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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="wikitable mw-collapsible mw-collapsed"
{| class="mw-collapsible wikitable"
|+ Intervals of 53edo
|+ style="font-size: 105%; white-space: nowrap;" | Intervals of 53edo
|-
! Steps
! Steps
! Cents
! Cents
Line 12: Line 14:
| 1
| 1
| 22.642
| 22.642
| 1/1
| 531441/524288, 81/80, 64/63, 50/49
|-
|-
| 2
| 2
| 45.283
| 45.283
| 1/1
| 36/35, 49/48, 128/125, 250/243
|-
|-
| 3
| 3
| 67.925
| 67.925
| 1/1
| 28/27, 25/24
|-
|-
| 4
| 4
| 90.566
| 90.566
| 1/1
| 256/243, 135/128, 21/20
|-
|-
| 5
| 5
| 113.208
| 113.208
| 1/1
| 16/15, 15/14, 2187/2048
|-
|-
| 6
| 6
| 135.849
| 135.849
| 1/1
| 27/25
|-
|-
| 7
| 7
| 158.491
| 158.491
| 1/1
| 35/32
|-
|-
| 8
| 8
| 181.132
| 181.132
| 1/1
| 10/9
|-
|-
| 9
| 9
| 203.774
| 203.774
| 1/1
| 9/8
|-
|-
| 10
| 10
| 226.415
| 226.415
| 1/1
| 8/7
|-
|-
| 11
| 11
| 249.057
| 249.057
| 1/1
| 81/70, 125/108, 144/125, 147/128
|-
|-
| 12
| 12
| 271.698
| 271.698
| 1/1
| 7/6, 75/64
|-
|-
| 13
| 13
| 294.340
| 294.340
| 1/1
| 32/27
|-
|-
| 14
| 14
| 316.981
| 316.981
| 1/1
| 6/5
|-
|-
| 15
| 15
| 339.623
| 339.623
| 1/1
| 105/64, 243/200
|-
|-
| 16
| 16
| 362.264
| 362.264
| 1/1
| 100/81, 315/256
|-
|-
| 17
| 17
| 384.906
| 384.906
| 1/1
| 5/4
|-
|-
| 18
| 18
| 407.547
| 407.547
| 1/1
| 81/64
|-
|-
| 19
| 19
| 430.189
| 430.189
| 1/1
| 9/7, 32/25
|-
|-
| 20
| 20
| 452.830
| 452.830
| 1/1
| 64/49, 35/27
|-
|-
| 21
| 21
| 475.472
| 475.472
| 1/1
| 21/16
|-
|-
| 22
| 22
Line 100: Line 102:
| 23
| 23
| 520.755
| 520.755
| 1/1
| 27/20
|-
|-
| 24
| 24
| 543.396
| 543.396
| 1/1
| 48/35
|-
|-
| 25
| 25
Line 112: Line 114:
| 26
| 26
| 588.679
| 588.679
| 1/1
| 1024/729, 7/5, 45/32
|-
|-
| 27
| 27
| 611.321
| 611.321
| 1/1
| 729/512, 10/7, 64/45
|-
|-
| 28
| 28
Line 221: Line 223:
| 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)
|
|
|
|
|-
|
|
|
|
|-
|
|
|
|
|}
{| class="wikitable"
|+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
|}
{| class="wikitable"
|+Interval table (far fifthward)
|
|
|
|
|-
|
|
|
|
|-
|
|
|
|
|}
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