65edo: Difference between revisions

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{| class="wikitable"
{| class="wikitable"
|-
|-
! | [[Degree|Degree]]
! rowspan="2"| [[Degree|Degree]]
! | Size ([[cent|Cents]])
! colspan="3"| Size
|-
|-
| style="text-align:center;" | 0
![[cent|Cents]]
| style="text-align:right;" | 0.0000
!pions
!7mus
|-
| colspan="4" style="text-align:center;" | 0
|-
|-
| style="text-align:center;" | 1
| style="text-align:center;" | 1
| style="text-align:right;" | 18.4615
| style="text-align:right;" | 18.4615
|19.5692
|23.6308 (17.A17A<sub>16</sub>)
|-
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 2
| style="text-align:right;" | 36.9231
| style="text-align:right;" | 36.9231
|39.1385
|47.2615 (2F.42F4<sub>16</sub>)
|-
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 3
| style="text-align:right;" | 55.3846
| style="text-align:right;" | 55.3846
|58.7077
|70.8923 (46.E46E<sub>16</sub>)
|-
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 4
| style="text-align:right;" | 73.8462
| style="text-align:right;" | 73.84615
|78.2769
|94.5231 (5E.85E8<sub>16</sub>)
|-
|-
| style="text-align:center;" | 5
| style="text-align:center;" | 5
| style="text-align:right;" | 92.3077
| style="text-align:right;" | 92.3077
|97.84615
|118.15385 (76.2762<sub>16</sub>)
|-
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 6
| style="text-align:right;" | 110.7692
| style="text-align:right;" | 110.7692
|117.4154
|141.7846 (8D.C8DD<sub>16</sub>)
|-
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 7
| style="text-align:right;" | 129.2308
| style="text-align:right;" | 129.2308
|136.9846
|165.4154 (A5.6A57<sub>16</sub>)
|-
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 8
| style="text-align:right;" | 147.6923
| style="text-align:right;" | 147.6923
|156.55385
|189.04615 (BD.0BD1<sub>16</sub>)
|-
|-
| style="text-align:center;" | 9
| style="text-align:center;" | 9
| style="text-align:right;" | 166.1538
| style="text-align:right;" | 166.15385
|176.1231
|212.6769 (D4.AD4B<sub>16</sub>
|-
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 10
| style="text-align:right;" | 184.6154
| style="text-align:right;" | 184.6154
|195.6923
|236.3077 (EC.4EC5<sub>16</sub>)
|-
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 11
| style="text-align:right;" | 203.0769
| style="text-align:right;" | 203.0769
|215.2615
|259.9385 (103.F03F<sub>16</sub>)
|-
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 12
| style="text-align:right;" | 221.5385
| style="text-align:right;" | 221.5385
|234.8308
|283.5692 (11B.91B9<sub>16</sub>)
|-
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 13
| style="text-align:right;" | 240.0000
| style="text-align:right;" | 240
|254.4
|307.2 (133.3333<sub>16</sub>)
|-
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 14
| style="text-align:right;" | 258.4615
| style="text-align:right;" | 258.4615
|273.9692
|330.8308 (14A.D4AD<sub>16</sub>)
|-
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 15
| style="text-align:right;" | 276.9231
| style="text-align:right;" | 276.9231
|293.5385
|354.4615 (162.7627<sub>16</sub>)
|-
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 16
| style="text-align:right;" | 295.3846
| style="text-align:right;" | 295.3846
|313.1077
|378.0923 (17A.17A18<sub>16</sub>)
|-
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 17
| style="text-align:right;" | 313.8462
| style="text-align:right;" | 313.84615
|332.6769
|401.7231 (191.B91C<sub>16</sub>)
|-
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 18
| style="text-align:right;" | 332.3077
| style="text-align:right;" | 332.3077
|352.24615
|425.35385 (1A9.5A96<sub>16</sub>)
|-
|-
| style="text-align:center;" | 19
| style="text-align:center;" | 19
| style="text-align:right;" | 350.7692
| style="text-align:right;" | 350.7692
|371.8154
|448.9846 (1C0.FC1<sub>16</sub>)
|-
|-
| style="text-align:center;" | 20
| style="text-align:center;" | 20
| style="text-align:right;" | 369.2308
| style="text-align:right;" | 369.2308
|391.3846
|472.6154 (1D8.9D8A<sub>16</sub>)
|-
|-
| style="text-align:center;" | 21
| style="text-align:center;" | 21
| style="text-align:right;" | 387.6923
| style="text-align:right;" | 387.6923
|410.95385
|496.24615 (1F0.3F04<sub>16</sub>)
|-
|-
| style="text-align:center;" | 22
