11-limit rational and 72-TET chords
Hans Aberg
not difficult to figure how to play these chords in 72-TET. Is the
following a known theory?
So the fifth is 3/2, as before. The specifically major keys are associated
with the partial 5, and the specifically minor keys with the partial 7.
The partial 11 is only used for the diminutive chord. One tries to avoid
5, 7, 11 denominators, and mixing these partials, the reason will be
understood in 72-TET, where these partials will have unique offsets
relative the nearest semitone (see below), thus such mixing not giving
anything new that cannot be done without it. This gives the following
chord combinations (M = major, m = minor):
chord intervals
M7 1 5/4 3/2 7/4
Mmaj7 1 5/4 3/2 3*5/8
m7 1 7/3*2 3/2 7/4
dim 1 7/3*2 11/8 3*5/8
As for the keys to play in 72-TET, the partials 2, 3 get duodecitone
(72-TET tonestep) offset 0 relative the nearest semitone, the partial 5
get duodecitone offset -1, the partial 7 duodecitone offset -2, and the
partial 11 get a -1 quartertone offset = -3 duodecitones. So one just has
to look at the table above, to get the following 72-TET table, where x_y
means lowering the semitone x with y 72-TET tonesteps:
chord intervals
M7 1 4_1 7 10_2
Mmaj7 1 4_1 7 11_1
m7 1 3_2 7 10_2
dim 1 3_2 6_3 9_1
I wonder what it might sound like. :-) It would be nice if somebody would
report back.
--
Hans Aberg
Bob Pease
"Hans Aberg" <hab...@math.su.se> wrote in message
news:haberg-0504...@c83-250-195-81.bredband.comhem.se...
This sounds interesting.
However, I feel that your use of terms is beyond the scope of almost all
poster/readers here.
Can you direct us to a source which deals in some basics of 12-tone equal
temperment.
Thanks
Bob Pease
Norman Clark Stewart Jr
wasn't he talking 72 not 12?
"Bob Pease" <Pope...@pope.com> wrote in message
news:e11fcq$j...@dispatch.concentric.net...
Hans Aberg
<Pope...@pope.com> wrote:
> However, I feel that your use of terms is beyond the scope of almost all
> poster/readers here.
The components have been extensively discussed here, though perhaps no recently.
> Can you direct us to a source which deals in some basics of 12-tone equal
> temperment.
http://en.wikipedia.org/wiki/Equal_temperament
http://en.wikipedia.org/wiki/72_tone_equal_temperament
http://en.wikipedia.org/wiki/Limit_%28music%29
What do you want to know?
--
Hans Aberg
Hans Aberg
Clark Stewart Jr" <geoba...@sbcglobal.net> wrote:
> 72 or 12?
> wasn't he talking 72 not 12?
You got both. :-) 12-TET is good for the 3-limit, but not higher n-limits,
which unfortunately excludes the partial 5 used in the Pythagorean scale
and Just intonation. 72-TET is good for the 11-limit, which makes it a
good candidate for a universal n-TET extending 12-TET. It divides each
semitone into six tonesteps.
--
Hans Aberg
Norman Clark Stewart Jr
Hans Aberg
Clark Stewart Jr" <geoba...@sbcglobal.net> 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.
--
Hans Aberg
Hans Aberg
standard 12-TET button keyboard layout, often used on chromatic
accordions, and which is the most efficient type of keyboard layouts.
This keyboard layout can also be used to visualize and understand the
72-TET system. First learn the positions of the 12-TET semitones, written
s0, ..., s12 below. Then view the 12-TET semitones as each divided into
six duodecitones. In the keyboard layout, one lowers the pitch in
duodecitones by going up-left. The keyboard could be extended by adding
duplicate rows, just as is common on 5-row accordions, as many as one
finds suitable. Right in the middle of each of these duodecitonesteps,
there is the quartertone. Adding the quartertones themselves only gives a
good rational approximation of the partial 11. So if one also get
the partials 5 and 7 well approximated, one has to add these duodecitones
as well.
Keyboard dudecitone vectors:
y^
\
-1 \
---->
9 x
y ^
8 | .. .. .. .. .. .. .. .. ..
7 | .. .. .. .. .. .. .. .. 3*5/8
6 | .. .. s2 .. s5 .. s8 .. s11 3/2 at s7
5 | .. .. .. .. .. .. .. 7/4 ..
4 | .. .. .. 5/4 .. .. .. .. ..
3 | .. s1 .. s4 11/8 s7 .. s10 ..
2 | .. .. 7/3*2 .. .. .. .. .. ..
1 | .. .. .. .. .. .. .. .. ..
0 | s0 .. s3 .. s6 .. s9 .. s12
------------------------------------------------------------->
0 1 2 3 4 5 6 7 8 x
where s = semitone (12-TET tonestep), and rational intervals are indicated
by fractions.
Rational chords:
chord intervals
M7 1 5/4 3/2 7/4
Mmaj7 1 5/4 3/2 3*5/8
m7 1 7/3*2 3/2 7/4
dim 1 7/3*2 11/8 3*5/8
72-TET chords (x_y denotes lowering the semitone x with y 72-TET tonesteps):
chord intervals
M7 1 4_1 7 10_2
Mmaj7 1 4_1 7 11_1
m7 1 3_2 7 10_2
dim 1 3_2 6_3 9_1
--
Hans Aberg
Bob Pease
"Hans Aberg" <hab...@math.su.se> wrote in message
news:haberg-0604...@c83-250-195-81.bredband.comhem.se...
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.
RJ P .
Bob Pease
Hans Aberg
<Pope...@pope.com> 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
Graham Breed
> I am experimenting with a theory for 11-limit rational chords, and it is
> not difficult to figure how to play these chords in 72-TET. Is the
> following a known theory?
11-limit harmony isn't generally used with traditional diatonic
thinking, so if you elaborate on this you can probably claim it as your
own.
> So the fifth is 3/2, as before. The specifically major keys are associated
> with the partial 5, and the specifically minor keys with the partial 7.
> The partial 11 is only used for the diminutive chord. One tries to avoid
> 5, 7, 11 denominators, and mixing these partials, the reason will be
> understood in 72-TET, where these partials will have unique offsets
> relative the nearest semitone (see below), thus such mixing not giving
> anything new that cannot be done without it. This gives the following
> chord combinations (M = major, m = minor):
> chord intervals
> M7 1 5/4 3/2 7/4
> Mmaj7 1 5/4 3/2 3*5/8
> m7 1 7/3*2 3/2 7/4
> dim 1 7/3*2 11/8 3*5/8
You're favouring otonal to utonal in Partch's terminology, and some of
us like that when you get to the 11-limit. It makes the notes simpler
subsets of the harmonic series, and maybe clearer if you write them as
extended ratios:
M7 4:5:6:7
Mmaj7 8:10:12:15
m7 12:14:18:21
dim 24:28:33:45
I don't see where this "diminutive" chord comes from, or why you need
to bring in 11 to get it. How about 15:18:21:28?
> As for the keys to play in 72-TET, the partials 2, 3 get duodecitone
> (72-TET tonestep) offset 0 relative the nearest semitone, the partial 5
> get duodecitone offset -1, the partial 7 duodecitone offset -2, and the
> partial 11 get a -1 quartertone offset = -3 duodecitones. So one just has
> to look at the table above, to get the following 72-TET table, where x_y
> means lowering the semitone x with y 72-TET tonesteps:
> chord intervals
> M7 1 4_1 7 10_2
> Mmaj7 1 4_1 7 11_1
> m7 1 3_2 7 10_2
> dim 1 3_2 6_3 9_1
>
> I wonder what it might sound like. :-) It would be nice if somebody would
> report back.
Why don't you try it? There will likely be problems with voice leading
in normal diatonic music, but perhaps you can work them out. 6:7:9
sounds like a bluesey minor third, at any rate.
Graham
Hans Aberg
Breed" <x31eq...@gmail.com> wrote:
> > I am experimenting with a theory for 11-limit rational chords, and it is
> > not difficult to figure how to play these chords in 72-TET. Is the
> > following a known theory?
>
> 11-limit harmony isn't generally used with traditional diatonic
> thinking, so if you elaborate on this you can probably claim it as your
> own.
I just used the principles: first, to avoid mixing the partial 5, 7, 11,
and second, to avoid using them in the denominators, if possible. The
musical motivation of these principles would be that it would somehow be
more harmonic, by avoiding inversions. Then 11 came into play simply
because it was available nearby. In addition, it causes the diminutive
chord to pleasingly use all partials. The main motivation, though, would
be the ears, and I do not know that yet. :-)
>
> > So the fifth is 3/2, as before. The specifically major keys are associated
> > with the partial 5, and the specifically minor keys with the partial 7.
> > The partial 11 is only used for the diminutive chord. One tries to avoid
> > 5, 7, 11 denominators, and mixing these partials, the reason will be
> > understood in 72-TET, where these partials will have unique offsets
> > relative the nearest semitone (see below), thus such mixing not giving
> > anything new that cannot be done without it. This gives the following
> > chord combinations (M = major, m = minor):
> > chord intervals
> > M7 1 5/4 3/2 7/4
> > Mmaj7 1 5/4 3/2 3*5/8
> > m7 1 7/3*2 3/2 7/4
> > dim 1 7/3*2 11/8 3*5/8
>
> You're favouring otonal to utonal in Partch's terminology, and some of
> us like that when you get to the 11-limit. It makes the notes simpler
> subsets of the harmonic series, and maybe clearer if you write them as
> extended ratios:
>
> M7 4:5:6:7
> Mmaj7 8:10:12:15
> m7 12:14:18:21
> dim 24:28:33:45
Yes, one can fiddle around with that. One hidden principle in use is that
the octaves are essentially equivalent. In the 73-TET, the partial 3 get 0
offset from the semitones, so it can be used almost as freely as octave
combinations, though higher exponents 3^k should be avoided.
