User:Overthink/The 7-limit in 53edo: Difference between revisions

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; Development of this page is paused indefinitely. 171edo is much more interesting.
In 53edo, the [[7-limit]] is well-approximated, and especially the 5- and 3-limits. On this page, we will analyze the approximations and structures of 53edo in the 7-limit.
In 53edo, the [[7-limit]] is well-approximated, and especially the 5- and 3-limits. On this page, we will analyze the approximations and structures of 53edo in the 7-limit.
{| class="mw-collapsible wikitable"
{| class="mw-collapsible wikitable"
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| 2
| 2
| 45.283
| 45.283
| 36/35, 49/48, 128/125
| 36/35, 49/48, 128/125, 250/243
|-
|-
| 3
| 3
Line 37: Line 38:
| 7
| 7
| 158.491
| 158.491
| 35/32,
| 35/32
|-
|-
| 8
| 8
Line 53: Line 54:
| 11
| 11
| 249.057
| 249.057
| 81/70
| 81/70, 125/108, 144/125, 147/128
|-
|-
| 12
| 12
Line 81: Line 82:
| 18
| 18
| 407.547
| 407.547
| 81/64,
| 81/64
|-
|-
| 19
| 19
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| 24
| 24
| 543.396
| 543.396
| 2187/1600, 48/35
| 48/35
|-
|-
| 25
| 25
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| 26
| 26
| 588.679
| 588.679
| 729/512, 7/5, 45/32
| 1024/729, 7/5, 45/32
|-
|-
| 27
| 27
| 611.321
| 611.321
| 1024/729, 10/7, 64/45
| 729/512, 10/7, 64/45
|-
|-
| 28
| 28
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| 1200.000
| 1200.000
| 2/1
| 2/1
|}
The [[81/80]] and [[64/63]] commas translate pythagorean intervals into nearby pental and septimal intervals respectively. Considering them seperately is too complex, so we conflate them into one comma step, tempering out [[5120/5103]]. Here's a table of intervals organized using tempering of 5120/5103. Each interval is a fifth above the interval to the left of it, and a comma above the interval below it. Not all ratios are shown, or else the table will be too complex.
{| class="wikitable"
|+Interval table (far fourthward)
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|-
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|-
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{| class="wikitable"
|+Hemifamity interval table (middle)
|
|256/175
|
|
|
|
|
|256/245,
729/700
|384/245
|
|
|
|
|512/343
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|-
|48/25
|36/25
|27/25
|81/50
|128/105,
243/200
|64/35
|48/35
|36/35
|54/35
|81/70,
|243/140,
256/147
|64/49
|96/49
|72/49
|54/49
|-
|135/128
|64/45
|16/15
|8/5
|6/5
|9/5
|27/20
|64/63,
81/80
|32/21
|8/7
|12/7
|9/7
|27/14
|81/56
|243/224
|-
|28/15,
4096/2187
|7/5,
1024/729
|21/20,
256/243
|63/40,
128/81
|32/27
|16/9
|4/3
|1/1
|3/2
|9/8
|27/16
|80/63,
81/64
|40/21,
243/128
|10/7,
729/512
|15/14,
2187/2048
|-
|448/243
|112/81
|28/27
|14/9
|7/6
|7/4
|21/16
|63/32,
160/81
|40/27
|10/9
|5/3
|5/4
|15/8
|45/32
|135/128
|-
|49/27
|49/36
|49/48
|49/32
|147/128,
280/243
|140/81
|35/27
|35/18
|35/24
|35/32
|105/64,
400/243
|315/256,
100/81
|50/27
|25/18
|25/24
|-
|343/192
|343/256,
980/729
|1029/1024,
245/243
|245/162
|245/216
|245/144
|245/192
|245/128,
1400/729
|735/512,
350/243
|175/162
|175/108
|175/144
|175/96
|175/128
|250/243,
525/512
|}
{| class="wikitable"
|+Interval table (far fifthward)
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|}

Latest revision as of 09:27, 25 December 2025

Development of this page is paused indefinitely. 171edo is much more interesting.

In 53edo, the 7-limit is well-approximated, and especially the 5- and 3-limits. On this page, we will analyze the approximations and structures of 53edo in the 7-limit.

Intervals of 53edo
Steps Cents Approximate ratios
0 0.000 1/1
1 22.642 531441/524288, 81/80, 64/63, 50/49
2 45.283 36/35, 49/48, 128/125, 250/243
3 67.925 28/27, 25/24
4 90.566 256/243, 135/128, 21/20
5 113.208 16/15, 15/14, 2187/2048
6 135.849 27/25
7 158.491 35/32
8 181.132 10/9
9 203.774 9/8
10 226.415 8/7
11 249.057 81/70, 125/108, 144/125, 147/128
12 271.698 7/6, 75/64
13 294.340 32/27
14 316.981 6/5
15 339.623 105/64, 243/200
16 362.264 100/81, 315/256
17 384.906 5/4
18 407.547 81/64
19 430.189 9/7, 32/25
20 452.830 64/49, 35/27
21 475.472 21/16
22 498.113 4/3
23 520.755 27/20
24 543.396 48/35
25 566.038 1/1
26 588.679 1024/729, 7/5, 45/32
27 611.321 729/512, 10/7, 64/45
28 633.962 1/1
29 656.604 1/1
30 679.245 1/1
31 701.887 3/2
32 724.528 1/1
33 747.170 1/1
34 769.811 1/1
35 792.453 1/1
36 815.094 1/1
37 837.736 1/1
38 860.377 1/1
39 883.019 1/1
40 905.660 1/1
41 928.302 1/1
42 950.943 1/1
43 973.585 1/1
44 996.226 1/1
45 1018.868 1/1
46 1041.509 1/1
47 1064.151 1/1
48 1086.792 1/1
49 1109.434 1/1
50 1132.075 1/1
51 1154.717 1/1
52 1177.358 1/1
53 1200.000 2/1

The 81/80 and 64/63 commas translate pythagorean intervals into nearby pental and septimal intervals respectively. Considering them seperately is too complex, so we conflate them into one comma step, tempering out 5120/5103. Here's a table of intervals organized using tempering of 5120/5103. Each interval is a fifth above the interval to the left of it, and a comma above the interval below it. Not all ratios are shown, or else the table will be too complex.

Interval table (far fourthward)
Hemifamity interval table (middle)
256/175 256/245,

729/700

384/245 512/343
48/25 36/25 27/25 81/50 128/105,

243/200

64/35 48/35 36/35 54/35 81/70, 243/140,

256/147

64/49 96/49 72/49 54/49
135/128 64/45 16/15 8/5 6/5 9/5 27/20 64/63,

81/80

32/21 8/7 12/7 9/7 27/14 81/56 243/224
28/15,

4096/2187

7/5,

1024/729

21/20,

256/243

63/40,

128/81

32/27 16/9 4/3 1/1 3/2 9/8 27/16 80/63,

81/64

40/21,

243/128

10/7,

729/512

15/14,

2187/2048

448/243 112/81 28/27 14/9 7/6 7/4 21/16 63/32,

160/81

40/27 10/9 5/3 5/4 15/8 45/32 135/128
49/27 49/36 49/48 49/32 147/128,

280/243

140/81 35/27 35/18 35/24 35/32 105/64,

400/243

315/256,

100/81

50/27 25/18 25/24
343/192 343/256,

980/729

1029/1024,

245/243

245/162 245/216 245/144 245/192 245/128,

1400/729

735/512,

350/243

175/162 175/108 175/144 175/96 175/128 250/243,

525/512

Interval table (far fifthward)
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