User:Moremajorthanmajor/22wtn: Difference between revisions

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A '''22 tone well tempered nonet''' (22WTN) is related to well temperaments of [[18edo]] and [[19edo]], but with a nonet rather than the 2/1 being constant. It alternates [[EDF|division of the just perfect fifth]] into 11 equal parts (11EDF) with sequences of eleven equal steps of 54.54545 to 72.7273 cents each. This will create eleven equally-spaced intervals between an interval between 600 and 800 cents and the just perfect fifth.
A '''22 tone well tempered nonet''' (22WTN) is related to well temperaments of [[18edo]] and [[19edo]], but with a nonet rather than the 2/1 being constant. Its patent tunings optimally alternate [[EDF|division of the just perfect fifth]] into 11 equal parts (11EDF) with sequences of eleven equal steps of 54.54545 to 72.7273 cents each. This will create eleven equally-spaced intervals between an interval between 600 and 800 cents and the just perfect fifth.


==Optimal Intervals==
==Patent Intervals==
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!
!Minimum a
!Target a
!Maximum a
!Target b
!Minimum b
!Alternating [[edo]] and *ed(9/8) a
!Maximum b
!Alternating edo and *ed(9/8) b
|-
|-
!1
|1
! colspan="4" |63.8141
! colspan="2" |63.8141
|66.667
|63.1579
|-
|-
!2
|2
! colspan="4" |127.6282
! colspan="2" |127.6282
|133.333
|126.3157
|-
|-
!3
|3
! colspan="4" |191.4423
! colspan="2" |191.4423
|200.000
|189.4737
|-
|-
!4
|4
! colspan="4" |255.2564
! colspan="2" |255.2564
|266.667
|252.6316
|-
|-
!5
|5
! colspan="4" |319.07045
! colspan="2" |319.07045
|333.333
|315.7895
|-
|-
!6
|6
! colspan="4" |382.88455
! colspan="2" |382.88455
|400.000
|378.9474
|-
|-
!7
|7
! colspan="4" |446.6986
! colspan="2" |446.6986
|466.667
|442.1053
|-
|-
!8
|8
! colspan="4" |510.5127
! colspan="2" |510.5127
|533.333
|505.2632
|-
|-
!9
|9
! colspan="4" |574.3268
! colspan="2" |574.3268
|600.000
|568.42105
|-
|-
!10
|10
! colspan="4" |638.1409
! colspan="2" |638.1409
|666.667
|631.57895
|-
|-
!11
|11
! colspan="4" |701.955
! colspan="2" |701.955
|733.333
|694.7368
|-
|-
|12
|12
|770.5264
|773.1043
|775.4244
|764.2106
|761.955
|800.000
|766.2407
|757.8947
|-
|-
|13
|13
|839.0979
|844.2536
|848.8938
|826.466
|821.955
|866.667
|830.5264
|821.0526
|-
|-
|14
|14
|907.6692
|915.4029
|921.3632
|888.7219
|881.955
|933.333
|894.8121
|884.2105
|-
|-
|15
|15
|976.2407
|986.5522
|995.83255
|950.9775
|941.955
|1000.000
|959.0979
|947.3684
|-
|-
|16
|16
|1044.8121
|1057.7014
|1069.3019
|1013.2331
|1001.955
|1066.667
|1023.3836
|1010.5263
|-
|-
|17
|17
|1113.3836
|1128.8507
|1142.7713
|1075.48875
|1061.955
|1133.333
|1087.6692
|1073.6843
|-
|-
|18
|18
|1181.955
|1200
|1216.2407
|1137.7444
|1121.955
|1200.000
|1151.955
|1136.8421
|-
|-
|19
|19
|1250.5264
|1271.1493
|1289.7101
|1200
|1181.955
|1250.9775
|1216.2407
|1200.000
|-
|-
|20
|20
|1319.0979
|1342.2986
|1363.1795
|1262.2556
|1241.955
|1301.955
|1280.5264
|1267.97
|-
|-
|21
|21
|1387.6692
|1413.4479
|1436.6489
|1324.511
|1301.955
|1352.9325
|1344.8121
|1333.94
|-
|-
|22
|22
|1456.2407
|1484.5971
|1510.1183
|1386.7669
|1361.955
! colspan="2" |1403.91
|1409.0907
|}
|}
[[Category:Well tempered nonet]]

Latest revision as of 09:31, 1 September 2025

A 22 tone well tempered nonet (22WTN) is related to well temperaments of 18edo and 19edo, but with a nonet rather than the 2/1 being constant. Its patent tunings optimally alternate division of the just perfect fifth into 11 equal parts (11EDF) with sequences of eleven equal steps of 54.54545 to 72.7273 cents each. This will create eleven equally-spaced intervals between an interval between 600 and 800 cents and the just perfect fifth.

Patent Intervals

Target a Target b Alternating edo and *ed(9/8) a Alternating edo and *ed(9/8) b
1 63.8141 66.667 63.1579
2 127.6282 133.333 126.3157
3 191.4423 200.000 189.4737
4 255.2564 266.667 252.6316
5 319.07045 333.333 315.7895
6 382.88455 400.000 378.9474
7 446.6986 466.667 442.1053
8 510.5127 533.333 505.2632
9 574.3268 600.000 568.42105
10 638.1409 666.667 631.57895
11 701.955 733.333 694.7368
12 773.1043 764.2106 800.000 757.8947
13 844.2536 826.466 866.667 821.0526
14 915.4029 888.7219 933.333 884.2105
15 986.5522 950.9775 1000.000 947.3684
16 1057.7014 1013.2331 1066.667 1010.5263
17 1128.8507 1075.48875 1133.333 1073.6843
18 1200 1137.7444 1200.000 1136.8421
19 1271.1493 1200 1250.9775 1200.000
20 1342.2986 1262.2556 1301.955 1267.97
21 1413.4479 1324.511 1352.9325 1333.94
22 1484.5971 1386.7669 1403.91
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