32edt: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''32EDT''' is the [[Edt|equal division of the third harmonic]] into 32 parts of 59.4361 [[cent|cents]] each, corresponding to 20.1898 [[edo]]. It has a distinct sharp tendency, in the sense that if 3 is pure, 5, 7, 11, 13, 17, and 19 are all sharp. It tempers out 3125/3087 and 885735/823543 in the 7-limit; 891/875, 1331/1323, and 2475/2401 in the 11-limit; 275/273, 351/343, 729/715, and 847/845 in the 13-limit; 121/119, 189/187, 225/221, 459/455, and 845/833 in the 17-limit; 135/133, 171/169, 247/245, 325/323, and 363/361 in the 19-limit (no-twos subgroup). It is the eighth [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|zeta peak tritave division]].
'''32EDT''' is the [[Edt|equal division of the third harmonic]] into 32 parts of 59.4361 [[cent]]s each, corresponding to 20.1898 [[edo]]. It has a distinct sharp tendency, in the sense that if 3 is pure, 5, 7, 11, 13, 17, and 19 are all sharp. It tempers out 3125/3087 and 885735/823543 in the 7-limit; 891/875, 1331/1323, and 2475/2401 in the 11-limit; 275/273, 351/343, 729/715, and 847/845 in the 13-limit; 121/119, 189/187, 225/221, 459/455, and 845/833 in the 17-limit; 135/133, 171/169, 247/245, 325/323, and 363/361 in the 19-limit (no-twos subgroup). It is the eighth [[the Riemann zeta function and tuning#Removing primes|zeta peak tritave division]].


==Harmonics==
== Harmonics ==
{{Harmonics in equal
{{Harmonics in equal
| steps = 19
| steps = 32
| num = 3
| num = 3
| denom = 1
| denom = 1
| columns = 9
| intervals = prime
| intervals = prime
}}
}}
{{Harmonics in equal
{{Harmonics in equal
| steps = 19
| steps = 32
| num = 3
| num = 3
| denom = 1
| denom = 1
| start = 12
| start = 12
| collapsed = 1
| collapsed = 1
| intervals = prime
| intervals = odd
}}
}}


== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! step
! Step
! cents
! [[Cent]]s
!hekts
! [[Hekt]]s
|-
|-
| | 1
| 1
| | 59.436
| 59.436
|40.625
| 40.625
|-
|-
| | 2
| 2
| | 118.872
| 118.872
|81.25
| 81.25
|-
|-
| | 3
| 3
| | 178.308
| 178.308
|121.875
| 121.875
|-
|-
| | 4
| 4
| | 237.744
| 237.744
|162.5
| 162.5
|-
|-
| | 5
| 5
| | 297.180
| 297.180
|203.125
| 203.125
|-
|-
| | 6
| 6
| | 356.617
| 356.617
|243.75
| 243.75
|-
|-
| | 7
| 7
| | 416.053
| 416.053
|284.375
| 284.375
|-
|-
| | 8
| 8
| | 475.489
| 475.489
|325
| 325
|-
|-
| | 9
| 9
| | 534.925
| 534.925
|365.625
| 365.625
|-
|-
| | 10
| 10
| | 594.361
| 594.361
|406.25
| 406.25
|-
|-
| | 11
| 11
| | 653.797
| 653.797
|446.875
| 446.875
|-
|-
| | 12
| 12
| | 713.233
| 713.233
|487.5
| 487.5
|-
|-
| | 13
| 13
| | 772.669
| 772.669
|528.125
| 528.125
|-
|-
| | 14
| 14
| | 832.105
| 832.105
|568.75
| 568.75
|-
|-
| | 15
| 15
| | 891.541
| 891.541
|609.375
| 609.375
|-
|-
| | 16
| 16
| | 950.978
| 950.978
|650
| 650
|-
|-
| | 17
| 17
| | 1010.414
| 1010.414
|690.625
| 690.625
|-
|-
| | 18
| 18
| | 1069.85
| 1069.85
|731.25
| 731.25
|-
|-
| | 19
| 19
| | 1129.286
| 1129.286
|774.875
| 774.875
|-
|-
| | 20
| 20
| | 1188.722
| 1188.722
|812.5
| 812.5
|-
|-
| | 21
| 21
| | 1248.158
| 1248.158
|853.125.
| 853.125.
|-
|-
| | 22
| 22
| | 1307.594
| 1307.594
|893.75
| 893.75
|-
|-
| | 23
| 23
| | 1367.03
| 1367.03
|934.375
| 934.375
|-
|-
| | 24
| 24
| | 1426.466
| 1426.466
|975
| 975
|-
|-
| | 25
| 25
| | 1485.902
| 1485.902
|1015.625
| 1015.625
|-
|-
| | 26
| 26
| | 1545.338
| 1545.338
|1056.25
| 1056.25
|-
|-
| | 27
| 27
| | 1604.775
| 1604.775
|1096.875
| 1096.875
|-
|-
| | 28
| 28
| | 1664.211
| 1664.211
|1137.5
| 1137.5
|-
|-
| | 29
| 29
| | 1723.647
| 1723.647
|1178.125
| 1178.125
|-
|-
| | 30
| 30
| | 1783.083
| 1783.083
|1218.75
| 1218.75
|-
|-
| | 31
| 31
| | 1842.519
| 1842.519
|1259.375
| 1259.375
|-
|-
| | 32
| 32
| | 1901.955
| 1901.955
|1300
| 1300
|}
|}