> I don't see where this "diminutive" chord comes from, or why you need
> to bring in 11 to get it. How about 15:18:21:28?
I gave some motivations above, which you have read by now. In addition, in
the 72-TET, one can give good approximations of the partial 11, which
shows up as a quartertone offset from the semitone. So it seems natural,
in view of the motivations above, to put it in.
By no means I try to create a universal harmonic theory, that
should supersede other ones. I fiddled around with different principles,
and this is what I arrived at. It looked interesting. I would be happy if
it produces good sounds for use in music as well.
> > As for the keys to play in 72-TET, the partials 2, 3 get duodecitone
> > (72-TET tonestep) offset 0 relative the nearest semitone, the partial 5
> > get duodecitone offset -1, the partial 7 duodecitone offset -2, and the
> > partial 11 get a -1 quartertone offset = -3 duodecitones. So one just has
> > to look at the table above, to get the following 72-TET table, where x_y
> > means lowering the semitone x with y 72-TET tonesteps:
> > chord intervals
> > M7 1 4_1 7 10_2
> > Mmaj7 1 4_1 7 11_1
> > m7 1 3_2 7 10_2
> > dim 1 3_2 6_3 9_1
> >
> > I wonder what it might sound like. :-) It would be nice if somebody would
> > report back.
>
> Why don't you try it?
Right now, I do not have the musical software to do it.
> There will likely be problems with voice leading
> in normal diatonic music, but perhaps you can work them out. 6:7:9
> sounds like a bluesey minor third, at any rate.
There are problems with fitting the stuff into a scale, but this is a
problem of any rational based scale. I think that people in reality will
perform against certain chords, with a certain series of base notes. Then
the base notes may be fit into a scale. So one style of playing might be
to use some more standard scale, and then slip into these chords for
suitable musical color.
I am though only at the very beginning of the scale issue, which is a lot
more difficult than the chord issue. So starting off with suitable set of
chords seems to be the way to go.
--
Hans Aberg
Norman Clark Stewart Jr
with. I'm sure you've heard the term hamiltonian where math is added to a
formula that doesn't quite work. If it's been discussed before, I'm sorry,
but we have software not just piano forte.
YOu have ansered that equal temperment is the direction toward the future.
I don't see much justification for it to be. So, I was asking from you why
you think ET is worth
all these hamitonians?
Not to say I don't want to hear about your work...cus i do.
"Hans Aberg" <hab...@math.su.se> wrote in message
news:haberg-0604...@c83-250-195-81.bredband.comhem.se...
J. B. Wood
hab...@math.su.se (Hans Aberg) wrote:
> 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.
Hello, Hans, and I have to ask what you mean by "approximate well."
Perfect 5ths in 12-TET are only ~2 cents narrow of their just (3/2)
value. Major thirds OTOH are wide of their just (5/4)) value by ~14
cents. Like a lot of folks I have trouble discerning tones that are less
than a ET semitone (100 cents), my typical reaction being, "Well, that
sounds like a slighty sharp C or a slightly flat C#." Which is probably
one of the reasons why using more than 12 tones per octave has not caught
on (it just HAS to be a conspiracy of some sort ;-)). Capabilities of the
human singing voice must also be considered. The other reasons that come
to mind are acoustic instrument design, the desire for unlimited key
transposition and how to keep written music within manageable bounds. You
really have two discussions - the number of tones comprising the
octave/scale and how they are to be spaced.
Speaking of N-TET I did find Paul Erlich's Xenharmonikon paper "Tuning,
Tonality, and Twenty-Two-Tone Temperament" an interesting read. Don't
know if he ever built the piano-like keyboard described in that paper,
though. Paul has recorded pieces like "Decatonic Swing" that I find take
some getting used to. Sincerely,
John Wood (Code 5550) e-mail: wo...@itd.nrl.navy.mil
Naval Research Laboratory
4555 Overlook Avenue, SW
Washington, DC 20375-5337
Hans Aberg
the partial 2 should be approximated exactly, which leads to the set of
relative pitches 2^(k/n), where k is an integer. One can make a "partial p
n-TET", where the partial p is approximated exactly, which leads to the
set of relative pitches p^(k/n), where k is an integer. People
are experimenting with such n-TET's as well (see below). One can
generalize this further, by observing that the set of relative pitches so
produced constitute a (mathematical) subgroup of the group
of nonnegative invertible elements R^x_+ (in print, typically a
double-struck R with a superscript cross and a subscript plus) in the set
of real numbers R. The fact that the set of relative pitches
should constitute a subgroup is a reformulation of the musical property
that one should be able to play equally well in any key. Then the original
musical problem can be formulated as this: Given a set of relative pitches
P, find a subgroup of R^x_+ that approximates it well, according to some
norm of approximations.
As for the partial p n-TET, it does not seem to lead to anything desirable
from this general musical point of view. though it may have more special
uses. I post some approximation tables for p = 3, 5, so that you can check
if there is something useful to you. The tables are truncated at n = 100,
because larger numbers are less useful in a musical context. I have made
tables for p = 7, 11, 13, 17, as well, it is written by a program that
could can write for other values of p as well; but again, this is not so
useful in a musical context, it seems.
Offsets in cents of smallest rational approximation with denominator n
(= the number of tones in an equally tempered chromatic scale) of log_3 p:
n, p: 2 3 5 7 11 13 17
1 -442.88 0.00 557.97 -274.51 219.19 401.66 -505.32
2 157.12 0.00 -42.03 -274.51 219.19 -198.34 94.68
3 -42.88 0.00 157.97 125.49 -180.81 1.66 -105.32
4 -142.88 0.00 -42.03 25.49 -80.81 101.66 94.68
5 37.12 0.00 77.97 -34.51 -20.81 -78.34 -25.32
6 -42.88 0.00 -42.03 -74.51 19.19 1.66 94.68
7 71.40 0.00 43.68 68.35 47.76 58.80 8.97
8 7.12 0.00 -42.03 25.49 69.19 -48.34 -55.32
9 -42.88 0.00 24.63 -7.84 -47.48 1.66 28.02
10 37.12 0.00 -42.03 -34.51 -20.81 41.66 -25.32
11 -6.52 0.00 12.51 52.77 1.01 -34.70 40.14
12 -42.88 0.00 -42.03 25.49 19.19 1.66 -5.32
13 18.65 0.00 4.12 2.42 34.57 32.43 -43.78
14 -14.31 0.00 -42.03 -17.36 -37.95 -26.91 8.97
15 37.12 0.00 -2.03 -34.51 -20.81 1.66 -25.32
16 7.12 0.00 32.97 25.49 -5.81 26.66 19.68
17 -19.35 0.00 -6.74 7.85 7.43 -21.87 -11.20
18 23.78 0.00 24.63 -7.84 19.19 1.66 28.02
19 -0.78 0.00 -10.45 -21.88 29.72 22.71 -0.05
20 -22.88 0.00 17.97 25.49 -20.81 -18.34 -25.32
21 14.26 0.00 -13.46 11.21 -9.38 1.66 8.97
22 -6.52 0.00 12.51 -1.78 1.01 19.84 -14.41
23 -25.49 0.00 -15.94 -13.64 10.49 -15.73 16.42
24 7.12 0.00 7.97 -24.51 19.19 1.66 -5.32
25 -10.88 0.00 -18.03 13.49 -20.81 17.66 22.68
26 18.65 0.00 4.12 2.42 -11.58 -13.72 2.37
27 1.56 0.00 -19.81 -7.84 -3.03 1.66 -16.43
28 -14.31 0.00 0.83 -17.36 4.90 15.95 8.97
29 12.29 0.00 20.04 15.15 12.29 -12.13 -8.77
30 -2.88 0.00 -2.03 5.49 19.19 1.66 14.68
31 -17.08 0.00 16.03 -3.54 -13.07 14.56 -2.09
32 7.12 0.00 -4.53 -12.01 -5.81 -10.84 -17.82
33 -6.52 0.00 12.51 16.40 1.01 1.66 3.77
34 15.94 0.00 -6.74 7.85 7.43 13.43 -11.20
35 2.83 0.00 9.40 -0.22 13.48 -9.77 8.97
36 -9.55 0.00 -8.70 -7.84 -14.14 1.66 -5.32
37 11.17 0.00 6.62 -15.05 -7.84 12.47 13.60
