67edt: Difference between revisions

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{{Infobox ET}}
'''67EDT''' is the [[Edt|equal division of the third harmonic]] into 67 parts of 28.3874 [[cent|cents]] each, corresponding to 42.2723 [[edo]]. It is related to the regular temperament which tempers out 2100875/2097152 and |-36 45 -14 -1> in the 7-limit, which is supported by 296, 465, 761, and 1057 EDOs among others.
'''67EDT''' is the [[Edt|equal division of the third harmonic]] into 67 parts of 28.3874 [[cent|cents]] each, corresponding to 42.2723 [[edo]]. It is related to the regular temperament which tempers out 2100875/2097152 and |-36 45 -14 -1> in the 7-limit, which is supported by 296, 465, 761, and 1057 EDOs among others.


=Related regular temperaments=
==Related regular temperaments==
==296&465 temperament==
===296&465 temperament===
===7-limit===
====7-limit====
Commas: 2100875/2097152, |-36 45 -14 -1>
Commas: 2100875/2097152, |-36 45 -14 -1>


POTE generator: ~64/63 = 28.3825
POTE generator: ~64/63 = 28.3825


Map: [<1 0 -3 6|, <0 67 225 -135|]
Mapping: [<1 0 -3 6|, <0 67 225 -135|]


EDOs: 296, 465, 761, 1057
EDOs: {{EDOs|296, 465, 761, 1057}}


===11-limit===
====11-limit====
Commas: 46656/46585, 2100875/2097152, 21437500/21434787
Commas: 46656/46585, 2100875/2097152, 21437500/21434787


POTE generator: ~64/63 = 28.3824
POTE generator: ~64/63 = 28.3824


Map: [<1 0 -3 6 1|, <0 67 225 -135 104|]
Mapping: [<1 0 -3 6 1|, <0 67 225 -135 104|]


EDOs: 296, 465, 761, 1057
EDOs: 296, 465, 761, 1057


===13-limit===
====13-limit====
Commas: 1575/1573, 46656/46585, 199927/199650, 216513/216320
Commas: 1575/1573, 46656/46585, 199927/199650, 216513/216320


POTE generator: ~64/63 = 28.3825
POTE generator: ~64/63 = 28.3825


Map: [<1 0 -3 6 1 -2|, <0 67 225 -135 104 241|]
Mapping: [<1 0 -3 6 1 -2|, <0 67 225 -135 104 241|]


EDOs: 296, 465, 761, 1057
EDOs: 296, 465, 761, 1057


[[Category:Edt]]
== Intervals ==
[[Category:Edonoi]]
{{Interval table}}
 
==Harmonics==
{{Harmonics in equal
| steps = 67
| num = 3
| denom = 1
| intervals = integer
}}
{{Harmonics in equal
| steps = 67
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = integer
}}

Latest revision as of 19:23, 1 August 2025

← 66edt 67edt 68edt →
Prime factorization 67 (prime)
Step size 28.3874 ¢ 
Octave 42\67edt (1192.27 ¢)
Consistency limit 3
Distinct consistency limit 3

67EDT is the equal division of the third harmonic into 67 parts of 28.3874 cents each, corresponding to 42.2723 edo. It is related to the regular temperament which tempers out 2100875/2097152 and |-36 45 -14 -1> in the 7-limit, which is supported by 296, 465, 761, and 1057 EDOs among others.

Related regular temperaments

296&465 temperament

7-limit

Commas: 2100875/2097152, |-36 45 -14 -1>

POTE generator: ~64/63 = 28.3825

Mapping: [<1 0 -3 6|, <0 67 225 -135|]

EDOs: 296, 465, 761, 1057

11-limit

Commas: 46656/46585, 2100875/2097152, 21437500/21434787

POTE generator: ~64/63 = 28.3824

Mapping: [<1 0 -3 6 1|, <0 67 225 -135 104|]

EDOs: 296, 465, 761, 1057

13-limit

Commas: 1575/1573, 46656/46585, 199927/199650, 216513/216320

POTE generator: ~64/63 = 28.3825

Mapping: [<1 0 -3 6 1 -2|, <0 67 225 -135 104 241|]

EDOs: 296, 465, 761, 1057

Intervals

Steps Cents Hekts Approximate ratios
0 0 0 1/1
1 28.4 19.4
2 56.8 38.8 30/29, 31/30
3 85.2 58.2
4 113.5 77.6 31/29
5 141.9 97 13/12, 25/23
6 170.3 116.4 21/19
7 198.7 135.8 28/25
8 227.1 155.2 33/29
9 255.5 174.6 29/25, 36/31
10 283.9 194 33/28
11 312.3 213.4 6/5
12 340.6 232.8 28/23
13 369 252.2 21/17, 31/25
14 397.4 271.6 29/23, 34/27
15 425.8 291 23/18
16 454.2 310.4 13/10
17 482.6 329.9 33/25
18 511 349.3
19 539.4 368.7 15/11
20 567.7 388.1 25/18
21 596.1 407.5 31/22
22 624.5 426.9 33/23
23 652.9 446.3
24 681.3 465.7
25 709.7 485.1
26 738.1 504.5 23/15
27 766.5 523.9 14/9
28 794.8 543.3
29 823.2 562.7 29/18
30 851.6 582.1 18/11
31 880 601.5
32 908.4 620.9 22/13
33 936.8 640.3
34 965.2 659.7
35 993.6 679.1
36 1021.9 698.5
37 1050.3 717.9 11/6
38 1078.7 737.3 28/15
39 1107.1 756.7
40 1135.5 776.1 25/13, 27/14
41 1163.9 795.5
42 1192.3 814.9
43 1220.7 834.3
44 1249 853.7 35/17
45 1277.4 873.1 23/11
46 1305.8 892.5
47 1334.2 911.9
48 1362.6 931.3 11/5
49 1391 950.7 29/13
50 1419.4 970.1 25/11, 34/15
51 1447.8 989.6 30/13
52 1476.1 1009
53 1504.5 1028.4 31/13
54 1532.9 1047.8 17/7
55 1561.3 1067.2
56 1589.7 1086.6 5/2
57 1618.1 1106 28/11
58 1646.5 1125.4 31/12
59 1674.9 1144.8 29/11
60 1703.2 1164.2
61 1731.6 1183.6 19/7
62 1760 1203 36/13
63 1788.4 1222.4
64 1816.8 1241.8
65 1845.2 1261.2 29/10
66 1873.6 1280.6
67 1902 1300 3/1

Harmonics

Approximation of harmonics in 67edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -7.7 +0.0 +12.9 -4.3 -7.7 +9.3 +5.2 +0.0 -12.1 -6.8 +12.9
Relative (%) -27.2 +0.0 +45.5 -15.3 -27.2 +32.7 +18.3 +0.0 -42.6 -23.8 +45.5
Steps
(reduced) 		42
(42) 		67
(0) 		85
(18) 		98
(31) 		109
(42) 		119
(52) 		127
(60) 		134
(0) 		140
(6) 		146
(12) 		152
(18)
Approximation of harmonics in 67edt
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -12.1 +1.5 -4.3 -2.5 +6.1 -7.7 +12.2 +8.6 +9.3 +13.9 -6.3
Relative (%) -42.6 +5.4 -15.3 -8.9 +21.4 -27.2 +43.0 +30.2 +32.7 +49.0 -22.1
Steps
(reduced) 		156
(22) 		161
(27) 		165
(31) 		169
(35) 		173
(39) 		176
(42) 		180
(46) 		183
(49) 		186
(52) 		189
(55) 		191
(57)
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