Re: Why equal temperament

In article <e12qtk$jka@xxxxxxxxxxxxxxxxxxxxxxx>, "Bob Pease"
<Popester@xxxxxxxx> wrote:

Why equal temperment at all?

This has been extensively discussed in this newsgroup. You are essentially
asking for reworking the historical development towards a temperament
playable in any key, in order to admit arbitrary modulations. In this
picture, n-TET for various n, come in as approximations of different
rational based tunings. IN another picture, in modern music, n-TET's may
be used in their own right, not directly connected to rational intervals.

I did not ask for a re=work of amything
That's why I asked for references or sources
It is unreasonable to expext everyome on this ng to know the specialized
jargon of every subject which has been discussed.

I did then not make sufficiently explicit what I meant: the answer to the
question "why ET" is very historically complicated, simply, and you will
not arrive at the full answer easily. That is why has been extensively
discussed in the past in this newsgroup, trying to pin those things down.
So nobody is expected to know the full details, because the full details
are not fully known.

In brief, the awareness of a potential ET is very old, remnants of it
tracing back to the Ancient Greek. But it came into full use in Western
music much later than most would think, in the 19th century Romantic era.
The reason as to is unclear, one could have been the emergence of new
concert pianos with steel frames, being able to dominate the music much
more. Another could have been the emergence of large orchestras, making it
more difficult for musicians to play individually.

One strange thing in this development and gradual use of 12-TET, though,
is that it does not approximate well the underlying Pythagorean and Just
intonations, specifically the partial 5 they make use of. There the offset
is one full duodecitone. If you check the Wikipedia, it gives articles to
other n-TET which historically attempts to address this issue.

One easy way to get an idea of which is n-TET is suitable as
rational approximation is simply to write these approximation out (see
below). Then, a quick scan shows that 72-TET is a candidate for partials
2, 3, 5, 7, 11, but not 13. Strictly speaking, this is not the 11-limit
approximation, which throws in some other intervals as well, that should
be approximated. In general, one can take a set of rational intervals, or
any intervals whatsoever, and try to find an n-TET that approximates just
that set of intervals well. Some people into tunings do that, and in the
past, they have discussed their work in this newsgroup as well.

--------------------------------------------------------------
Offsets are counted of partial relative n-equal approximation.
+ rational is sharp relative n-equal note =
played n-equal is flat relative intended rational
- rational is flat relative n-equal note =
played n-equal is sharp relative intended rational

One of the most interesting candidates is 72-equal:
n, p: 2 3 5 7 11 13 17 19
72 0.00 1.96 2.98 2.16 1.32 7.19 4.96 -2.49