| style="text-align:center;" | 22
| style="text-align:right;" | 406.1538
| style="text-align:right;" | 406.15385
|430.5231
|519.8769 (207.E07E<sub>16</sub>)
|-
|-
| style="text-align:center;" | 23
| style="text-align:center;" | 23
| style="text-align:right;" | 424.6154
| style="text-align:right;" | 424.6154
|450.0923
|543.5077 (21F.81F8<sub>16</sub>)
|-
|-
| style="text-align:center;" | 24
| style="text-align:center;" | 24
| style="text-align:right;" | 443.0769
| style="text-align:right;" | 443.0769
|469.6615
|567.1385 (237.2372<sub>16</sub>)
|-
|-
| style="text-align:center;" | 25
| style="text-align:center;" | 25
| style="text-align:right;" | 461.5385
| style="text-align:right;" | 461.5385
|489.2308
|590.7692 (24E.C4ED<sub>16</sub>)
|-
|-
| style="text-align:center;" | 26
| style="text-align:center;" | 26
| style="text-align:right;" | 480.0000
| style="text-align:right;" | 480
|508.8
|614.4 (266.6666<sub>16</sub>)
|-
|-
| style="text-align:center;" | 27
| style="text-align:center;" | 27
| style="text-align:right;" | 498.4615
| style="text-align:right;" | 498.4615
|528.3692
|638.0308 (27E07E08<sub>16</sub>)
|-
|-
| style="text-align:center;" | 28
| style="text-align:center;" | 28
| style="text-align:right;" | 516.9231
| style="text-align:right;" | 516.9231
|547.9385
|661.6615 (295.A95B<sub>16</sub>)
|-
|-
| style="text-align:center;" | 29
| style="text-align:center;" | 29
| style="text-align:right;" | 535.3846
| style="text-align:right;" | 535.3846
|567.5077
|685.2923 (2AD.4AD4<sub>16</sub>)
|-
|-
| style="text-align:center;" | 30
| style="text-align:center;" | 30
| style="text-align:right;" | 553.8462
| style="text-align:right;" | 553.84615
|587.0769
|708.9231 (2C4.EC4F<sub>16</sub>)
|-
|-
| style="text-align:center;" | 31
| style="text-align:center;" | 31
| style="text-align:right;" | 572.3077
| style="text-align:right;" | 572.3077
|606.64615
|732.55385 (2DC.8DC9<sub>16</sub>)
|-
|-
| style="text-align:center;" | 32
| style="text-align:center;" | 32
| style="text-align:right;" | 590.7692
| style="text-align:right;" | 590.7692
|626.2154
|756.1846 (2F4.2F42<sub>16</sub>)
|-
|-
| style="text-align:center;" | 33
| style="text-align:center;" | 33
| style="text-align:right;" | 609.2308
| style="text-align:right;" | 609.2308
|645.7846
|779.8154 (30B.D0BD<sub>16</sub>)
|-
|-
| style="text-align:center;" | 34
| style="text-align:center;" | 34
| style="text-align:right;" | 627.6923
| style="text-align:right;" | 627.6923
|665.35385
|803.44615 (323.7237<sub>16</sub>)
|-
|-
| style="text-align:center;" | 35
| style="text-align:center;" | 35
| style="text-align:right;" | 646.1538
| style="text-align:right;" | 646.1538
|684.9231
|827.0769 (33B.13B1<sub>16</sub>)
|-
|-
| style="text-align:center;" | 36
| style="text-align:center;" | 36
| style="text-align:right;" | 664.6154
| style="text-align:right;" | 664.6154
|704.4923
|850.7077 (352.B52B<sub>16</sub>)
|-
|-
| style="text-align:center;" | 37
| style="text-align:center;" | 37
| style="text-align:right;" | 683.0769
| style="text-align:right;" | 683.0769
|724.0615
|874.3385 (36A.56A5<sub>16</sub>)
|-
|-
| style="text-align:center;" | 38
| style="text-align:center;" | 38
| style="text-align:right;" | 701.5385
| style="text-align:right;" | 701.5385
|743.6308
|897.9692 (381.F81F8<sub>16</sub>)
|-
|-
| style="text-align:center;" | 39
| style="text-align:center;" | 39
| style="text-align:right;" | 720.0000
| style="text-align:right;" | 720
|763.2
|921.6 (399.999A<sub>16</sub>)
|-
|-
| style="text-align:center;" | 40
| style="text-align:center;" | 40
| style="text-align:right;" | 738.4615
| style="text-align:right;" | 738.4615
|782.7692
|945.2308 (3B1.3B13<sub>16</sub>)
|-
|-
| style="text-align:center;" | 41
| style="text-align:center;" | 41
| style="text-align:right;" | 756.9231
| style="text-align:right;" | 756.9231
|802.3385
|968.8615 (3C8.DC8E<sub>16</sub>)
|-
|-
| style="text-align:center;" | 42
| style="text-align:center;" | 42
| style="text-align:right;" | 775.3846
| style="text-align:right;" | 775.3846
|821.9077
|992.4923 (3E0.7E08<sub>16</sub>)
|-
|-
| style="text-align:center;" | 43
| style="text-align:center;" | 43
| style="text-align:right;" | 793.8462
| style="text-align:right;" | 793.84615
|841.4769
|1016.1231 (3F8.1F82<sub>16</sub>)
|-
|-
| style="text-align:center;" | 44
