User:Moremajorthanmajor/32wtn: Difference between revisions

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A '''32 tone well tempered nonet''' (32WTN) is related to well temperaments of [[27edo]] and [[28edo]], but with a nonet rather than the 2/1 being constant. Its patent tunings optimally alternate [[EDF|division of the just perfect fifth]] into 16 equal parts (16EDF) with sequences of sixteen equal steps of 37.5 to 50 cents each. This will create sixteen equally-spaced intervals between an interval between 600 and 800 cents and the just perfect fifth.
A '''32 tone well tempered nonet''' (32WTN) is related to well temperaments of [[27edo]] and [[28edo]], but with a nonet rather than the 2/1 being constant. Its patent tunings optimally alternate [[EDF|division of the just perfect fifth]] into 16 equal parts (16EDF) with sequences of sixteen equal steps of 37.5 to 50 cents each. This will create sixteen equally-spaced intervals between an interval between 600 and 800 cents and the just perfect fifth.


== Optimal Intervals ==
== Patent Intervals ==
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 7: Line 7:
!Target a
!Target a
!Target b
!Target b
!Alternating [[edo]] and *ed(9/8) a
!Alternating edo and *ed(9/8) b
|-
|-
!1
|1
! colspan="2" |43.8722
! colspan="2" |43.8722
|44.4444
|42.8571
|-
|-
!2
|2
! colspan="2" |87.7444
! colspan="2" |87.7444
|88.8889
|85.7129
|-
|-
!3
|3
! colspan="2" |131.6166
! colspan="2" |131.6166
|133.3333
|128.5714
|-
|-
!4
|4
! colspan="2" |175.48875
! colspan="2" |175.48875
|177.7778
|171.4286
|-
|-
!5
|5
! colspan="2" |219.3609
! colspan="2" |219.3609
|222.2222
|214.2857
|-
|-
!6
|6
! colspan="2" |263.2331
! colspan="2" |263.2331
|266.6667
|257.1429
|-
|-
!7
|7
! colspan="2" |307.1053
! colspan="2" |307.1053
|311.1112
|300.00
|-
|-
!8
|8
! colspan="2" |350.9775
! colspan="2" |350.9775
|355.5556
|342.8571
|-
|-
!9
|9
! colspan="2" |394.8497
! colspan="2" |394.8497
|400.00
|385.7129
|-
|-
!10
|10
! colspan="2" |438.7219
! colspan="2" |438.7219
|444.4444
|428.5714
|-
|-
!11
|11
! colspan="2" |482.5941
! colspan="2" |482.5941
|488.8889
|471.4286
|-
|-
!12
|12
! colspan="2" |526.46625
! colspan="2" |526.46625
|533.3333
|514.2857
|-
|-
!13
|13
! colspan="2" |570.3384
! colspan="2" |570.3384
|577.7778
|557.1429
|-
|-
!14
|14
! colspan="2" |614.2106
! colspan="2" |614.2106
|622.2222
|600.00
|-
|-
!15
|15
! colspan="2" |658.0828
! colspan="2" |658.0828
|666.6667
|642.8571
|-
|-
!16
|16
! colspan="2" |701.9550
! colspan="2" |701.9550
|711.1111
|685.7129
|-
|-
|17
|17
|747.232
|747.232
|743.459
|743.459
|755.5556
|728.5714
|-
|-
|18
|18
|792.509
|792.509
|784.9625
|784.9625
|800.00
|771.4286
|-
|-
|19
|19
|837.7855
|837.7855
|826.466
|826.466
|844.4444
|814.2857
|-
|-
|20
|20
|883.062
|883.062
|867.97
|867.97
|888.8889
|857.1429
|-
|-
|21
|21
|928.339
|928.339
|909.474
|909.474
|933.3333
|900.00
|-
|-
|22
|22
|973.616
|973.616
|950.9775
|950.9775
|977.7778
|942.8571
|-
|-
|23
|23
|1018.938
|1018.938
|993.481
|993.481
|1022.2222
|985.7129
|-
|-
|24
|24
|1064.1695
|1064.1695
|1033.985
|1033.985
|1066.6667
|1028.5714
|-
|-
|25
|25
|1109.446
|1109.446
|1075.489
|1075.489
|1111.1111
|1071.4286
|-
|-
|26
|26
|1154.723
|1154.723
|1116.9925
|1116.9925
|1155.5556
|1114.2857
|-
|-
|27
|27
|1200
|1200
|1158.496
|1158.496
|1200
|1157.1429
|-
|-
|28
|28
|1245.277
|1245.277
|1200
|1240.782
|1200
|1200
|-
|-
Line 107: Line 165:
|1290.554
|1290.554
|1241.504
|1241.504
|1281.564
|1250.9775
|-
|-
|30
|30
|1335.8305
|1335.8305
|1283.0075
|1283.0075
|1322.346
|1301.955
|-
|-
|31
|31
|1381.107
|1381.107
|1324.511
|1324.511
|1363.128
|1352.9325
|-
|-
|32
|32
|1426.384
|1426.384
|1366.015
|1366.015
! colspan="2" |1403.91
|}
|}


[[Category:Well tempered nonet]]
[[Category:Well tempered nonet]]
{{todo|clarify|unify precision|inline=1}}

Latest revision as of 09:31, 1 September 2025

A 32 tone well tempered nonet (32WTN) is related to well temperaments of 27edo and 28edo, but with a nonet rather than the 2/1 being constant. Its patent tunings optimally alternate division of the just perfect fifth into 16 equal parts (16EDF) with sequences of sixteen equal steps of 37.5 to 50 cents each. This will create sixteen 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 43.8722 44.4444 42.8571
2 87.7444 88.8889 85.7129
3 131.6166 133.3333 128.5714
4 175.48875 177.7778 171.4286
5 219.3609 222.2222 214.2857
6 263.2331 266.6667 257.1429
7 307.1053 311.1112 300.00
8 350.9775 355.5556 342.8571
9 394.8497 400.00 385.7129
10 438.7219 444.4444 428.5714
11 482.5941 488.8889 471.4286
12 526.46625 533.3333 514.2857
13 570.3384 577.7778 557.1429
14 614.2106 622.2222 600.00
15 658.0828 666.6667 642.8571
16 701.9550 711.1111 685.7129
17 747.232 743.459 755.5556 728.5714
18 792.509 784.9625 800.00 771.4286
19 837.7855 826.466 844.4444 814.2857
20 883.062 867.97 888.8889 857.1429
21 928.339 909.474 933.3333 900.00
22 973.616 950.9775 977.7778 942.8571
23 1018.938 993.481 1022.2222 985.7129
24 1064.1695 1033.985 1066.6667 1028.5714
25 1109.446 1075.489 1111.1111 1071.4286
26 1154.723 1116.9925 1155.5556 1114.2857
27 1200 1158.496 1200 1157.1429
28 1245.277 1200 1240.782 1200
29 1290.554 1241.504 1281.564 1250.9775
30 1335.8305 1283.0075 1322.346 1301.955
31 1381.107 1324.511 1363.128 1352.9325
32 1426.384 1366.015 1403.91
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