38 -0.78 0.00 -10.45 9.70 -1.86 -8.87 -0.05
39 -12.12 0.00 4.12 2.42 3.81 1.66 -13.01
40 7.12 0.00 -12.03 -4.51 9.19 11.66 4.68
41 -3.86 0.00 1.87 -11.09 14.31 -8.10 -7.76
42 14.26 0.00 -13.46 11.21 -9.38 1.66 8.97
43 3.63 0.00 -0.17 4.56 -4.07 10.96 -2.99
44 -6.52 0.00 12.51 -1.78 1.01 -7.43 12.86
45 10.45 0.00 -2.03 -7.84 5.86 1.66 1.35
46 0.59 0.00 10.14 12.45 10.49 10.36 -9.67
47 -8.84 0.00 -3.73 6.34 -10.60 -6.85 5.32
48 7.12 0.00 7.97 0.49 -5.81 1.66 -5.32
49 -2.07 0.00 -5.30 -5.12 -1.22 9.82 8.97
50 -10.88 0.00 5.97 -10.51 3.19 -6.34 -1.32
51 4.17 0.00 -6.74 7.85 7.43 1.66 -11.20
52 -4.42 0.00 4.12 2.42 11.50 9.35 2.37
53 9.95 0.00 -8.07 -2.81 -7.23 -5.89 -7.20
54 1.56 0.00 2.41 -7.84 -3.03 1.66 5.79
55 -6.52 0.00 -9.30 9.13 1.01 8.93 -3.50
56 7.12 0.00 0.83 4.06 4.90 -5.48 8.97
57 -0.78 0.00 -10.45 -0.82 8.66 1.66 -0.05
58 -8.40 0.00 -0.65 -5.54 -8.40 8.56 -8.77
59 4.57 0.00 8.82 -10.10 -4.54 -5.12 3.16
60 -2.88 0.00 -2.03 5.49 -0.81 1.66 -5.32
61 9.57 0.00 7.15 0.90 2.80 8.22 6.16
62 2.28 0.00 -3.32 -3.54 6.29 -4.79 -2.09
63 -4.79 0.00 5.59 -7.84 -9.38 1.66 8.97
64 7.12 0.00 -4.53 6.74 -5.81 7.91 0.93
65 0.19 0.00 4.12 2.42 -2.35 -4.49 -6.86
66 -6.52 0.00 -5.67 -1.78 1.01 1.66 3.77
67 4.88 0.00 2.74 -5.85 4.26 7.63 -3.83
68 -1.71 0.00 -6.74 7.85 7.43 -4.22 6.45
69 -8.10 0.00 1.45 3.75 -6.90 1.66 -0.97
70 2.83 0.00 -7.75 -0.22 -3.67 7.38 -8.17
71 -3.45 0.00 0.22 -4.08 -0.53 -3.97 1.72
72 7.12 0.00 7.97 -7.84 2.52 1.66 -5.32
73 0.95 0.00 -0.94 4.94 5.49 7.14 4.27
74 -5.05 0.00 6.62 1.17 -7.84 -3.74 -2.61
75 5.12 0.00 -2.03 -2.51 -4.81 1.66 6.68
76 -0.78 0.00 5.34 -6.09 -1.86 6.92 -0.05
77 -6.52 0.00 -3.07 6.01 1.01 -3.53 -6.62
78 3.27 0.00 4.12 2.42 3.81 1.66 2.37
79 -2.38 0.00 -4.06 -1.09 6.53 6.72 -4.05
80 7.12 0.00 2.97 -4.51 -5.81 -3.34 4.68
81 1.56 0.00 -4.99 6.97 -3.03 1.66 -1.61
82 -3.86 0.00 1.87 3.54 -0.32 6.54 6.88
83 5.31 0.00 -5.89 0.19 2.32 -3.16 0.71
84 -0.03 0.00 0.83 -3.08 4.90 1.66 -5.32
85 -5.24 0.00 -6.74 -6.27 -6.69 6.37 2.92
86 3.63 0.00 -0.17 4.56 -4.07 -2.99 -2.99
87 -1.50 0.00 6.24 1.35 -1.50 1.66 5.03
88 -6.52 0.00 -1.12 -1.78 1.01 6.21 -0.77
89 2.06 0.00 5.16 -4.84 3.46 -2.83 -6.44
90 -2.88 0.00 -2.03 5.49 5.86 1.66 1.35
91 5.47 0.00 4.12 2.42 -4.99 6.06 -4.22
92 0.59 0.00 -2.90 -0.59 -2.55 -2.69 3.38
93 -4.17 0.00 3.13 -3.54 -0.16 1.66 -2.09
94 3.92 0.00 -3.73 6.34 2.17 5.92 5.32
95 -0.78 0.00 2.18 3.39 4.45 -2.55 -0.05
96 -5.38 0.00 -4.53 0.49 -5.81 1.66 -5.32
97 2.48 0.00 1.27 -2.34 -3.49 5.78 1.90
98 -2.07 0.00 -5.30 -5.12 -1.22 -2.42 -3.28
99 5.60 0.00 0.39 4.28 1.01 1.66 3.77
100 1.12 0.00 5.97 1.49 3.19 5.66 -1.32
Offsets in cents of smallest rational approximation with denominator n
(= the number of tones in an equally tempered chromatic scale) of log_5 p:
n, p: 2 3 5 7 11 13 17
1 516.81 -380.87 0.00 250.87 587.88 -487.57 -287.55
2 -83.19 219.13 0.00 250.87 -12.12 112.43 -287.55
3 116.81 19.13 0.00 -149.13 187.88 -87.57 112.45
4 -83.19 -80.87 0.00 -49.13 -12.12 112.43 12.45
5 36.81 99.13 0.00 10.87 107.88 -7.57 -47.55
6 -83.19 19.13 0.00 50.87 -12.12 -87.57 -87.55
7 2.53 -38.02 0.00 79.45 73.59 26.72 55.31
8 66.81 69.13 0.00 -49.13 -12.12 -37.57 12.45
9 -16.52 19.13 0.00 -15.79 54.54 45.76 -20.88
10 36.81 -20.87 0.00 10.87 -12.12 -7.57 -47.55
11 -28.64 -53.60 0.00 32.69 42.42 -51.21 39.72
12 16.81 19.13 0.00 -49.13 -12.12 12.43 12.45
13 -37.03 -11.64 0.00 -26.05 34.03 -26.03 -10.63
14 2.53 -38.02 0.00 -6.27 -12.12 26.72 -30.41
15 36.81 19.13 0.00 10.87 27.88 -7.57 32.45
16 -8.19 -5.87 0.00 25.87 -12.12 37.43 12.45
17 22.69 -27.93 0.00 -31.48 23.17 6.55 -5.20
18 -16.52 19.13 0.00 -15.79 -12.12 -20.90 -20.88
19 11.55 -1.93 0.00 -1.76 19.45 17.69 28.24
20 -23.19 -20.87 0.00 10.87 -12.12 -7.57 12.45
21 2.53 19.13 0.00 22.30 16.45 26.72 -1.84
22 25.90 0.95 0.00 -21.85 -12.12 3.34 -14.82
23 -4.93 -15.66 0.00 -10.00 13.96 -18.00 25.49
24 16.81 19.13 0.00 0.87 -12.12 12.43 12.45
25 -11.19 3.13 0.00 10.87 11.88 -7.57 0.45
26 9.12 -11.64 0.00 20.11 -12.12 20.12 -10.63
27 -16.52 19.13 0.00 -15.79 10.10 1.32 -20.88
28 2.53 4.84 0.00 -6.27 -12.12 -16.14 12.45
29 20.26 -8.46 0.00 2.60 8.56 8.98 2.10
30 -3.19 19.13 0.00 10.87 -12.12 -7.57 -7.55
31 13.59 6.22 0.00 18.62 7.23 15.66 -16.58
32 -8.19 -5.87 0.00 -11.63 -12.12 -0.07 12.45
33 7.72 -17.24 0.00 -3.67 6.06 -14.84 3.36
34 -12.60 7.36 0.00 3.82 -12.12 6.55 -5.20
35 2.53 -3.73 0.00 10.87 5.02 -7.57 -13.26
36 -16.52 -14.21 0.00 -15.79 -12.12 12.43 12.45
37 -2.11 8.32 0.00 -8.59 4.09 -1.08 4.34
38 11.55 -1.93 0.00 -1.76 -12.12 -13.88 -3.34
39 -6.27 -11.64 0.00 4.72 3.26 4.74 -10.63
40 6.81 9.13 0.00 10.87 -12.12 -7.57 12.45
41 -10.02 -0.38 0.00 -12.54 2.51 9.99 5.13
42 2.53 -9.44 0.00 -6.27 -12.12 -1.85 -1.84
43 -13.42 9.83 0.00 -0.29 1.83 -13.15 -8.48
44 -1.37 0.95 0.00 5.42 -12.12 3.34 12.45
45 10.15 -7.54 0.00 10.87 1.21 -7.57 5.78
46 -4.93 10.43 0.00 -10.00 -12.12 8.08 -0.59
47 6.17 2.11 0.00 -4.44 0.64 -2.46 -6.70
48 -8.19 -5.87 0.00 0.87 -12.12 12.43 12.45
49 2.53 10.96 0.00 5.98 0.12 2.23 6.33
50 -11.19 3.13 0.00 10.87 11.88 -7.57 0.45
51 -0.84 -4.40 0.00 -7.95 -0.36 6.55 -5.20
52 9.12 11.44 0.00 -2.97 10.95 -2.95 -10.63
53 -3.94 4.03 0.00 1.82 -0.80 10.54 6.79
54 5.70 -3.09 0.00 6.43 10.10 1.32 1.34
55 -6.82 -9.96 0.00 10.87 -1.22 -7.57 -3.91
56 2.53 4.84 0.00 -6.27 9.30 5.29 -8.98
57 -9.50 -1.93 0.00 -1.76 -1.60 -3.36 7.19
58 -0.43 -8.46 0.00 2.60 8.56 8.98 2.10
59 8.34 5.57 0.00 6.81 -1.96 0.57 -2.80
60 -3.19 -0.87 0.00 -9.13 7.88 -7.57 -7.55
61 5.34 -7.10 0.00 -4.86 -2.29 4.23 7.53
62 -5.77 6.22 0.00 -0.74 7.23 -3.70 2.77
63 2.53 0.08 0.00 3.26 -2.60 7.67 -1.84
64 -8.19 -5.87 0.00 7.12 6.63 -0.07 -6.30
65 -0.11 6.82 0.00 -7.59 -2.89 -7.57 7.83
66 7.72 0.95 0.00 -3.67 6.06 3.34 3.36
67 -2.59 -4.75 0.00 0.13 -3.17 -3.99 -0.98
68 5.05 7.36 0.00 3.82 5.52 6.55 -5.20
69 -4.93 1.74 0.00 7.40 -3.43 -0.61 8.10
70 2.53 -3.73 0.00 -6.27 5.02 -7.57 3.88
71 -7.13 7.86 0.00 -2.65 -3.67 2.57 -0.23
72 0.15 2.46 0.00 0.87 4.54 -4.24 -4.22
73 7.22 -2.79 0.00 4.30 -3.91 5.58 -8.10
74 -2.11 -7.90 0.00 7.63 4.09 -1.08 4.34
75 4.81 3.13 0.00 -5.13 -4.12 -7.57 0.45
76 -4.24 -1.93 0.00 -1.76 3.66 1.90 -3.34
77 2.53 -6.85 0.00 1.52 -4.33 -4.45 -7.03
78 -6.27 3.74 0.00 4.72 3.26 4.74 4.76
79 0.36 -1.13 0.00 -7.35 -4.53 -1.49 1.06
80 6.81 -5.87 0.00 -4.13 2.88 7.43 -2.55
81 -1.71 4.31 0.00 -0.98 -4.72 1.32 -6.07
82 4.62 -0.38 0.00 2.09 2.51 -4.64 5.13
83 -3.67 -4.97 0.00 5.09 -4.90 4.00 1.61
84 2.53 4.84 0.00 -6.27 2.16 -1.85 -1.84
85 -5.54 0.30 0.00 -3.24 -5.07 6.55 -5.20
86 0.53 -4.13 0.00 -0.29 1.83 0.80 5.47
87 6.47 5.33 0.00 2.60 -5.23 -4.81 2.10
88 -1.37 0.95 0.00 5.42 1.51 3.34 -1.19
89 4.45 -3.34 0.00 -5.31 -5.38 -2.18 -4.40
90 -3.19 5.79 0.00 -2.46 1.21 5.76 5.78
91 2.53 1.55 0.00 0.32 -5.53 0.34 2.56
92 -4.93 -2.61 0.00 3.05 0.92 -4.96 -0.59
93 0.68 6.22 0.00 5.71 -5.67 2.75 -3.68
94 6.17 2.11 0.00 -4.44 0.64 -2.46 6.07
95 -1.08 -1.93 0.00 -1.76 -5.81 5.06 2.98
96 4.31 -5.87 0.00 0.87 0.38 -0.07 -0.05
97 -2.78 2.63 0.00 3.45 -5.94 -5.09 -3.01
98 2.53 -1.28 0.00 5.98 0.12 2.23 -5.92
99 -4.40 -5.11 0.00 -3.67 6.06 -2.72 3.36
100 0.81 3.13 0.00 -1.13 -0.12 4.43 0.45
--
Hans Aberg
Hans Aberg
wo...@itd.nrl.navy.mil (J. B. Wood) wrote:
> > 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.
>
> Hello, Hans, and I have to ask what you mean by "approximate well."
According to some norm you decide. Different norms will yield different results.
> Perfect 5ths in 12-TET are only ~2 cents narrow of their just (3/2)
> value.
It is the partial 5, which is a major third, I am speaking about.
> Major thirds OTOH are wide of their just (5/4)) value by ~14
> cents.
