh Cb
21 1160.7208 35/18, 43/22 maj mos10th narrow octave C
22 1215.9932 2/1 dim mos11th octave C#

These intervals are close to a few other related non-octave scales:

16ed16\22 7ed5/4 16ed5/3 9ed4/3 43ed4 16ed16\21
1 54.54545 55.188 55.2724 55.338 55.81395 57.1429
2 109.0909 110.375 110.5448 110.677 111.6729 114.2857
3 163.6364 165.563 165.8173 166.015 167.4419 171.4286
4 218.1818 220.751 221.0897 221.353 223.2558 228.5714
5 272.7273 275.938 276.3621 276.692 279.0698 285.7143
6 327.2727 331.126 331.6345 332.030 334.8837 342.8571
7 381.8182 386.314 386.9069 387.368 390.6977 400
8 436.3636 441.501 442.1794 442.707 446.5116 457.1429
9 490.9091 496.689 497.4517 498.045 502.3256 514.2857
10 545.54545 551.877 552.7242 553.383 558.1395 571.4286
11 600 607.064 607.9966 608.722 613.9535 628.5714
12 654.54545 662.252 663.269 664.060 669.7674 685.7143
13 709.0909 717.440 718.54145 719.398 725.5814 742.8571
14 763.6364 772.627 773.8129 774.737 781.39535 800
15 818.1818 827.815 829.0863 830.075 837.7209 857.1429
16 872.7273 883.003 884.3587 885.413 893.0233 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 (pathological)
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:

B# F# C# G# D# A# E# B F C G D A E Bb Fb Cb Gb Db Ab Eb Bbb Fbb Cbb Gbb
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

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 ((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). 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. I propose the name tristone for this temperament, as 3 semitones make a period-reduced octave, and it alludes to the tritone tempering.

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