| style="text-align:center;" | 44
| style="text-align:right;" | 812.3077
| style="text-align:right;" | 812.3077
|861.04615
|1039.75385 (40F.C0FC<sub>16</sub>)
|-
|-
| style="text-align:center;" | 45
| style="text-align:center;" | 45
| style="text-align:right;" | 830.7692
| style="text-align:right;" | 830.7692
|880.6154
|1063.3846 (427.6276<sub>16</sub>)
|-
|-
| style="text-align:center;" | 46
| style="text-align:center;" | 46
| style="text-align:right;" | 849.2308
| style="text-align:right;" | 849.2308
|900.1846
|1087.0154 (43F.03F<sub>16</sub>)
|-
|-
| style="text-align:center;" | 47
| style="text-align:center;" | 47
| style="text-align:right;" | 867.6923
| style="text-align:right;" | 867.6923
|919.75385
|1110.64615 (456.A56A<sub>16</sub>)
|-
|-
| style="text-align:center;" | 48
| style="text-align:center;" | 48
| style="text-align:right;" | 886.1538
| style="text-align:right;" | 886.15385
|939.3231
|1134.2769 (46E.46E4<sub>16</sub>)
|-
|-
| style="text-align:center;" | 49
| style="text-align:center;" | 49
| style="text-align:right;" | 904.6154
| style="text-align:right;" | 904.6154
|958.8923
|1157.9077 (485.E85E8<sub>16</sub>)
|-
|-
| style="text-align:center;" | 50
| style="text-align:center;" | 50
| style="text-align:right;" | 923.0769
| style="text-align:right;" | 923.0769
|978.4615
|1181.53845 (49D.89D9<sub>16</sub>)
|-
|-
| style="text-align:center;" | 51
| style="text-align:center;" | 51
| style="text-align:right;" | 941.5385
| style="text-align:right;" | 941.5385
|998.0308
|1205.1692 (4B5.2B53<sub>16</sub>)
|-
|-
| style="text-align:center;" | 52
| style="text-align:center;" | 52
| style="text-align:right;" | 960.0000
| style="text-align:right;" | 960
|1017.6
|1228.8 (4CC.CCCD<sub>16</sub>)
|-
|-
| style="text-align:center;" | 53
| style="text-align:center;" | 53
| style="text-align:right;" | 978.4615
| style="text-align:right;" | 978.4615
|1037.1692
|1252.4308 (4E4.6E47<sub>16</sub>)
|-
|-
| style="text-align:center;" | 54
| style="text-align:center;" | 54
| style="text-align:right;" | 996.9231
| style="text-align:right;" | 996.9231
|1056.7385
|1276.0615 (4FC.0FC1<sub>16</sub>)
|-
|-
| style="text-align:center;" | 55
| style="text-align:center;" | 55
| style="text-align:right;" | 1015.3846
| style="text-align:right;" | 1015.3846
|1076.3077
|1299.6923 (513.B13B<sub>16</sub>)
|-
|-
| style="text-align:center;" | 56
| style="text-align:center;" | 56
| style="text-align:right;" | 1033.8462
| style="text-align:right;" | 1033.84615
|1095.8769
|1323.3231 (52B.52B5<sub>16</sub>)
|-
|-
| style="text-align:center;" | 57
| style="text-align:center;" | 57
| style="text-align:right;" | 1052.3077
| style="text-align:right;" | 1052.3077
|1115.44615
|1346.95385 (542,F42F<sub>16</sub>)
|-
|-
| style="text-align:center;" | 58
| style="text-align:center;" | 58
| style="text-align:right;" | 1070.7692
| style="text-align:right;" | 1070.7692
|1135.0154
|1370.5846 (55A.95A9<sub>16</sub>)
|-
|-
| style="text-align:center;" | 59
| style="text-align:center;" | 59
| style="text-align:right;" | 1089.2308
| style="text-align:right;" | 1089.2308
|1154.5846
|1394.2154 (572.3723<sub>16</sub>)
|-
|-
| style="text-align:center;" | 60
| style="text-align:center;" | 60
| style="text-align:right;" | 1107.6923
| style="text-align:right;" | 1107.6923
|1174.1535
|1417.84615 (589.D89E<sub>16</sub>)
|-
|-
| style="text-align:center;" | 61
| style="text-align:center;" | 61
| style="text-align:right;" | 1126.1538
| style="text-align:right;" | 1126.15385
|1193.7231
|1441.4769 (5A1.7A18<sub>16</sub>)
|-
|-
| style="text-align:center;" | 62
| style="text-align:center;" | 62
| style="text-align:right;" | 1144.6154
| style="text-align:right;" | 1144.6154
|1213.2923
|1465.1077 (5B9.1B92<sub>16</sub>)
|-
|-
| style="text-align:center;" | 63
| style="text-align:center;" | 63
| style="text-align:right;" | 1163.0769
| style="text-align:right;" | 1163.0769
|1232.8615
|1488.7385 (5D0.BD0C<sub>16</sub>)
|-
|-
| style="text-align:center;" | 64
| style="text-align:center;" | 64
| style="text-align:right;" | 1181.5385
| style="text-align:right;" | 1181.5385
|1252.4308
|1512.3692 (5E8.5E86<sub>16</sub>)
|-
|-
| style="text-align:center;" | 65
| style="text-align:center;" | 65
| style="text-align:right;" | 1200.0000
| style="text-align:right;" | 1200
|1272
|1536 (600<sub>16</sub>)
|}
|}