Right. So these are a duodecitone off relative 12-TET. In 72-TET, the
partials 2, 3, 5, 7, 11 are approximated within 3 cents.
> Like a lot of folks I have trouble discerning tones that are less
> than a ET semitone (100 cents), my typical reaction being, "Well, that
> sounds like a slighty sharp C or a slightly flat C#."
And to this add all the musical pitchbending.
> Which is probably
> one of the reasons why using more than 12 tones per octave has not caught
> on (it just HAS to be a conspiracy of some sort ;-)).
What do you mean here? People do play a lot out of the 12-TET, via
pitchbending, scalestretching and various other types of reference scales.
The question is really: are there people actually playing 12-TET (see
below)? :-)
> Capabilities of the
> human singing voice must also be considered. The other reasons that come
> to mind are acoustic instrument design, the desire for unlimited key
> transposition and how to keep written music within manageable bounds. You
> really have two discussions - the number of tones comprising the
> octave/scale and how they are to be spaced.
It seems that 12-TET, in actual musical performance, is only used as a key
reference pitches, used for key transpositions. As long musicians are not
playing in unison, it seems that different pitch systems can live together
in the same performance.
> Speaking of N-TET I did find Paul Erlich's Xenharmonikon paper "Tuning,
> Tonality, and Twenty-Two-Tone Temperament" an interesting read. Don't
> know if he ever built the piano-like keyboard described in that paper,
> though. Paul has recorded pieces like "Decatonic Swing" that I find take
> some getting used to.
Do you have URL with a picture of the keyboard?
The only keyboard layout that seems feasible for the 72-TET is the one I
posted. With this one, one can on the one hand develop a performance on
a traditional chromatic button keyboard. Then, when switching to 72-TET,
the new performance will merely be access to some key offsets, but
fingering will otherwise be the same.
I can also think of a traditional chromatic button keyboard, where one
adds additional key switches selecting the 72-TET offsets.
--
Hans Aberg
Hans Aberg
Clark Stewart Jr" <geoba...@sbcglobal.net> wrote:
> I'm enjoying your posts and want to learn more about what you are working
> with. I'm sure you've heard the term hamiltonian where math is added to a
> formula that doesn't quite work. If it's been discussed before, I'm sorry,
> but we have software not just piano forte.
>
> YOu have ansered that equal temperment is the direction toward the future.
Not really. It's there as a basis for key selection and modulation
(transposition) in musical performance, it seems. A musical performance
contains a lot of other things.
> I don't see much justification for it to be. So, I was asking from you why
> you think ET is worth
> all these hamitonians?
I do not know of any relevance of Hamiltonians in this context.
--
Hans Aberg
J. B. Wood
> > Which is probably
> > one of the reasons why using more than 12 tones per octave has not caught
> > on (it just HAS to be a conspiracy of some sort ;-)).
>
> What do you mean here? People do play a lot out of the 12-TET, via
> pitchbending, scalestretching and various other types of reference scales.
> The question is really: are there people actually playing 12-TET (see
> below)? :-)
>
Hello, and again one has to mention what the variation is and what people
will tolerate. The inharmonicity of piano strings results in the
stretching of octaves in order to keep the beats of the partials at the
rate required by ET. The unisons are not exactly at the calculated ET
pitch and can be off as much as 15-20 cents at the edge of the bass and
treble even though the piano is considered to be in tune. So in the piano
case I guess you could say it's not exactly ET. But it's close enough and
that's the whole point. Even if you consider an ideal P5 and M3 to have
pitch ratios of 3/2 and 5/4, respectively, the human ear/brain allows a
generous +/- variance from these ideals. Personally I'm quite happy with
"bright" ET major thirds but a Pythagorean major third (~408 cents) I find
harsh.
> It seems that 12-TET, in actual musical performance, is only used as a key
> reference pitches, used for key transpositions. As long musicians are not
> playing in unison, it seems that different pitch systems can live together
> in the same performance.
>
Sure, as long as we don't stray too far in pitch between
instruments/voices that are intended to be in unison. But it applies to
harmony as well. Which makes incorporating Gamelan instruments into
Western orchestras a wee bit problematic.
> Do you have URL with a picture of the keyboard?
>
> The only keyboard layout that seems feasible for the 72-TET is the one I
> posted. With this one, one can on the one hand develop a performance on
> a traditional chromatic button keyboard. Then, when switching to 72-TET,
> the new performance will merely be access to some key offsets, but
> fingering will otherwise be the same.
>
Try this link, Hans:
http://eceserv0.ece.wisc.edu/~sethares/paperspdf/erlich-22.pdf
Hans Aberg
wo...@itd.nrl.navy.mil (J. B. Wood) wrote:
> > > Which is probably
> > > one of the reasons why using more than 12 tones per octave has not caught
> > > on (it just HAS to be a conspiracy of some sort ;-)).
> >
> > What do you mean here? People do play a lot out of the 12-TET, via
> > pitchbending, scalestretching and various other types of reference scales.
> > The question is really: are there people actually playing 12-TET (see
> > below)? :-)
> Hello, and again one has to mention what the variation is and what people
> will tolerate.
Music performers will just choose those factors according to musical
tradition, context and needs, and will not know it in the scientific
manner you propose.
> The inharmonicity of piano strings results in the
> stretching of octaves in order to keep the beats of the partials at the
> rate required by ET.
That is just a theory which hardly can be right, as the same kind of piano
can be tuned with several different kinds of scalestretching, depending on
the music style it should support. For example, a concert grand piano, for
use in an orchestra will have a no-stretch area in the middle, with some
stretch towards the bass and the treble. It will not fit any mathematical
formula, but will be tuned just according some musical needs.
> The unisons are not exactly at the calculated ET
> pitch and can be off as much as 15-20 cents at the edge of the bass and
> treble even though the piano is considered to be in tune. So in the piano
> case I guess you could say it's not exactly ET.
In addition, I think the pitch will change with how hard the keys are
struck. When a piano is tuned, I would think one has to decide how hard
the keys should be struck. The music performer will choose this factor in
coordination with other factors when deciding how hard the key should be
struck.
> But it's close enough and
> that's the whole point.
In a music performance, one does not try to play the pitch "close enough",
but one is using the scale pitch as a mental reference pitch. WHen
training for learning the scale pitches, one plays a piece, and stops at
critical point and tries to figure out, say using chromatic tuner or a
reference musician or another music instrument, what may be right, but one
is not in the actual performance using any such tools. So then the
musician will have to rely on intuition, and it is not clear what is
actually being played.