[[Category:Edt]]
{{todo|expand}}
[[Category:Edonoi]]

Latest revision as of 19:21, 1 August 2025

← 31edt 32edt 33edt →
Prime factorization 25
Step size 59.4361 ¢ 
Octave 20\32edt (1188.72 ¢) (→ 5\8edt)
Consistency limit 4
Distinct consistency limit 4

32EDT is the equal division of the third harmonic into 32 parts of 59.4361 cents each, corresponding to 20.1898 edo. It has a distinct sharp tendency, in the sense that if 3 is pure, 5, 7, 11, 13, 17, and 19 are all sharp. It tempers out 3125/3087 and 885735/823543 in the 7-limit; 891/875, 1331/1323, and 2475/2401 in the 11-limit; 275/273, 351/343, 729/715, and 847/845 in the 13-limit; 121/119, 189/187, 225/221, 459/455, and 845/833 in the 17-limit; 135/133, 171/169, 247/245, 325/323, and 363/361 in the 19-limit (no-twos subgroup). It is the eighth zeta peak tritave division.

Harmonics

Approximation of prime harmonics in 32edt
Harmonic 2 3 5 7 11 13 17 19 23
Error Absolute (¢) -11.3 +0.0 +7.2 +19.0 +9.2 +17.2 +28.2 +14.0 -19.6
Relative (%) -19.0 +0.0 +12.1 +32.0 +15.5 +28.9 +47.5 +23.5 -33.0
Steps
(reduced) 		20
(20) 		32
(0) 		47
(15) 		57
(25) 		70
(6) 		75
(11) 		83
(19) 		86
(22) 		91
(27)
Approximation of odd harmonics in 32edt
Harmonic 25 27 29 31 33 35 37 39 41 43 45
Error Absolute (¢) +14.4 +0.0 -4.8 -1.4 +9.2 +26.2 -10.6 +17.2 -10.0 +26.5 +7.2
Relative (%) +24.2 +0.0 -8.1 -2.4 +15.5 +44.1 -17.8 +28.9 -16.8 +44.5 +12.1
Steps
(reduced) 		94
(30) 		96
(0) 		98
(2) 		100
(4) 		102
(6) 		104
(8) 		105
(9) 		107
(11) 		108
(12) 		110
(14) 		111
(15)

Intervals

Step Cents Hekts
1 59.436 40.625
2 118.872 81.25
3 178.308 121.875
4 237.744 162.5
5 297.180 203.125
6 356.617 243.75
7 416.053 284.375
8 475.489 325
9 534.925 365.625
10 594.361 406.25
11 653.797 446.875
12 713.233 487.5
13 772.669 528.125
14 832.105 568.75
15 891.541 609.375
16 950.978 650
17 1010.414 690.625
18 1069.85 731.25
19 1129.286 774.875
20 1188.722 812.5
21 1248.158 853.125.
22 1307.594 893.75
23 1367.03 934.375
24 1426.466 975
25 1485.902 1015.625
26 1545.338 1056.25
27 1604.775 1096.875
28 1664.211 1137.5
29 1723.647 1178.125
30 1783.083 1218.75
31 1842.519 1259.375
32 1901.955 1300
Retrieved from "https://en.xen.wiki/index.php?title=32edt&oldid=205995"