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 making its best tunings very nearly [[88cET]]. Its best tunings are related to the [[Tetracot family|octacot temperament]].
'''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 making its best tunings very nearly [[88cET]]. Its best tunings are related to the [[Tetracot family|octacot temperament]].


== Intervals ==
== Optimal 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"
|+
|+
!
!
!''“fifth” 2 = 600¢''
!Minimum a
!“nonet” = 1351.429¢
!Maximum a
!“nonet” = 1460¢
!Minimum b
!''“fifth” 2 = 800¢''
!Maximum b
!Annapolis[6L 4s]
|-
|-
!1
!1
! colspan="4" |87.7444
! colspan="4" |87.7444
|Α#
|-
|-
!2
!2
! colspan="4" |175.48875
! colspan="4" |175.48875
|Β
|-
|-
!3
!3
! colspan="4" |263.2331
! colspan="4" |263.2331
|Γ
|-
|-
!4
!4
! colspan="4" |350.9775
! colspan="4" |350.9775
|Γ#
|-
|-
!5
!5
! colspan="4" |438.7219
! colspan="4" |438.7219
|Δ
|-
|-
!6
!6
! colspan="4" |526.46625
! colspan="4" |526.46625
|Δ#
|-
|-
!7
!7
! colspan="4" |614.2106
! colspan="4" |614.2106
|Ε
|-
|-
!'''8'''
!'''8'''
! colspan="4" |'''701.955'''
! colspan="4" |'''701.955'''
|-
|9
|''776.955''
|783.1392
|796.7106
|''801.955''
|-
|10
|''851.955''
|864.3234
|891.46625
|''901.955''
|-
|11
|''925.955''
|945.5076
|986.2219
|''1001.955''
|-
|12
|''1001.955''
|1026.692
|1080.9775
|''1101.955''
|-
|13
|''1076.955''
|1107.876
|1175.7331
|''1201.955''
|-
|14
|''1151.955''
|1189.0602
|1270.48875
|''1301.955''
|-
|15
|''1225.955''
|1270.2444
|1365.2444
|''1401.955''
|-
|16
|''1301.955''
|1351.4286
|1460
|''1501.955''
|}
== Common Tunings ==
{| class="wikitable"
|+
!
!Fifth 2 = 4\7
!step 2 = 88¢
!Fifth 2 = 8\13
!Annapolis[6L 4s]
|-
!1
! colspan="3" |87.7444
|Α#
|-
!2
! colspan="3" |175.48875
|Β
|-
!3
! colspan="3" |263.2331
|Γ
|-
!4
! colspan="3" |350.9775
|Γ#
|-
!5
! colspan="3" |438.7219
|Δ
|-
!6
! colspan="3" |526.46625
|Δ#
|-
!7
! colspan="3" |614.2106
|Ε
|-
!'''8'''
! colspan="3" |'''701.955'''
|'''Ϛ/Ϝ'''
|'''Ϛ/Ϝ'''
|-
|-
|9
|9
|797.955
|804.8121
|781.955
|787.6692
|787.6692
|789.955
|794.263
|Ϛ#/Ϝ#
|Ϛ#/Ϝ#
|-
|-
|10
|10
|873.3835
|893.955
|877.955
|907.6692
|886.57
|861.955
|873.3836
|Ζ
|Ζ
|-
|-
|11
|11
|989.955
|1010.5264
|941.955
|959.0979
|959.0979
|965.955
|978.878
|Η
|Η
|-
|-
|12
|12
|1085.955
|1113.3836
|1021.955
|1044.8121
|1044.8121
|1053.955
|1071.186
|Η#
|Η#
|-
|-
|13
|13
|1181.955
|1216.2407
|1101.955
|1130.5264
|1130.5264
|1141.955
|1163.4935
|Θ
|Θ
|-
|-
|14
|14
|1277.955
|1319.0979
|1181.955
|1216.2407
|1216.2407
|1229.955
|1255.801
|Θ#
|Θ#
|-
|-
|15
|15
|1373.955
|1421.955
|1261.955
|1301.955
|1301.955
|1317.955
|1348.109
|I (J)
|I (J)
|-
|-
|16
|16
|1387.6693
|1469.955
|1405.955
|1524.8121
|1440.4165
|1341.955
|'''A'''
|1387.6792
|-
|A
!
! colspan="3" |Fifths
|
|-
|1
|699.9249
|702.2016
|706.5183
|Α#-Ϛ#/Ϝ#
|-
|2
|697.8948
|702.46625
|711.0816
|Β-Z
|-
|3
|695.8647
|702.7219
|715.645
|Γ-H
|-
|4
|693.8346
|702.9775
|720.2083
|Γ#-H#
|-
|5
|691.80455
|703.2331
|724.7716
|Δ-Θ
|-
|6
|689.7745
|703.48875
|729.3349
|Δ#-Θ#
|-
|7
|687.7444
|703.7448
|733.8982
|Ε-I (J)
|-
|8
|685.7143
|704
|738.4615
|'''Ϛ/Ϝ-A'''
|-
|9
|687.7444
|703.7448
|733.8982
|Ϛ#/Ϝ#-A#
|-
|10
|689.7745
|703.48875
|729.3349
|Ζ-B
|-
|11
|691.80455
|703.2331
|724.7716
|Η-Γ
|-
|12
|693.8346
|702.9775
|720.2083
|Η#-Γ#
|-
|13
|695.8647
|702.7219
|715.645
|Θ-Δ
|-
|14
|697.8948
|702.46625
|711.0816
|Θ#-Δ#
|-
|15
|699.9249
|702.2016
|706.5183
|I (J)-Ε
|-
!16
! colspan="3" |'''701.955'''
|'''A-Ϛ/Ϝ'''
|}
|}

Revision as of 06:05, 30 May 2021

16WTN is 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 making its best tunings very nearly 88cET. Its best tunings are related to the octacot temperament.

Optimal 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]).

Minimum a Maximum a Minimum b Maximum b Annapolis[6L 4s]
1 87.7444 Α#
2 175.48875 Β
3 263.2331 Γ
4 350.9775 Γ#
5 438.7219 Δ
6 526.46625 Δ#
7 614.2106 Ε
8 701.955 Ϛ/Ϝ
9 797.955 804.8121 781.955 787.6692 Ϛ#/Ϝ#
10 893.955 907.6692 861.955 873.3836 Ζ
11 989.955 1010.5264 941.955 959.0979 Η
12 1085.955 1113.3836 1021.955 1044.8121 Η#
13 1181.955 1216.2407 1101.955 1130.5264 Θ
14 1277.955 1319.0979 1181.955 1216.2407 Θ#
15 1373.955 1421.955 1261.955 1301.955 I (J)
16 1469.955 1524.8121 1341.955 1387.6792 A
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