> Even if you consider an ideal P5 and M3 to have
> pitch ratios of 3/2 and 5/4, respectively, the human ear/brain allows a
> generous +/- variance from these ideals.
The pitch differences need, in order to be heard as absolute, independent
pitches, to be something like 4 cents. (If played simultaneously, much
less can be detected through beats; but that is not discussed here.).
A duodecitone is 100/6 cents, or about 17 cents, which is many times that.
So it should be quite significant for a trained musician. But it will the
ears that decide, of course.
> Personally I'm quite happy with
> "bright" ET major thirds but a Pythagorean major third (~408 cents) I find
> harsh.
One can probably get used to whatever in music style. Then even other
forms may sound wrong, especially in one is just working with one or a few
styles. But there are a lot of variation out there, not only in regional
and ancient music, but probably also in music that is thought to be
12-TET: As long as a musicians in a performance do not play in unison,
which is quite rare except for certain groups of music instruments, they
can quite freely choose factors such as pitch bends, scalestretch, and
even scale.
> > It seems that 12-TET, in actual musical performance, is only used as a key
> > reference pitches, used for key transpositions. As long musicians are not
> > playing in unison, it seems that different pitch systems can live together
> > in the same performance.
> Sure, as long as we don't stray too far in pitch between
> instruments/voices that are intended to be in unison. But it applies to
> harmony as well. Which makes incorporating Gamelan instruments into
> Western orchestras a wee bit problematic.
You have an article here:
http://en.wikipedia.org/wiki/Gamelan#Tuning
One can note that the gamelan practice of tuning instruments in pairs a
bit off in order to produce beats is used in modern accordions. And if the
an gamelan orchestra tries in its particular tuning, intuitively tries
to match the the partials 2, 3, 7, 11, then that can be approximated well
using 72-TET. One has to check some gamelan orchestra tunings, though. But
with some luck, perhaps 72-TET can be used to describe gamelan tunings,
though it has to be augmented with some scale stretch descriptions.
> > Do you have URL with a picture of the keyboard?
> >
> > The only keyboard layout that seems feasible for the 72-TET is the one I
> > posted. With this one, one can on the one hand develop a performance on
> > a traditional chromatic button keyboard. Then, when switching to 72-TET,
> > the new performance will merely be access to some key offsets, but
> > fingering will otherwise be the same.
> >
>
> Try this link, Hans:
>
> http://eceserv0.ece.wisc.edu/~sethares/paperspdf/erlich-22.pdf
Thank you, I got it. I will check it out.
I think, though, that at least one type of 72-TET performance will be
merely based on 12-TET, but with suitable duodecitone alterations for
suitable musical colorations. This way, one will avoid developing an
elaborate theory for the full 72-TET. But this would only be one musical
style, clearly. Others may want to do something more complicated.
--
Hans Aberg
Graham Breed
> In article <1144328738....@i39g2000cwa.googlegroups.com>, "Graham
> Breed" <x31eq...@gmail.com> wrote:
>
> > > I am experimenting with a theory for 11-limit rational chords, and it is
> > > not difficult to figure how to play these chords in 72-TET. Is the
> > > following a known theory?
> >
> > 11-limit harmony isn't generally used with traditional diatonic
> > thinking, so if you elaborate on this you can probably claim it as your
> > own.
>
> I just used the principles: first, to avoid mixing the partial 5, 7, 11,
> and second, to avoid using them in the denominators, if possible. The
> musical motivation of these principles would be that it would somehow be
> more harmonic, by avoiding inversions. Then 11 came into play simply
> because it was available nearby. In addition, it causes the diminutive
> chord to pleasingly use all partials. The main motivation, though, would
> be the ears, and I do not know that yet. :-)
So first you want to avoid mixing partials, and then you pleasingly use
all partials?
You mean 72-TET?
> > I don't see where this "diminutive" chord comes from, or why you need
> > to bring in 11 to get it. How about 15:18:21:28?
>
> I gave some motivations above, which you have read by now. In addition, in
> the 72-TET, one can give good approximations of the partial 11, which
> shows up as a quartertone offset from the semitone. So it seems natural,
> in view of the motivations above, to put it in.
If you're building chords with thirds, the 11-limit will typically give
you neutral thirds, but you aren't using these. They will, of course,
drag you far from the 12-TET skeleton. 7:9:11 makes a good substitute
for an augmented triad, for example. There are also some diminished
sevenths in 72-TET that don't translate properly into just intonation
(hence giving a reason to use an equal temperament in the first place).
And how about 6:7:8:11?
> By no means I try to create a universal harmonic theory, that
> should supersede other ones. I fiddled around with different principles,
> and this is what I arrived at. It looked interesting. I would be happy if
> it produces good sounds for use in music as well.
There are lots of sounds out there. What sounds good depends on the
context you use them in. If factors of 11 are only used as
elaborations on a diatonic framework, it's difficult to get them to
work melodically.
> > > I wonder what it might sound like. :-) It would be nice if somebody would
> > > report back.
> >
> > Why don't you try it?
>
> Right now, I do not have the musical software to do it.
It really isn't that difficult to get hold of. There's Csound, Scala,
my own MIDI Relay, and a load of soft synths. See if you can get this
site to work:
http://www.microtonal-synthesis.com/
> > There will likely be problems with voice leading
> > in normal diatonic music, but perhaps you can work them out. 6:7:9
> > sounds like a bluesey minor third, at any rate.
>
> There are problems with fitting the stuff into a scale, but this is a
> problem of any rational based scale. I think that people in reality will
> perform against certain chords, with a certain series of base notes. Then
> the base notes may be fit into a scale. So one style of playing might be
> to use some more standard scale, and then slip into these chords for
> suitable musical color.
There are scales which are designed to handle different chords, like
the decimal tuning mentioned in these parts not long ago (and
compatible with 72-TET). It does make the melodies more comprehensible
to have some idea of a base scale. You're obviously taking a
chords-first rather than melody-first approach here, which may be
because you're starting from the theory rather than sound.
One interesting idea is that some chord progressions that don't work in
5-limit meantone do work if you use the occasional 6:7:9 and its
reflection. I don't think anybody's really explored it yet. And
there's no need for 11.
> I am though only at the very beginning of the scale issue, which is a lot
> more difficult than the chord issue. So starting off with suitable set of
> chords seems to be the way to go.
Starting off with being able to hear what you're doing is the way to
go!
Graham
Graham Breed
> > Speaking of N-TET I did find Paul Erlich's Xenharmonikon paper "Tuning,
> > Tonality, and Twenty-Two-Tone Temperament" an interesting read. Don't
> > know if he ever built the piano-like keyboard described in that paper,
> > though. Paul has recorded pieces like "Decatonic Swing" that I find take
> > some getting used to.
>
> Do you have URL with a picture of the keyboard?
There's a simple picture here:
http://www.microtonal.co.uk/instrum.htm#keyboard
I lost the pictures of Kalle's keyboard. The must have disappeared
when I lost my web host a couple of years back.
Graham
Hans Aberg
"Graham Breed" <x31eq...@gmail.com> wrote:
> > I just used the principles: first, to avoid mixing the partial 5, 7, 11,
> > and second, to avoid using them in the denominators, if possible. The
> > musical motivation of these principles would be that it would somehow be
> > more harmonic, by avoiding inversions. Then 11 came into play simply
> > because it was available nearby. In addition, it causes the diminutive
> > chord to pleasingly use all partials. The main motivation, though, would
> > be the ears, and I do not know that yet. :-)
>
> So first you want to avoid mixing partials, and then you pleasingly use
> all partials?
I avoided them in the same chord interval, but used the different partials
for different intervals in the same chord. The motivation is how the chord
should be used in music performance: There, it is common to switch between
3/2 intervals. So the basic chord with just two tones seems should be 1,
3/2. Then the partials 5, 7 came in naturally for major and minor triads
and tetrads. But adding these fro chord alteration scale runs into
problems. After that, 11 came in as a close, simple approximation for
the diminished chord.
> > In the 73-TET, the partial 3 get 0
> > offset from the semitones, so it can be used almost as freely as octave
> > combinations, though higher exponents 3^k should be avoided.
>
> You mean 72-TET?
Yes, in 72-TET, the partial 3 isn't exact, so the offset will grow towards
a duodecitone offset. So therefore, I reasoned in the construction that
small positive and negative k would be OK, but not larger ones. This then
promotes using the partial 5, and 7 instead, whenever possible. Of course,
this is not a formal theory, but just some empiric.
> > > I don't see where this "diminutive" chord comes from, or why you need
> > > to bring in 11 to get it. How about 15:18:21:28?
> >
> > I gave some motivations above, which you have read by now. In addition, in
> > the 72-TET, one can give good approximations of the partial 11, which
> > shows up as a quartertone offset from the semitone. So it seems natural,
> > in view of the motivations above, to put it in.
>
> If you're building chords with thirds, the 11-limit will typically give
> you neutral thirds, but you aren't using these.
One reason that I avoided 11 in 72-TET is that it is a full quartertone,
or 3 duodecitones, so that one cannot distinguish positive and negative k
in 11^k.
> They will, of course,
> drag you far from the 12-TET skeleton. 7:9:11 makes a good substitute
> for an augmented triad, for example. There are also some diminished
> sevenths in 72-TET that don't translate properly into just intonation
> (hence giving a reason to use an equal temperament in the first place).
>
> And how about 6:7:8:11?
