User:Moremajorthanmajor/8wtn: Difference between revisions

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8WTN is the [[EDF|equal division of the just perfect fifth]] into four parts of 175.48875 [[cent|cents]] each alternated with a sequence of four equal steps of another interval between 150 and 200 cents. Its best tunings are related to the [[Tetracot family|tetracot temperament]], which tempers out 20000/19683 in the 5-limit.
'''8WTN''' is optimally the [[EDF|equal division of the just perfect fifth]] into four parts of 175.48875 [[cent]]s each alternated with a sequence of four equal steps of another interval between 150 and 200 cents in its patent tunings. Its best tunings are related to the [[Tetracot family|tetracot temperament]], which tempers out 20000/19683 in the 5-limit.
 
== Patent Intervals<ref>a WT approach by J. Ruhf</ref> ==


== Intervals ==
{| class="wikitable"
{| class="wikitable"
!
|
!''“fifth” 2 = 600¢''
! Optimal<ref>[[4 EDF]] + 3ED(4/3)</ref>
!“nonet” = 1351.429¢
! Alternating EDO and *ed(9/8)<ref>[[7 EDO]] + 9/8</ref>
!step 2 = 10/9
!“nonet” = 1460¢
!''“fifth” 2 = 800¢''
|-
|-
!1
| 1
! colspan="5" |175.48875
| 175.48875
| 171.429
|-
|-
!2
| 2
! colspan="5" |350.9775
| 350.9775
| 342.857
|-
|-
!3
| 3
! colspan="5" |526.46625
| 526.46625
| 514.286
|-
|-
!4
| 4
! colspan="5" |701.955
| 701.955
| 685.714
|-
|-
|5
| 5
|''851.955''
| 867.97
|864.3234
| 857.143
|884.3587
|891.46625
|''901.955''
|-
|-
|6
| 6
|''1001.955''
| 1032.985
|1026.692
| 1028.571
|1066.7624
|1080.9775
|''1101.955''
|-
|-
|7
| 7
|''1151.955''
| colspan="2" style="text-align:center" | 1200
|1189.0602
|1249.1661
|1270.48875
|''1301.955''
|-
|-
|8
| 8
|''1301.955''
| 1366.015
|1351.429
| 1403.91
|1431.56985
|1460
|''1501.955''
|}
|}


== Common Tuning ==
<references/>
{| class="wikitable"
 
|1
[[Category:Well tempered nonet]]
|175.48875
 
|-
{{todo|unify precision}}
|2
|350.9775
|-
|3
|526.46625
|-
|4
|701.955
|-
|5
|873.3836
|-
|6
|1044.8121
|-
|7
|1216.2407
|-
|8
|1387.6693
|-
!
!Fifths
|-
|1
|697.8948
|-
|2
|693.8346
|-
|3
|689.7745
|-
|4
|685.7143
|-
|5
|689.7745
|-
|6
|693.8346
|-
|7
|697.8948
|-
|8
|701.955
|}

Latest revision as of 17:16, 31 March 2025

8WTN is optimally the equal division of the just perfect fifth into four parts of 175.48875 cents each alternated with a sequence of four equal steps of another interval between 150 and 200 cents in its patent tunings. Its best tunings are related to the tetracot temperament, which tempers out 20000/19683 in the 5-limit.

Patent Intervals[1]

Optimal[2] Alternating EDO and *ed(9/8)[3]
1 175.48875 171.429
2 350.9775 342.857
3 526.46625 514.286
4 701.955 685.714
5 867.97 857.143
6 1032.985 1028.571
7 1200
8 1366.015 1403.91
  1. ↑ a WT approach by J. Ruhf
  2. ↑ 4 EDF + 3ED(4/3)
  3. ↑ 7 EDO + 9/8
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