A list of Blackwood's diatonic behaviour of equal temperament

 In the succession to the previous article on Blackwood's traditional formation the relationship of a diatonic scale can be written as 5w+2h=n-TET. w, h and n must be an integer and n>w>h>0,, as the diatonic behaviour.

I calculated that from 5 to 60-TET by a Python program to see which number of equal temperaments satisfy or not the diatonic behaviour. The table below is the result. If it is labelled as False, it means it does satisfy the diatonic behaviour. If it satisfies, the chart will display number of microstep of a whole tone and half tone respectively.

Based on the finding below, we can see that 12-TET is the smallest amount of steps the perform the diatonic behaviour. This is the reason 12 is the most used number. The 35-TET is the last one which does not perform in diatonicly, larger than that, the number are all satisfy diatonic behaviour. The 47-TET is the smallest n-TET can perform bi-diatonic behaviour, 7 microsteps for a whole step and 6 microsteps for a whole step; or 9 for a whole step and 1 for a whole step. Although the n-TET fits the diatonic behaviour, when spelling chords and modes in diatonic pattern, each TET can sound different of each others, which the ratio of half:whole step describes the quality. If the ratio is the same, for example, the ratios of 12 and 24 is the same, the sounding quality of chords and modes can be similar.

n-TETn-step whole t.n-step half t.ratio of h:w
5FALSEN/A
6FALSEN/A
7FALSEN/A
8FALSEN/A
9FALSEN/A
10FALSEN/A
11FALSEN/A
12210.5
13FALSEN/A
14FALSEN/A
15FALSEN/A
16FALSEN/A
17310.3333333333
18FALSEN/A
19320.6666666667
20FALSEN/A
21FALSEN/A
22410.25
23FALSEN/A
24420.5
25FALSEN/A
26430.75
27510.2
28FALSEN/A
29520.4
30FALSEN/A
31530.6
32610.1666666667
33540.8
34620.3333333333
35FALSEN/A
36630.5
37710.1428571429
38640.6666666667
39720.2857142857
40650.8333333333
41730.4285714286
42810.125
43740.5714285714
44820.25
45750.7142857143
46830.375
47760.8571428571
47910.1111111111
48840.5
49920.2222222222
50850.625
51930.3333333333
52860.75
521010.1
53940.4444444444
54870.875
541020.2
55950.5555555556
561030.3
57960.6666666667
571110.09090909091
581040.4
59970.7777777778
591120.1818181818
601050.5

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