The choice will depends on how one sees the diminished cord, and how it
should be used in musical performance. When I made the variation, I
thought of it as an alteration of the minor chord. I do not claim that
there should be a unique diminished chord. Perhaps more than one should be
used depending on musical context. I will have to think of some musical
example using diminished chords fro this one.
> > By no means I try to create a universal harmonic theory, that
> > should supersede other ones. I fiddled around with different principles,
> > and this is what I arrived at. It looked interesting. I would be happy if
> > it produces good sounds for use in music as well.
>
> There are lots of sounds out there. What sounds good depends on the
> context you use them in.
Right.
> If factors of 11 are only used as
> elaborations on a diatonic framework, it's difficult to get them to
> work melodically.
The partial 11 is also rather high as number, so it may not have played
not so much role musically. For the diminished chord construction, I just
put it in, because it was close-by. Of course, only the ears can tell what
is right.
> > > > I wonder what it might sound like. :-) It would be nice if
somebody would
> > > > report back.
> > >
> > > Why don't you try it?
> >
> > Right now, I do not have the musical software to do it.
>
> It really isn't that difficult to get hold of. There's Csound, Scala,
> my own MIDI Relay, and a load of soft synths. See if you can get this
> site to work:
>
> http://www.microtonal-synthesis.com/
Thank you. I got it.
> > > There will likely be problems with voice leading
> > > in normal diatonic music, but perhaps you can work them out. 6:7:9
> > > sounds like a bluesey minor third, at any rate.
> >
> > There are problems with fitting the stuff into a scale, but this is a
> > problem of any rational based scale. I think that people in reality will
> > perform against certain chords, with a certain series of base notes. Then
> > the base notes may be fit into a scale. So one style of playing might be
> > to use some more standard scale, and then slip into these chords for
> > suitable musical color.
>
> There are scales which are designed to handle different chords, like
> the decimal tuning mentioned in these parts not long ago (and
> compatible with 72-TET). It does make the melodies more comprehensible
> to have some idea of a base scale. You're obviously taking a
> chords-first rather than melody-first approach here, which may be
> because you're starting from the theory rather than sound.
No, when trying to make a scale, I found it was easier to first focus on
chords only first.
Then I was lead to thoughts abut how performance is actually being done, a
thought process that is not yet finished. Part of the input is from music
that mixes 12-TET musical instruments with scales that do not fit into
this picture, for example, from Balkan. Perhaps this might be an example:
http://www.balkanfolk.com/cd_185500
A music performance seems to have different components, like a key to
coordinate against, chords that form a background carpet, and one or at
most a few music lines in the foreground. Then these components need not
be strictly coordinated within one single theoretical framework, if they
do not clash too much, it seems. So this opens up for more possibilities
than what is possible within a single theoretical framework.
> One interesting idea is that some chord progressions that don't work in
> 5-limit meantone do work if you use the occasional 6:7:9 and its
> reflection. I don't think anybody's really explored it yet. And
> there's no need for 11.
As for 72-TET, I think the partials 5, 7 will be in most use, because they
do get a full quartertone offset, and therefore positive and negative
exponents will be distinguishable, As for 11, it gets a full quartertone
offset. Take 11 away, and I think that full quartertone offsets will be
problematic when describing natural music that has offsets relative
12-TET. So also that suggests that the partial 11 will have limited use.
> > I am though only at the very beginning of the scale issue, which is a lot
> > more difficult than the chord issue. So starting off with suitable set of
> > chords seems to be the way to go.
>
> Starting off with being able to hear what you're doing is the way to
> go!
Sure. Most of all, I would want to have the t2-TET chromatic keyboard I
described, as it will admit hands-on experimentation and experience. I can
though imagine the kind of music performances I would make on such a
keyboard, and to some extent imagine what it would be like. But nothing
would be like the real thing! :-)
--
Hans Aberg
Hans Aberg
Breed" <x31eq...@gmail.com> wrote:
> > > Speaking of N-TET I did find Paul Erlich's Xenharmonikon paper "Tuning,
> > > Tonality, and Twenty-Two-Tone Temperament" an interesting read. Don't
> > > know if he ever built the piano-like keyboard described in that paper,
> > > though. Paul has recorded pieces like "Decatonic Swing" that I find take
> > > some getting used to.
> >
> > Do you have URL with a picture of the keyboard?
>
> There's a simple picture here:
>
> http://www.microtonal.co.uk/instrum.htm#keyboard
>
> I lost the pictures of Kalle's keyboard. The must have disappeared
> when I lost my web host a couple of years back.
Thank you. I saw it. The chromatic button keyboard layout I posted has the
advantage of extending the common 12-TET chromatic button keyboard layout
in a way that music performances can be ported fairly simple between
12-TET and 72-TET - both directions should work. In addition, chromatic
button keyboards have the advantage over piano keyboards of being more
compact, in addition to requiring fewer fingering patterns to be learned.
And when extended with additional duplicate rows, transpositions can be
made without any fingering changes at all. So, when moving to 72-TET, I
think those advantages will grow, making a 72-TET piano keyboard less
usable in practial musical performance. But it is best to investigate all
possibilities.
--
Hans Aberg
Bob Pease
"Graham Breed" <x31eq...@gmail.com> wrote in message
news:1144384392....@u72g2000cwu.googlegroups.com...
I find microtonal.co very instructive at a level which is accessable to
anyone with some foundation in the basics of music theory.
Thanks
Bob Pease
J. B. Wood
hab...@math.su.se (Hans Aberg) wrote:
> > The inharmonicity of piano strings results in the
> > stretching of octaves in order to keep the beats of the partials at the
> > rate required by ET.
>
> That is just a theory which hardly can be right, as the same kind of piano
> can be tuned with several different kinds of scalestretching, depending on
> the music style it should support. For example, a concert grand piano, for
> use in an orchestra will have a no-stretch area in the middle, with some
> stretch towards the bass and the treble. It will not fit any mathematical
> formula, but will be tuned just according some musical needs.
It's not a theory, Hans. Stretch AFAIK is a byproduct of inharmonicity.
IOW you'd rather not have it but vibrating string mechanics dictates
otherwise. The goal is to adjust the beats of partials (overtones) of an
interval (usually a fifth or third) to a rate that is consistent with the
tuning/temperament desired. Piano tuners thus establish a "temperament
octave" in the piano mid-range. The remaining notes are then tuned by
octaves to their corresponding ones in the temperament octave and this
technique results in the stretch. There's no mathematical formula that
dictates the exact tuning of any of the 87 (one note tuned to reference
pitch) notes except that (for ET) the reference (tuning fork) pitch is
usually A = 440 Hz or C = 261.63 Hz. However, the beat rate required for
a particular tuning/temperament IS calculated (You might want to consult
Larry Fine's book on piano rebuilding and tuning.) If a piano is to be
tuned to other than ET (e.g. a meantone or an irregular (well)
temperament)) then a tuner would use beat rates appropriate to that
tuning.
And after having said all that, a tuner certainly may provide a few
embellishments here and there but these would have to be subtle less the
piano be judged out-of-tune by a number of listeners that have been
conditioned to a particular temperament.
> In addition, I think the pitch will change with how hard the keys are
> struck. When a piano is tuned, I would think one has to decide how hard
> the keys should be struck. The music performer will choose this factor in
> coordination with other factors when deciding how hard the key should be
> struck.
The timbre (not counting loudness) can be slightly different but the
fundamental pitch is basically the same regardless of key pressure. Most
piano tuners also play some or all of the tuning intervals loud to check
for beats associated with higher-order partials and to make sure the
strings are set and don't wonder off pitch.
>As long as a musicians in a performance do not play in unison,
>which is quite rare except for certain groups of music instruments, they
>can quite freely choose factors such as pitch bends, scalestretch, and
>even scale.
If I read the above correctly are you saying that musicians usually play
in unison? If you meant that playing in unison is rare then I would take
issue with that. Sincerely,
Hans Aberg
wo...@itd.nrl.navy.mil (J. B. Wood) wrote:
> > > The inharmonicity of piano strings results in the
> > > stretching of octaves in order to keep the beats of the partials at the
> > > rate required by ET.á
> >
> > That is just aátheory which hardly can be right, as the same kind of piano
> > can be tuned witháseveral different kinds of scalestretching, depending on
> > the music style it should support. For example, a concert grand piano, for
> > use in an orchestra will have aáno-stretch area in the middle, with some
> > stretch towards the bass and the treble. It will not fit any mathematical
> > formula, but will be tuned just according someámusical needs.
>
> It's not a theory, Hans. Stretch AFAIK is a byproduct of inharmonicity.
If it is not used that way in piano tuning, how can it be?
> IOW you'd rather not have it but vibrating string mechanics dictates
> otherwise.á The goal is to adjust the beats of partials (overtones) of an
> interval (usually a fifth or third) to a rate that is consistent with the
> tuning/temperament desired.á
That seems to not be the way performers use it. They seem to use more
scale stretch when not playing in unison, at least in one example I heard:
a flute and a singer, each performing singly. There are no beats to adjust
for.
> Piano tuners thus establish a "temperament
> octave" in the piano mid-range.á The remaining notes are then tuned by
> octaves to their corresponding ones in the temperament octave and this
> technique results in the stretch.á There's no mathematical formula that
> dictates the exact tuning of any of the 87 (one note tuned to reference
> pitch) notes except that (for ET) the reference (tuning fork) pitch is
> usually A = 440 Hz or C = 261.63 Hz.á
But if one should adjust theáinharmonicity beats, there should be.
Foráeach musical instrument the correct mathematical scale stretch could
be computed, and no other tuning could be used, as the partials must
match.
