User:Moremajorthanmajor/6wtn: Difference between revisions

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'''6WTN''', if the attempt is made to use it as an actual scale, would divide the [[just perfect fifth]] into three equal parts, each of size 233.985 cents, which is to say (3/2)^(1/3) as a frequency ratio, and three equal steps of another interval between 200 and 266.667 cents. This will create three equally-spaced intervals between an interval between 600 and 800 cents and the just perfect fifth. If we want to consider it to be a temperament, it tempers out [[16/15]], [[21/20]], [[28/27]], [[81/80]], and [[256/243]] as well as [[5edo]].
'''6WTN''', if the attempt is made to use it as an actual scale, would optimally divide the [[just perfect fifth]] into three equal parts, each of size 233.985 cents, which is to say (3/2)^(1/3) as a frequency ratio, alternating these with three equal steps of another interval between 200 and 266.667 cents in its patent tunings. This will create three equally-spaced intervals between an interval between 600 and 800 cents and the just perfect fifth. If we want to consider it to be a temperament, it tempers out [[16/15]], [[21/20]], [[28/27]], [[81/80]], and [[256/243]] as well as [[5edo]].


==Factoids about 6WTN==
==Patent Intervals==
The best tunings of 6WTN are related to the [[Gamelismic clan|gamelismic temperaments]], which temper out 1029/1024 in the 7-limit.
==Intervals==
{| class="wikitable"
{| class="wikitable"
!
!
!''“fifth” 2 = 600¢''
!Optimal
!step 2 = 9/8
!Alternating [[EDO|edo]] and *ed(9/8)
!“nonet” = 1351.429¢
!step 2 = 8/7
!“nonet” = 1460¢
!''“fifth” 2 = 800¢''
|-
|-
|1
| colspan="6" |233.985
|-
|2
| colspan="6" |467.97
|-
|3
| colspan="6" |701.955
|-
|4
|''901.955''
|905.865
|918.446
|933.129
|954.637
|''968.662''
|-
|5
|''1101.955''
|1109.775
|1134.937
|1164.303
|1207.318
|''1235.328''
|-
|6
|''1301.955''
|1313.685
|1351.429
|1395.477
|1460
|''1501.955''
|}
== Common Tuning ==
{| class="wikitable"
|1
|1
|233.985
|233.985
|240
|-
|-
|2
|2
|467.97
|467.97
|480
|-
|-
|3
|3
|701.955
|701.955
|-
|4
|941.955
|-
|5
|1181.955
|-
|6
|1421.955
|-
!
!Fifths
|-
|1
|707.97
|-
|2
|713.985
|-
|3
|720
|720
|-
|-
|4
|4
|713.985
|950.9775
|960
|-
|-
|5
!5
|707.97
! colspan="2" |1200
|-
|-
|6
|6
|701.955
|1449.0225
|1403.91
|}
|}
[[Category:Well tempered nonet]]

Latest revision as of 09:31, 1 September 2025

6WTN, if the attempt is made to use it as an actual scale, would optimally divide the just perfect fifth into three equal parts, each of size 233.985 cents, which is to say (3/2)^(1/3) as a frequency ratio, alternating these with three equal steps of another interval between 200 and 266.667 cents in its patent tunings. This will create three equally-spaced intervals between an interval between 600 and 800 cents and the just perfect fifth. If we want to consider it to be a temperament, it tempers out 16/15, 21/20, 28/27, 81/80, and 256/243 as well as 5edo.

Patent Intervals

Optimal Alternating edo and *ed(9/8)
1 233.985 240
2 467.97 480
3 701.955 720
4 950.9775 960
5 1200
6 1449.0225 1403.91
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