Line 216: Line 349:
[[photia7|photia7]]
[[photia7|photia7]]


[[photia12|photia12]]      [[Category:11/8]]
[[photia12|photia12]]       
[[Category:11/8]]
[[Category:13/8]]
[[Category:13/8]]
[[Category:17/16]]
[[Category:17/16]]

Revision as of 01:48, 19 May 2019

65 tone equal temperament

65edo divides the octave into 65 equal parts of 18.4615 cents each. It can be characterized as the temperament which tempers out the schisma, 32805/32768, the sensipent comma, 78732/78125, and the wuerschmidt comma. In the 7-limit, there are two different maps; the first is <65 103 151 182|, tempering out 126/125, 245/243 and 686/675, so that 65edo supports sensi temperament, and the second is <65 103 151 183|, tempering out 225/224, 3125/3097, 4000/3969 and 5120/5103, so that 65edo supports garibaldi temperament. In both cases, the tuning privileges the 5-limit over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit wuerschmidt temperament (wurschmidt and worschmidt) these two mappings provide.

65edo approximates the intervals 3/2, 5/4, 11/8 and 19/16 well, so that it does a good job representing the 2.3.5.11.19 just intonation subgroup. To this one may want to add 13/8 and 17/16, giving the 19-limit no-sevens subgroup 2.3.5.11.13.17.19. Also of interest is the 19-limit 2*65 subgroup 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as 130edo.

65edo contains 13edo as a subset. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see Rubble: a Xenuke Unfolded.