> However, the beat rate required for
> a particular tuning/temperament IS calculated (You might want to consult
> Larry Fine's book on piano rebuilding and tuning.)á
The question is if this presents aáuniversal tuning theory in widespread use.
> If a piano is to be
> tuned to other than ET (e.g. a meantone or an irregular (well)
> temperament)) then a tuner would use beat rates appropriate to that
> tuning.
Some may have the idea that scalestretch should be tied to inharmonicity,
others not, following some empirical formula instead.
> And after having said all that, a tuner certainly may provide a few
> embellishments here and there but these would have to be subtle less the
> piano be judged out-of-tune by a number of listeners that have been
> conditioned to a particular temperament.
And others used to it, may have the opposite view. Matt Fields said once
something to the effect that Scottish bagpipes have
anáenormousáscalestretch. Are these out of tune? And is the tuning
associated with bagpipe inharmonicity, which I think goes the opposite way
relative stringed instruments?
> > In addition, I think the pitch will change with how hard the keys are
> > struck. When a piano is tuned, I would think one has to decide how hard
> > the keys should be struck. The music performer will choose this factor in
> > coordination with other factors when deciding how hard the key should be
> > struck.
>
> The timbre (not counting loudness) can be slightly different but the
> fundamental pitch is basically the same regardless of key pressure.á
What is the change in cents?
> Most
> piano tuners also play some or all of the tuning intervals loud to check
> for beats associated with higher-order partials and to make sure the
> strings are set and don't wonder off pitch.
But that would be a final check, not a tuning, right? When I tuned a
piano, any other stringed instruments, I finish off with hard pulls just
to make sure the strings are properly set,ánot to check partials, which I
cannot control anyway. are you saying that if the partials are right at
mf, the strings are properly set at fff, but the partials areáwrong there,
the tuning somehow can be adjusted? And one tunes it in mf or something?
> >As long as a musicians in a performance do not play in unison,
> >which is quite rare except for certain groups of music instruments, they
> >can quite freely choose factors such asápitch bends, scalestretch, and
> >even scale.
>
> If I read the above correctly are you saying that musicians usually play
> in unison?á If you meant that playing in unison is rare then I would take
> issue with that.á
Playing in unison seems rare, except for certain doubling like in violin
sections and the like. Flutes do not playáwell in unison, due to beats. Of
course, it becomes rarer with smaller groups of musical instruments. The
idea is that ináensembles where musical instruments have to play more in
unison together, likeámodernáorchestras,áscalestretch will be avoided. But
in other quarters, smaller,áregional andáancientáorchestras, there should
then be more scalestretch. For example, gamelan orchestras use
scalestretch:
á http://en.wikipedia.org/wiki/Gamelan#Tuning
But each orchestra has its own tuning. So it does not seem t be associated
witháinstrumentáinharmonicity, but merely to give each orchestra a
distinctive musical color. This is how I thinkáscalestretch is actually
used in general. It does not mean that some vanádevelop
aáscalestretchátheory based on ties for use in more special situations.
But the main point is not the rarity, but if you play in unison, then
clearly you have to adjust pitches, but if you do not play in unison, that
is not necessary to the same extent. If such adjustment was always, then
theápresence of 12-TET instruments would forceáalláinstruments and singer
to use 12-TET, but that is not the case. I gave one example:
á http://www.balkanfolk.com/cd_185500
What do your ears tell you here, is everyone playing/singing in 12-TET here?
--
Hans Aberg
J. B. Wood
hab...@math.su.se (Hans Aberg) wrote:
> > However, the beat rate required for
> > a particular tuning/temperament IS calculated (You might want to consult
> > Larry Fine's book on piano rebuilding and tuning.)
>
>
Hello, and there have been a lot of books published on piano tuning
technique over the years. While the technique varies as to setting up the
temperament octave (e.g., whether to do it by fifths exclusively) the goal
is the same - to adjust the beat rates of specific partials to those
values required by the type of tuning (ET, meantone, well, etc) desired.
The "art" in piano tuning is how the tuner is able to trade-off all those
competing vibrational phenomena so that an 88-key "in tune" instrument
results. Tuning a guitar is child's play by comparison (my apologies to
Joey G.).
>Some may have the idea that scalestretch should be tied to inharmonicity,
>others not, following some empirical formula instead.
Perhaps. But if we're still talking about pianos, no two are alike even
if they are the same make and model. The scale stretch that resulted from
the tuning of one S&S model D could conceivably be ported with some
success to another S&S model D but I would not in general expect that
piano to be in tune if that's all that was done.
> And others used to it, may have the opposite view. Matt Fields said once
> something to the effect that Scottish bagpipes have
> an enormous scalestretch. Are these out of tune? And is the tuning
> associated with bagpipe inharmonicity, which I think goes the opposite way
> relative stringed instruments?
I'll defer to Matt on that one. But I am curious.
> are you saying that if the partials are right at
> mf, the strings are properly set at fff, but the partials are wrong there,
> the tuning somehow can be adjusted? And one tunes it in mf or something?
Sure, if required. The tuner pulls out his/her trusty tuning hammer
(wrench) and tweaks the tuning as required. However, a C-G fifth, for
example, when sounded at fff or mf should not display a significant
difference in beats on a piano in good operating order (although I'm at a
loss to say what that difference might be in cents). As I've stated, all
this stuff makes piano tuning an art rather than an exact science. And
tuners make it look so easy when they come to your house and do the whole
job in one or two hours.
> But each orchestra has its own tuning. So it does not seem t be associated
> with instrument inharmonicity, but merely to give each orchestra a
> distinctive musical color. This is how I think scalestretch is actually
> used in general. It does not mean that some van develop
> a scalestretch theory based on ties for use in more special situations.
I don't know what you mean here, Hans. If were talking about Western
orchestras, the general procedure is to have the first violin sound a 440
Hz A. The other instruments then key in on that. The "color" of an
orchestra seems more attributable to the conductor and his interpretation
of the score and the playing skills of the musicians. Probably why oft
heard Strauss waltzes always sound fresh when played by the VPO.
Hans Aberg
wo...@itd.nrl.navy.mil (J. B. Wood) wrote:
> > > However, the beat rate required for
> > > a particular tuning/temperament IS calculated (You might want to consult
> > > Larry Fine's book on piano rebuilding and tuning.)
> >
> > The question is if this presents a universal tuning theory in
widespread use.
> Hello, and there have been a lot of books published on piano tuning
> technique over the years. While the technique varies as to setting up the
> temperament octave (e.g., whether to do it by fifths exclusively) the goal
> is the same - to adjust the beat rates of specific partials to those
> values required by the type of tuning (ET, meantone, well, etc) desired.
But the question is how scalestretch enters this picture, not how to tune
a middle octave, which I think will be very accurate in frequency
according to the scale chosen.
> The "art" in piano tuning is how the tuner is able to trade-off all those
> competing vibrational phenomena so that an 88-key "in tune" instrument
> results. Tuning a guitar is child's play by comparison (my apologies to
> Joey G.).
Have you tried tuning a guitar with very soft strings?
> >Some may have the idea that scalestretch should be tied to inharmonicity,
> >others not, following some empirical formula instead.
>
> Perhaps. But if we're still talking about pianos, no two are alike even
> if they are the same make and model.
We are not speaking only about pianos here, but the use of scalestretch on
all instruments in use, and also how it may with the musical context.
If scalestretch is a phenomenon musically tied to inharmonicity, then on
musical instruments with negative inharmonicity, the scale should be
compressed instead.
> The scale stretch that resulted from
> the tuning of one S&S model D could conceivably be ported with some
> success to another S&S model D but I would not in general expect that
> piano to be in tune if that's all that was done.
I think one can only try with different degrees of scalestretch in
different kinds of musical contexts and see (or rather hear) what happens.
> > And others used to it, may have the opposite view. Matt Fields said once
> > something to the effect that Scottish bagpipes have
> > an enormous scalestretch. Are these out of tune? And is the tuning
> > associated with bagpipe inharmonicity, which I think goes the opposite way
> > relative stringed instruments?
>
> I'll defer to Matt on that one. But I am curious.
None of us here in this newsgroup seems really to know how scalestretch is
actually used out there. Musicians are not aware of scalestretch in a
conscience was "I am now using it so and so much".
If a note is increased in pitch, it will not sound immediately as out of
tune, but increased in intensity, if the underlying reference pitch
somehow is communicated. If the scale is stretched, and the underlying
reference pitches somehow can be communicated, I think that one will hear
an intensified scale, not a different scale. So this would give
explanation of it.
> > are you saying that if the partials are right at
> > mf, the strings are properly set at fff, but the partials are wrong there,
> > the tuning somehow can be adjusted? And one tunes it in mf or something?
>
> Sure, if required. The tuner pulls out his/her trusty tuning hammer
> (wrench) and tweaks the tuning as required.
But then the base note will change. If the string is properly set at mf,
you can't adjust it fff if you do not like the partials at this dynamics
alone, that is obvious.
> However, a C-G fifth, for
> example, when sounded at fff or mf should not display a significant
> difference in beats on a piano in good operating order (although I'm at a
> loss to say what that difference might be in cents).
It is not the question of an interval at mf of fff, but a note at mf and a
note at fff.
> > But each orchestra has its own tuning. So it does not seem t be associated
> > with instrument inharmonicity, but merely to give each orchestra a
> > distinctive musical color. This is how I think scalestretch is actually
> > used in general. It does not mean that some van develop
> > a scalestretch theory based on ties for use in more special situations.
> I don't know what you mean here, Hans.