Offsets in cents of smallest rational approximation with denominator n
(= the number of tones in an equally tempered chromatic scale) of log_2 p:
n, p: 2 3 5 7 11 13 17
1 0.00 -498.04 386.31 -231.17 551.32 -359.47 104.96
2 0.00 101.96 -213.69 -231.17 -48.68 240.53 104.96
3 0.00 -98.04 -13.69 168.83 151.32 40.53 104.96
4 0.00 101.96 86.31 68.83 -48.68 -59.47 104.96
5 0.00 -18.04 -93.69 8.83 71.32 -119.47 104.96
6 0.00 -98.04 -13.69 -31.17 -48.68 40.53 -95.04
7 0.00 16.24 43.46 -59.75 37.03 -16.62 -66.47
8 0.00 -48.04 -63.69 68.83 -48.68 -59.47 -45.04
9 0.00 35.29 -13.69 35.49 17.98 40.53 -28.38
10 0.00 -18.04 26.31 8.83 -48.68 0.53 -15.04
11 0.00 47.41 -50.05 -12.99 5.86 -32.20 -4.14
12 0.00 1.96 -13.69 -31.17 -48.68 40.53 4.96
13 0.00 -36.51 17.08 45.75 -2.53 9.76 12.65
14 0.00 16.24 -42.26 25.97 37.03 -16.62 19.24
15 0.00 -18.04 -13.69 8.83 -8.68 -39.47 24.96
16 0.00 26.96 11.31 -6.17 26.32 15.53 29.96
17 0.00 -3.93 33.37 -19.41 -13.39 -6.53 34.37
18 0.00 -31.38 -13.69 -31.17 17.98 -26.14 -28.38
19 0.00 7.22 7.37 21.46 -17.10 19.48 -21.36
20 0.00 -18.04 26.31 8.83 11.32 0.53 -15.04
21 0.00 16.24 -13.69 -2.60 -20.11 -16.62 -9.33
22 0.00 -7.14 4.50 -12.99 5.86 22.35 -4.14
23 0.00 23.69 21.10 -22.48 -22.60 5.75 0.61
24 0.00 1.96 -13.69 18.83 1.32 -9.47 4.96
25 0.00 -18.04 2.31 8.83 23.32 -23.47 8.96
26 0.00 9.65 17.08 -0.40 -2.53 9.76 12.65
27 0.00 -9.16 -13.69 -8.95 17.98 -3.92 16.07
28 0.00 16.24 0.60 -16.89 -5.82 -16.62 19.24
29 0.00 -1.49 13.90 17.10 13.39 12.94 -19.18
30 0.00 -18.04 -13.69 8.83 -8.68 0.53 -15.04
31 0.00 5.18 -0.78 1.08 9.38 -11.09 -11.17
32 0.00 -10.54 11.31 -6.17 -11.18 15.53 -7.54
33 0.00 11.05 -13.69 -12.99 5.86 4.16 -4.14
34 0.00 -3.93 -1.92 15.88 -13.39 -6.53 -0.93
35 0.00 16.24 9.17 8.83 2.75 -16.62 2.10
36 0.00 1.96 -13.69 2.16 -15.35 7.19 4.96
37 0.00 -11.56 -2.88 -4.15 -0.03 -2.72 7.66
38 0.00 7.22 7.37 -10.12 14.48 -12.10 10.22
39 0.00 -5.74 -13.69 14.98 -2.53 9.76 12.65
40 0.00 11.96 -3.69 8.83 11.32 0.53 14.96
41 0.00 -0.48 5.83 2.97 -4.78 -8.25 -12.12
42 0.00 -12.33 -13.69 -2.60 8.46 11.96 -9.33
43 0.00 4.28 -4.38 -7.92 -6.82 3.32 -6.67
44 0.00 -7.14 4.50 -12.99 5.86 -4.93 -4.14
45 0.00 8.62 12.98 8.83 -8.68 -12.81 -1.71
46 0.00 -2.39 -4.99 3.61 3.49 5.75 0.61
47 0.00 12.59 3.33 -1.39 -10.38 -2.03 2.83
48 0.00 1.96 11.31 -6.17 1.32 -9.47 4.96
49 0.00 -8.25 -5.52 -10.77 -11.95 7.87 7.00
50 0.00 5.96 2.31 8.83 -0.68 0.53 8.96
51 0.00 -3.93 9.84 4.12 10.14 -6.53 10.84
52 0.00 9.65 -5.99 -0.40 -2.53 9.76 -10.43
53 0.00 0.07 1.41 -4.76 7.92 2.79 -8.25
54 0.00 -9.16 8.54 -8.95 -4.24 -3.92 -6.16
55 0.00 3.77 -6.41 8.83 5.86 -10.38 -4.14
56 0.00 -5.19 0.60 4.54 -5.82 4.81 -2.19
57 0.00 7.22 7.37 0.40 3.95 -1.58 -0.31
58 0.00 -1.49 -6.79 -3.59 -7.30 -7.75 1.51
59 0.00 -9.91 -0.13 -7.45 2.17 6.63 3.26
60 0.00 1.96 6.31 8.83 -8.68 0.53 4.96
61 0.00 -6.24 -7.13 4.89 0.50 -5.37 6.59
62 0.00 5.18 -0.78 1.08 9.38 8.27 8.18
63 0.00 -2.81 5.36 -2.60 -1.06 2.43 -9.33
64 0.00 8.21 -7.44 -6.17 7.57 -3.22 -7.54
65 0.00 0.42 -1.38 8.83 -2.53 -8.70 -5.81
66 0.00 -7.14 4.50 5.19 5.86 4.16 -4.14
67 0.00 3.45 -7.72 1.66 -3.91 -1.26 -2.51
68 0.00 -3.93 -1.92 -1.76 4.26 -6.53 -0.93
69 0.00 6.30 3.71 -5.09 -5.20 5.75 0.61
70 0.00 -0.90 -7.97 -8.32 2.75 0.53 2.10
71 0.00 -7.90 -2.42 5.45 -6.43 -4.54 3.55
72 0.00 1.96 2.98 2.16 1.32 7.19 4.96
73 0.00 -4.89 -8.21 -1.04 -7.59 2.17 6.33
74 0.00 4.66 -2.88 -4.15 -0.03 -2.72 7.66
75 0.00 -2.04 2.31 -7.17 7.32 -7.47 -7.04
76 0.00 7.22 7.37 5.67 -1.31 3.69 -5.57
77 0.00 0.66 -3.30 2.59 5.86 -1.03 -4.14
78 0.00 -5.74 1.70 -0.40 -2.53 -5.63 -2.74
79 0.00 3.22 6.57 -3.33 4.48 5.08 -1.37
80 0.00 -3.04 -3.69 -6.17 -3.68 0.53 -0.04
81 0.00 5.66 1.13 5.86 3.17 -3.92 1.25
82 0.00 -0.48 5.83 2.97 -4.78 6.38 2.52
83 0.00 -6.48 -4.05 0.15 1.92 1.97 3.75
84 0.00 1.96 0.60 -2.60 -5.82 -2.33 4.96
85 0.00 -3.93 5.14 -5.29 0.73 -6.53 6.13
86 0.00 4.28 -4.38 6.04 -6.82 3.32 -6.67
87 0.00 -1.49 0.11 3.31 -0.41 -0.85 -5.39
88 0.00 6.50 4.50 0.64 5.86 -4.93 -4.14
89 0.00 0.83 -4.70 -1.96 -1.49 4.57 -2.91
90 0.00 -4.71 -0.35 -4.51 4.65 0.53 -1.71
91 0.00 3.05 3.90 6.19 -2.53 -3.43 -0.54
92 0.00 -2.39 -4.99 3.61 3.49 5.75 0.61
93 0.00 5.18 -0.78 1.08 -3.52 1.82 1.73
94 0.00 -0.17 3.33 -1.39 2.38 -2.03 2.83
95 0.00 -5.41 -5.27 -3.81 -4.47 -5.79 3.90
96 0.00 1.96 -1.19 -6.17 1.32 3.03 4.96
97 0.00 -3.20 2.81 3.88 -5.38 -0.71 5.99
98 0.00 4.00 -5.52 1.48 0.30 -4.37 -5.25
99 0.00 -1.08 -1.57 -0.87 5.86 4.16 -4.14
100 0.00 5.96 2.31 -3.17 -0.68 0.53 -3.04
------------------------------------------------------------

--
Hans Aberg
.