Intervals

Degree Size
Cents pions 7mus
0
1 18.4615 19.5692 23.6308 (17.A17A16)
2 36.9231 39.1385 47.2615 (2F.42F416)
3 55.3846 58.7077 70.8923 (46.E46E16)
4 73.84615 78.2769 94.5231 (5E.85E816)
5 92.3077 97.84615 118.15385 (76.276216)
6 110.7692 117.4154 141.7846 (8D.C8DD16)
7 129.2308 136.9846 165.4154 (A5.6A5716)
8 147.6923 156.55385 189.04615 (BD.0BD116)
9 166.15385 176.1231 212.6769 (D4.AD4B16
10 184.6154 195.6923 236.3077 (EC.4EC516)
11 203.0769 215.2615 259.9385 (103.F03F16)
12 221.5385 234.8308 283.5692 (11B.91B916)
13 240 254.4 307.2 (133.333316)
14 258.4615 273.9692 330.8308 (14A.D4AD16)
15 276.9231 293.5385 354.4615 (162.762716)
16 295.3846 313.1077 378.0923 (17A.17A1816)
17 313.84615 332.6769 401.7231 (191.B91C16)
18 332.3077 352.24615 425.35385 (1A9.5A9616)
19 350.7692 371.8154 448.9846 (1C0.FC116)
20 369.2308 391.3846 472.6154 (1D8.9D8A16)
21 387.6923 410.95385 496.24615 (1F0.3F0416)
22 406.15385 430.5231 519.8769 (207.E07E16)
23 424.6154 450.0923 543.5077 (21F.81F816)
24 443.0769 469.6615 567.1385 (237.237216)
25 461.5385 489.2308 590.7692 (24E.C4ED16)
26 480 508.8 614.4 (266.666616)
27 498.4615 528.3692 638.0308 (27E07E0816)
28 516.9231 547.9385 661.6615 (295.A95B16)
29 535.3846 567.5077 685.2923 (2AD.4AD416)
30 553.84615 587.0769 708.9231 (2C4.EC4F16)
31 572.3077 606.64615 732.55385 (2DC.8DC916)
32 590.7692 626.2154 756.1846 (2F4.2F4216)
33 609.2308 645.7846 779.8154 (30B.D0BD16)
34 627.6923 665.35385 803.44615 (323.723716)
35 646.1538 684.9231 827.0769 (33B.13B116)
36 664.6154 704.4923 850.7077 (352.B52B16)
37 683.0769 724.0615 874.3385 (36A.56A516)
38 701.5385 743.6308 897.9692 (381.F81F816)
39 720 763.2 921.6 (399.999A16)
40 738.4615 782.7692 945.2308 (3B1.3B1316)
41 756.9231 802.3385 968.8615 (3C8.DC8E16)
42 775.3846 821.9077 992.4923 (3E0.7E0816)
43 793.84615 841.4769 1016.1231 (3F8.1F8216)
44 812.3077 861.04615 1039.75385 (40F.C0FC16)
45 830.7692 880.6154 1063.3846 (427.627616)
46 849.2308 900.1846 1087.0154 (43F.03F16)
47 867.6923 919.75385 1110.64615 (456.A56A16)
48 886.15385 939.3231 1134.2769 (46E.46E416)
49 904.6154 958.8923 1157.9077 (485.E85E816)
50 923.0769 978.4615 1181.53845 (49D.89D916)
51 941.5385 998.0308 1205.1692 (4B5.2B5316)
52 960 1017.6 1228.8 (4CC.CCCD16)
53 978.4615 1037.1692 1252.4308 (4E4.6E4716)
54 996.9231 1056.7385 1276.0615 (4FC.0FC116)
55 1015.3846 1076.3077 1299.6923 (513.B13B16)
56 1033.84615 1095.8769 1323.3231 (52B.52B516)
57 1052.3077 1115.44615 1346.95385 (542,F42F16)
58 1070.7692 1135.0154 1370.5846 (55A.95A916)
59 1089.2308 1154.5846 1394.2154 (572.372316)
60 1107.6923 1174.1535 1417.84615 (589.D89E16)
61 1126.15385 1193.7231 1441.4769 (5A1.7A1816)
62 1144.6154 1213.2923 1465.1077 (5B9.1B9216)
63 1163.0769 1232.8615 1488.7385 (5D0.BD0C16)
64 1181.5385 1252.4308 1512.3692 (5E8.5E8616)
65 1200 1272 1536 (60016)

Scales

photia7

photia12

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