RE: Erlich's computation

Paul,

Thanks for running that calculation. Unfortunately, I'm not proficient
in Mathematica, and am currently don't have the time (or am I too lazy?)
to write the raw Java or C++ code to do that sort of calculation, nor
the patience to do it in a spreadsheet.

Also, note that I'm interested in NON-OCTAVE ETs --in other words, run
the calculation first doing only the 5 kinds of thirds; then run it for
the 5 kinds of sixths--I know that the regular ETs will dominate the
results if you run the calculation on all at once (this is easily proven
mathematically).

Could you give us a breakdown, for each of those non-octave ETs in your
last message, showing the size of the error for each interval in each
calculation? Could you do so in the format toneFromET - Ratio? This will
help me better decide on one or another such tuning for actual music.

I'll repeat the ratios of interest for the benefit of the wider
audience:

the 5 kinds of thirds:
7/6
6/5
11/9
5/4
9/7

the corresponding sixths
('inversions' according to the traditional music theory concept--that
is,
if the above intervals are a/b, the inversion is 2b/a--so the inversion
of 7/6 is 12/7)

14/9
8/5
18/11
5/3
12/7

Harold,

> Unfortunately, I'm not proficient
> in Mathematica, and am currently don't have the time (or am I too lazy?)
> to write the raw Java or C++ code to do that sort of calculation, nor
> the patience to do it in a spreadsheet.

If my understanding of what you want is right, you can also use Scala.
Do the FIT command. The output for your first set of intervals is

linear approximation: 102.9282 cents, 11.658609/oct. Std. dev: 54.4266 cents
1: 3 2: 4 3: 4 4: 5 5: 5
83.0747 cents, 14.4448/oct. Std. dev: 11.3196 cents
1: 4 2: 4 3: 5 4: 5 5: 6
72.9590 cents, 16.4476/oct. Std. dev: 11.0749 cents
1: 4 2: 5 3: 5 4: 6 5: 6
67.4719 cents, 17.7852/oct. Std. dev: 10.7235 cents
1: 4 2: 5 3: 5 4: 6 5: 7
64.5444 cents, 18.5919/oct. Std. dev: 7.9679 cents
1: 5 2: 6 3: 6 4: 7 5: 8
54.7501 cents, 21.9178/oct. Std. dev: 6.0625 cents
1: 6 2: 7 3: 8 4: 9 5: 10
43.6898 cents, 27.4663/oct. Std. dev: 3.2973 cents
1: 7 2: 8 3: 9 4: 10 5: 11
38.9614 cents, 30.7998/oct. Std. dev: 2.6659 cents
1: 9 2: 11 3: 12 4: 13 5: 15
29.1772 cents, 41.1280/oct. Std. dev: 2.6174 cents
1: 11 2: 13 3: 14 4: 16 5: 18
24.3107 cents, 49.3611/oct. Std. dev: 1.9942 cents
1: 15 2: 18 3: 20 4: 22 5: 25
17.4970 cents, 68.5833/oct. Std. dev: 1.4530 cents
1: 16 2: 19 3: 21 4: 23 5: 26
16.6859 cents, 71.9172/oct. Std. dev: 1.0876 cents
1: 22 2: 26 3: 29 4: 32 5: 36
12.0754 cents, 99.3757/oct. Std. dev: 0.8719 cents
1: 27 2: 32 3: 35 4: 39 5: 44
9.8951 cents, 121.2716/oct. Std. dev: 0.3962 cents
1: 49 2: 58 3: 64 4: 71 5: 80
5.4386 cents, 220.6452/oct. Std. dev: 0.2022 cents
1: 76 2: 90 3: 99 4: 110 5: 124
3.5096 cents, 341.9161/oct. Std. dev: 0.0965 cents

For the second set it's:

linear approximation: 239.1481 cents, 5.017811/oct. Std. dev: 173.4513 cents
1: 9 2: 10 3: 10 4: 11 5: 11
83.2493 cents, 14.4145/oct. Std. dev: 11.9648 cents
1: 10 2: 10 3: 11 4: 11 5: 12
78.6519 cents, 15.2571/oct. Std. dev: 10.7398 cents
1: 11 2: 11 3: 12 4: 12 5: 13
72.0153 cents, 16.6631/oct. Std. dev: 10.4778 cents
1: 11 2: 12 3: 13 4: 13 5: 14
67.3623 cents, 17.8141/oct. Std. dev: 9.0460 cents
1: 12 2: 13 3: 13 4: 14 5: 15
63.3650 cents, 18.9379/oct. Std. dev: 8.8824 cents
1: 12 2: 13 3: 14 4: 14 5: 15
62.4421 cents, 19.2178/oct. Std. dev: 7.2006 cents
1: 13 2: 14 3: 15 4: 15 5: 16
58.1885 cents, 20.6226/oct. Std. dev: 6.2223 cents
1: 14 2: 15 3: 16 4: 16 5: 17
54.4751 cents, 22.0284/oct. Std. dev: 6.0766 cents
1: 16 2: 17 3: 18 4: 19 5: 20
47.1677 cents, 25.4411/oct. Std. dev: 5.6013 cents
1: 17 2: 18 3: 19 4: 20 5: 21
44.7009 cents, 26.8451/oct. Std. dev: 3.8935 cents
1: 18 2: 19 3: 20 4: 21 5: 22
42.4780 cents, 28.2499/oct. Std. dev: 2.6670 cents
1: 19 2: 20 3: 21 4: 22 5: 23
40.4648 cents, 29.6554/oct. Std. dev: 2.2774 cents
1: 31 2: 33 3: 35 4: 36 5: 38
24.5554 cents, 48.8691/oct. Std. dev: 2.1209 cents
1: 32 2: 34 3: 36 4: 37 5: 39
23.8689 cents, 50.2747/oct. Std. dev: 1.8883 cents
1: 43 2: 46 3: 48 4: 50 5: 53
17.6990 cents, 67.8004/oct. Std. dev: 1.7480 cents
1: 44 2: 47 3: 49 4: 51 5: 54
17.3396 cents, 69.2057/oct. Std. dev: 1.2383 cents
1: 45 2: 48 3: 50 4: 52 5: 55
16.9945 cents, 70.6112/oct. Std. dev: 0.9786 cents
1: 63 2: 67 3: 70 4: 73 5: 77
12.1383 cents, 98.8603/oct. Std. dev: 0.9361 cents
1: 64 2: 68 3: 71 4: 74 5: 78
11.9682 cents, 100.2660/oct. Std. dev: 0.8313 cents
1: 76 2: 81 3: 85 4: 88 5: 93
10.0436 cents, 119.4793/oct. Std. dev: 0.5569 cents
1: 77 2: 82 3: 86 4: 89 5: 94
9.9268 cents, 120.8849/oct. Std. dev: 0.3876 cents
1: 95 2: 101 3: 106 4: 110 5: 116
8.0465 cents, 149.1339/oct. Std. dev: 0.3521 cents
1: 96 2: 102 3: 107 4: 111 5: 117
7.9713 cents, 150.5396/oct. Std. dev: 0.2553 cents
1: 141 2: 150 3: 157 4: 163 5: 172
5.4262 cents, 221.1505/oct. Std. dev: 0.1929 cents
1: 217 2: 231 3: 242 4: 251 5: 265
3.5229 cents, 340.6296/oct. Std. dev: 0.1616 cents
1: 218 2: 232 3: 243 4: 252 5: 266
3.5084 cents, 342.0353/oct. Std. dev: 0.0963 cents

If you want better fifths, you can add it also to the set to approximate.

Manuel Op de Coul coul@ezh.nl

Harold wrote,

>I'll repeat the ratios of interest for the benefit of the wider
>audience:

>the 5 kinds of thirds:
>7/6
>6/5
>11/9
>5/4
>9/7

>the corresponding sixths
>('inversions' according to the traditional music theory concept--that
>is,
>if the above intervals are a/b, the inversion is 2b/a--so the inversion
>of 7/6 is 12/7)

>14/9
>8/5
>18/11
>5/3
>12/7

Harold, that changes things! You previously said 4/3 times the 5 kinds of
thirds, giving 44/27 instrad of 18/11! But a bigger deal is this:

>Also, note that I'm interested in NON-OCTAVE ETs --in other words, run
>the calculation first doing only the 5 kinds of thirds; then run it for
>the 5 kinds of sixths--I know that the regular ETs will dominate the
>results if you run the calculation on all at once (this is easily proven
>mathematically).

OK -- here goes:

For the 5 thirds alone:

21.72-tET max. err. 15.92�
27.35-tET max. err. 8.57�
30.77-tET max. err. 6.12�
41.03-tET max. err. 6.10�
49.07-tET max. err. 5.10�
68.38-tET max. err. 3.64�
71.96-tET max. err. 2.79�
90.49-tET max. err. 2.63�
99.54-tET max. err. 2.20�

For the 5 sixths alone:

19.29-tET max. err. 18.41�
20.73-tET max. err. 16.05�
22.11-tET max. err. 15.97�
23.49-tET max. err. 15.90�
24.03-tET max. err. 15.85�
25.44-tET max. err. 11.87�
26.85-tET max. err. 9.50�
28.27-tET max. err. 7.18�
29.68-tET max. err. 5.13�
50.40-tET max. err. 4.56�
67.84-tET max. err. 4.37�
69.22-tET max. err. 3.13�
70.58-tET max. err. 2.49�
98.79-tET max. err. 2.37�

>Could you give us a breakdown, for each of those non-octave ETs in your
>last message, showing the size of the error for each interval in each
>calculation? Could you do so in the format toneFromET - Ratio? This will
>help me better decide on one or another such tuning for actual music.

I'd do this for all the ones above, though I'm not sure if that would be a
completely satisfactory answer for you. For example, if you are specifically
looking for certain of the intervals to be approximated better, then there
may be tunings missing from the above lists that would work best for you.
Can you describe in more detail what you're looking for?