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Determining Ideal Tempos:
A Unified Theory of Tempo Relativity


Table 1

Self-generating progression of possible theoretically correct tempos. “Moderate” is 3/2 times “slow”, “moderately-fast” is 7/6 times “moderate”, and “fast” is 8/7 times “moderately-fast”. The left column is a culmination of the tempos in the other three columns. Integers are shown in bold. Tempos that appear in all four columns are shown in red font. 21 is the smallest integer to appear in all four columns, thus, it and its multiples (42, 63, 84, etc.) are the only integers to appear in all four columns, shown in bold with red cell highlighting. Out of these special tempos, 42 is the only one to have adjacent tempos that appear in all four columns, making it the slowest tempo to possess this unique quality. This progression can continue infinitely.

slow moderate moderately fast fast
1 1.5 1.75 2
1.5 2.25 2.625 3
1.75 2.625 3.0625 3.5
2 3 3.5 4
2.25 3.375 3.9375 4.5
2.625 3.9375 4.59375 5.25
3 4.5 5.25 6
3.0625 4.59375 5.5359375 6.125
3.375 5.0625 5.90625 6.75
3.5 5.25 6.125 7
3.9375 5.90625 6.890625 7.875
4 6 7 8
4.5 6.75 7.875 9
4.59375 6.890625 8.0390625 9.1875
5.25 7.875 9.1875 10.5
5.359375 8.0390625 9.3789061 10.718749
5.90625 8.859375 10.335937 11.8125
6 9 10.5 12
6.125 9.1875 10.71875 12.25
6.75 10.125 11.8125 13.5
6.890625 10.335937 12.058593 13.78125
7 10.5 12.25 14
7.875 11.8125 13.78125 15.75
8 12 14 16
8.0390625 12.058593 14.068358 16.078122
8.859375 13.289062 15.503905 17.71875
9 13.5 15.75 18
9.1875 13.78125 16.078125 18.375
9.3789061 14.068359 16.413805 18.757811
10.125 15.1875 17.71875 20.25
10.335937 15.503905 18.187888 20.671871
10.5 15.75 18.375 21
10.718749 16.078122 18.75781 21.437497
11.8125 17.71875 20.671875 23.625
12 18 21 24
12.058593 18.087889 21.102536 24.117182
12.25 18.375 21.4375 24.5
13.289062 19.933593 23.255858 26.578122
13.5 20.25 23.625 27
13.78125 20.671875 24.117186 27.5625
14 21 24.5 28
14.068358 21.102537 24.619625 28.136714
15.1875 22.78125 26.578125 30.375
15.503905 23.255857 27.131831 31.00805
15.75 23.625 27.5625 31.5
16 24 28 32
16.078122 24.117183 28.136713 32.156242
16.413085 24.619627 28.722896 35.826165
17.71875 26.578125 31.007811 35.4375
18 27 31.5 36
18.087888 27.131832 31.653803 36.175774
18.375 27.5625 32.15625 36.75
18.757811 28.136716 32.826168 37.51562
19.933593 29.900389 34.383786 37.867182
20.25 30.375 35.4375 40.5
20.671871 31.007086 36.175773 41.34374
21 31.5 36.75 42
21.102537 31.653805 36.929438 42.205071
21.437497 32.15625 37.515618 42.874991
23.255857 34.883785 40.697748 46.511711
23.625 35.4375 41.34375 47.25
24 36 42 48
24.117183 36.175774 42.205068 48.234362
24.5 36.75 42.875 49
24.619627 36.92944 43.084346 49.239251
26.578125 39.867187 46.511716 53.156245
27 40.5 47.25 54
27.131832 40.697748 47.480705 55.394155
27.5625 41.34375 48.234375 55.125
28 42 49 56
28.136716 42.205074 49.239251 56.273428
28.722896 43.084344 50.265066 57.445788
29.900389 44.850583 52.32568 59.800777
30.375 45.5625 53.15625 60.75
31.007086 46.510629 54.2624 62.014171
31.007811 46.511716 54.263668 62.01562
31.00805 46.512075 54.264086 62.016097
31.5 47.25 55.125 63
31.653805 47.480707 55.394156 63.307605
32 48 56 64
32.15625 48.234367 56.273426 64.312485
32.826168 49.239252 57.445793 65.652334
34.383786 51.575675 60.17162 68.767565
34.883785 52.325675 61.04662 69.767565
35.4375 53.15625 62.015625 70.875
36 54 63 72
36.175774 54.26366 63.307603 72.351545
36.75 55.125 64.3125 73.5
36.92944 55.39416 64.62656 73.85888
37.51562 56.27343 65.652335 75.03124
37.867182 56.80077 66.267565 75.73436
39.867187 59.80078 69.767576 79.734371
40.5 60.75 70.875 81
40.697748 61.04662 71.221056 81.395491
41.34375 62.015625 72.351561 82.687497
42 63 73.5 84
42.205074 63.30761 73.858878 84.410145
42.875 64.3125 75.03125 85.75
43.084344 64.626515 75.3976 86.168685

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