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 in its patent tunings. 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> ==


== Optimal Intervals ==
{| class="wikitable"
{| class="wikitable"
!
|
!Target Minimum
! Optimal<ref>[[4 EDF]] + 3ED(4/3)</ref>
!Target Maximum
! Alternating EDO and *ed(9/8)<ref>[[7 EDO]] + 9/8</ref>
|-
|-
!1
| 1
! colspan="2" |175.48875
| 175.48875
| 171.429
|-
|-
!2
| 2
! colspan="2" |350.9775
| 350.9775
| 342.857
|-
|-
!3
| 3
! colspan="2" |526.46625
| 526.46625
| 514.286
|-
|-
!4
| 4
! colspan="2" |701.955
| 701.955
| 685.714
|-
|-
!5
| 5
! colspan="2" |867.97
| 867.97
| 857.143
|-
|-
!6
| 6
! colspan="2" |1032.985
| 1032.985
| 1028.571
|-
|-
!7
| 7
! colspan="2" |1200
| colspan="2" style="text-align:center" | 1200
|-
|-
|8
| 8
|1350
| 1366.015
|1400
| 1403.91
|}
|}
<references/>
[[Category:Well tempered nonet]]
{{todo|unify precision}}

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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