User:Moremajorthanmajor/16wtn: Difference between revisions

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'''16WTN''' is the equal division of the fifth into 8 steps of 87.7444 [[cent]]s each, alternated with a sequence of eight equal steps of another interval between 75 and 100 cents in its patent tunings making its best tunings very nearly [[88cET]].
'''16WTN''' is optimally the equal division of the fifth into 8 steps of 87.7444 [[cent]]s each, alternated with a sequence of eight equal steps of another interval between 75 and 100 cents in its patent tunings making its best tunings very nearly [[88cET]].


== Optimal Intervals ==
== Patent Intervals ==
The best tunings of 16WTN can be interpreted as tunings for the Natural and Harmonic Minor modes of Annapolis[6L 4s]. For the notation of these modes, Greek numerals 1-10 can be used (Α,Β,Γ,Δ,Ε,Ϛ/Ϝ,Ζ,Η,Θ,Ι [J]).
The best tunings of 16WTN can be interpreted as tunings for the Natural and Harmonic Minor modes of Annapolis[6L 4s]. For the notation of these modes, Greek numerals 1-10 can be used (Α,Β,Γ,Δ,Ε,Ϛ/Ϝ,Ζ,Η,Θ,Ι [J]).
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!
!Minimum a
!Target a
!Maximum a
!Target b
![8\13 99.609¢]
!Alternating [[EDO|edo]] and *ed(9/8) a
!Minimum b
!Alternating edo and *ed(9/8) b
!Maximum b
![4\7 83.0075¢]
!Annapolis[6L 4s]
!Annapolis[6L 4s]
|-
|-
!1
|1
! colspan="6" |87.7444
! colspan="2" |87.7444
|92.3077
|85.7143
|Α#
|Α#
|-
|-
!2
|2
! colspan="6" |175.48875
! colspan="2" |175.48875
|184.6154
|171.4286
|Β
|Β
|-
|-
!3
|3
! colspan="6" |263.2331
! colspan="2" |263.2331
|276.9231
|257.1429
|Γ
|Γ
|-
|-
!4
|4
! colspan="6" |350.9775
! colspan="2" |350.9775
|369.2308
|342.8571
|Γ#
|Γ#
|-
|-
!5
|5
! colspan="6" |438.7219
! colspan="2" |438.7219
|461.5385
|428.5714
|Δ
|Δ
|-
|-
!6
|6
! colspan="6" |526.46625
! colspan="2" |526.46625
|553.84615
|51<span style="line-height: 1.5;">4.2857</span>
|Δ#
|Δ#
|-
|-
!7
|7
! colspan="6" |614.2106
! colspan="2" |614.2106
|646.15385
|600.000
|Ε
|Ε
|-
|-
!'''8'''
|'''8'''
! colspan="6" |'''701.955'''
! colspan="2" |'''701.955'''
|738.4615
|685.7143
|'''Ϛ/Ϝ'''
|'''Ϛ/Ϝ'''
|-
|-
|9
|9
|797.955
|804.8121
|801.564
|801.564
|781.955
|787.6693
|784.9625
|784.9625
|830.7692
|771.4286
|Ϛ#/Ϝ#
|Ϛ#/Ϝ#
|-
|-
|10
|10
|893.955
|907.6692
|901.173
|901.173
|861.955
|873.3836
|867.97
|867.97
|923.0769
|857.1429
|Ζ
|Ζ
|-
|-
|11
|11
|989.955
|1010.5264
|1000.782
|1000.782
|941.955
|959.0979
|950.9775
|950.9775
|1015.3846
|942.8571
|Η
|Η
|-
|-
|12
|12
|1085.955
|1113.3836
|1101.391
|1101.391
|1021.955
|1044.8121
|1033.985
|1033.985
|1107.6923
|1028.5714
|Η#
|Η#
|-
|-
|13
|13
|1181.955
|1216.2407
|1200
|1200
|1101.955
|1130.5264
|1116.9925
|1116.9925
|1200
|1114.2857
|Θ
|Θ
|-
|-
|14
|14
|1277.955
|1299.609
|1319.0979
|1200
|1280.1389
|1267.97
|1181.955
|1216.2407
|1200
|1200
|Θ#
|Θ#
|-
|-
|15
|15
|1373.955
|1398.218
|1421.955
|1283.0075
|1360.2777
|1333.94
|1261.955
|1301.955
|1301.955
|1293.83465
|I (J)
|I (J)
|-
|-
|16
|16
|1469.955
|1497.827
|1524.8121
|1366.015
|1440.4165
! colspan="2" |1403.91
|1341.955
! colspan="2" |1387.6693
|A
|A
|}
|}
[[Category:Well tempered nonet]]

Latest revision as of 09:31, 1 September 2025

16WTN is optimally the equal division of the fifth into 8 steps of 87.7444 cents each, alternated with a sequence of eight equal steps of another interval between 75 and 100 cents in its patent tunings making its best tunings very nearly 88cET.

Patent Intervals

The best tunings of 16WTN can be interpreted as tunings for the Natural and Harmonic Minor modes of Annapolis[6L 4s]. For the notation of these modes, Greek numerals 1-10 can be used (Α,Β,Γ,Δ,Ε,Ϛ/Ϝ,Ζ,Η,Θ,Ι [J]).

Target a Target b Alternating edo and *ed(9/8) a Alternating edo and *ed(9/8) b Annapolis[6L 4s]
1 87.7444 92.3077 85.7143 Α#
2 175.48875 184.6154 171.4286 Β
3 263.2331 276.9231 257.1429 Γ
4 350.9775 369.2308 342.8571 Γ#
5 438.7219 461.5385 428.5714 Δ
6 526.46625 553.84615 514.2857 Δ#
7 614.2106 646.15385 600.000 Ε
8 701.955 738.4615 685.7143 Ϛ/Ϝ
9 801.564 784.9625 830.7692 771.4286 Ϛ#/Ϝ#
10 901.173 867.97 923.0769 857.1429 Ζ
11 1000.782 950.9775 1015.3846 942.8571 Η
12 1101.391 1033.985 1107.6923 1028.5714 Η#
13 1200 1116.9925 1200 1114.2857 Θ
14 1299.609 1200 1267.97 1200 Θ#
15 1398.218 1283.0075 1333.94 1301.955 I (J)
16 1497.827 1366.015 1403.91 A
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