Gamelan orchestras, John. Here is the reference once more:
http://en.wikipedia.org/wiki/Gamelan#Tuning
The method of tuning instruments in pairs in order to produce beats is
also used on accordions:
http://www.cs.cmu.edu/afs/cs/user/phoebe/accordion/accordion-tuning.html
> If were talking about Western
> orchestras, the general procedure is to have the first violin sound a 440
> Hz A. The other instruments then key in on that. The "color" of an
> orchestra seems more attributable to the conductor and his interpretation
> of the score and the playing skills of the musicians. Probably why oft
> heard Strauss waltzes always sound fresh when played by the VPO.
The idea is that later Western classical orchestras avoid colors, because
it is harder to let larger groups of instruments work together, especially
if groups of instruments play in unison. That is why a grand piano for use
with an orchestra would have less scalestretch than a piano used with a
smaller music band.
Then, as Western scholars use the classical Western orchestra as a basis
for their theories, they start to think that this is a universal theory,
for use in all music. It's like 12-TET.
--
Hans Aberg
Matthew Fields
Hans Aberg <hab...@math.su.se> wrote:
>But the question is how scalestretch enters this picture, not how to tune
>a middle octave, which I think will be very accurate in frequency
>according to the scale chosen.
>We are not speaking only about pianos here, but the use of scalestretch on
>all instruments in use, and also how it may with the musical context.
>If scalestretch is a phenomenon musically tied to inharmonicity, then on
>musical instruments with negative inharmonicity, the scale should be
>compressed instead.
See the work of Sethares.
>
>None of us here in this newsgroup seems really to know how scalestretch is
>actually used out there. Musicians are not aware of scalestretch in a
>conscience was "I am now using it so and so much".
>
>If a note is increased in pitch, it will not sound immediately as out of
>tune, but increased in intensity, if the underlying reference pitch
>somehow is communicated. If the scale is stretched, and the underlying
>reference pitches somehow can be communicated, I think that one will hear
>an intensified scale, not a different scale. So this would give
>explanation of it.
Try that with a piccolo and a bass clarinet playing sustained 22nds. I
think you're dead wrong. The players will compensate with embouchure
to keep the frequencies as close to 8:1 as possible.
--
Matthew H. Fields http://www.umich.edu/~fields
Music: Splendor in Sound
To be great, do better and better. Don't wait for talent: no such thing.
Brights have a naturalistic world-view. http://www.the-brights.net/
Hans Aberg
<sp...@uce.gov> wrote:
> In article <haberg-1004...@c83-250-195-81.bredband.comhem.se>,
> Hans Aberg <hab...@math.su.se> wrote:
>
> >But the question is how scalestretch enters this picture, not how to tune
> >a middle octave, which I think will be very accurate in frequency
> >according to the scale chosen.
>
> >We are not speaking only about pianos here, but the use of scalestretch on
> >all instruments in use, and also how it may with the musical context.
> >If scalestretch is a phenomenon musically tied to inharmonicity, then on
> >musical instruments with negative inharmonicity, the scale should be
> >compressed instead.
>
> See the work of Sethares.
Can you give a more specific reference? Or given an indication of what it
actually says?
> >None of us here in this newsgroup seems really to know how scalestretch is
> >actually used out there. Musicians are not aware of scalestretch in a
> >conscience was "I am now using it so and so much".
> >
> >If a note is increased in pitch, it will not sound immediately as out of
> >tune, but increased in intensity, if the underlying reference pitch
> >somehow is communicated. If the scale is stretched, and the underlying
> >reference pitches somehow can be communicated, I think that one will hear
> >an intensified scale, not a different scale. So this would give
> >explanation of it.
>
> Try that with a piccolo and a bass clarinet playing sustained 22nds. I
> think you're dead wrong. The players will compensate with embouchure
> to keep the frequencies as close to 8:1 as possible.
What I am saying that if two players somehow need to match up their both
playing, they will avoid scalestretch, as that will sound poorly. But if
they play their own music line, they do not need to, so then they can
exploit such a feature. So it seems you have misunderstood what I am
saying here. And by "communicating the reference pitch", I do not mean
that it should be actually played, but that the musical interpretation is
such that the listener can figure around which pitch it is played. This
can happen either by suitable picthbends by the performer, but also if
both player and listener happens to be a part of the same music culture
where scales are known.
--
Hans Aberg
Hans Aberg
"Graham Breed" <x31eq...@gmail.com> wrote:
> > Right now, I do not have the musical software to do it.
>
> It really isn't that difficult to get hold of. There's Csound, Scala,
> my own MIDI Relay, and a load of soft synths.
I picked down Scala, where one easily can create any type of just chords
and play, and also get to see the name of it, if it is in the list (i.e.,
just plug in the numbers, and the program will automatically give the name
of the chord, if it exists). With Mac OS X, one can also pick down a
program called SimpleSynth, and use the built in Mac OS X MIDI synth.
(When installing Scala on a Mac, the first time the program is being
started up, it takes a long time for its X Window window in X11 to appear;
so just start Scala once by launching the Scala Aqua program, and then
wait for some good time until this X11 window appears.) Is then, of course
possible to hook up more advanced MIDI equipment, but you do not need to,
in order to get started and work with it.
I recommend highly anyone that want to play around with chords to use this
(or any such) program.
Anyway, the chords I posted have registered names. So it is just a
question of picking ones own favorites, and fit them into
a musical context (such as scales or a theory).
And I can continue my playing around with just chords and 72-TET and
listening to it, as well. :-)
--
Hans Aberg
Matthew Fields
Hans Aberg <hab...@math.su.se> wrote:
>In article <P0u_f.961$hi2...@news.itd.umich.edu>, "Matthew Fields"
><sp...@uce.gov> wrote:
>
>> In article <haberg-1004...@c83-250-195-81.bredband.comhem.se>,
>> Hans Aberg <hab...@math.su.se> wrote:
>>
>> >But the question is how scalestretch enters this picture, not how to tune
>> >a middle octave, which I think will be very accurate in frequency
>> >according to the scale chosen.
>>
>> >We are not speaking only about pianos here, but the use of scalestretch on
>> >all instruments in use, and also how it may with the musical context.
>> >If scalestretch is a phenomenon musically tied to inharmonicity, then on
>> >musical instruments with negative inharmonicity, the scale should be
>> >compressed instead.
>>
>> See the work of Sethares.
>
>Can you give a more specific reference? Or given an indication of what it
>actually says?
http://eceserv0.ece.wisc.edu/~sethares/alternatetunings/alternatetuningsoverview.html
>> >None of us here in this newsgroup seems really to know how scalestretch is
>> >actually used out there. Musicians are not aware of scalestretch in a
>> >conscience was "I am now using it so and so much".
>> >
>> >If a note is increased in pitch, it will not sound immediately as out of
>> >tune, but increased in intensity, if the underlying reference pitch
>> >somehow is communicated. If the scale is stretched, and the underlying
>> >reference pitches somehow can be communicated, I think that one will hear
>> >an intensified scale, not a different scale. So this would give
>> >explanation of it.
>>
>> Try that with a piccolo and a bass clarinet playing sustained 22nds. I
>> think you're dead wrong. The players will compensate with embouchure
>> to keep the frequencies as close to 8:1 as possible.
>
>What I am saying that if two players somehow need to match up their both
>playing, they will avoid scalestretch, as that will sound poorly. But if
>they play their own music line, they do not need to, so then they can
>exploit such a feature. So it seems you have misunderstood what I am
>saying here. And by "communicating the reference pitch", I do not mean
>that it should be actually played, but that the musical interpretation is
>such that the listener can figure around which pitch it is played. This
>can happen either by suitable picthbends by the performer, but also if
>both player and listener happens to be a part of the same music culture
>where scales are known.
So you think folks play in poor intonation with their own echoes when
they're playing solo... my experience says otherwise. But I will go
along with claims that people go to such extremes as 19-limit just
intonation for "intensity".
Hans Aberg
"Graham Breed" <x31eq...@gmail.com> wrote:
> > > > ... This gives the following
> > > > chord combinations (M = major, m = minor):
> > > > chord intervals
> > > > M7 1 5/4 3/2 7/4
> > > > Mmaj7 1 5/4 3/2 3*5/8
> > > > m7 1 7/3*2 3/2 7/4
> > > > dim 1 7/3*2 11/8 3*5/8
> > >
> > > You're favouring otonal to utonal in Partch's terminology, and some of
> > > us like that when you get to the 11-limit. It makes the notes simpler
> > > subsets of the harmonic series, and maybe clearer if you write them as
> > > extended ratios:
> > >
> > > M7 4:5:6:7
> > > Mmaj7 8:10:12:15
> > > m7 12:14:18:21
> > > dim 24:28:33:45
> If you're building chords with thirds, the 11-limit will typically give
> you neutral thirds, but you aren't using these. They will, of course,
> drag you far from the 12-TET skeleton. 7:9:11 makes a good substitute
> for an augmented triad, for example. There are also some diminished
> sevenths in 72-TET that don't translate properly into just intonation
> (hence giving a reason to use an equal temperament in the first place).
>
> And how about 6:7:8:11?
It is interesting to fiddle around with this in Scala, listening to it.
One can then also approximate to ones favorite ET by the click of a
button. Then, as expected, with 12-TET, there is a significant difference,
and with 72-TET, only a slight change in texture.
Then I think that my suggestion of dim 24:28:33:45 sounds like a good
minor dim, much more so than the 12-TET dim. Your version of dim 6:7:8:11
produces a lot of beats, and does not sound as nice at all to me; it
sounds thin somehow. This difference of course invites to further
experimentation. The dim version I made, in its conception, is an
alteration of a minor chord. But one could, say, take the top three notes
of a major seventh, and extend it with one note. These dim variations
would then be used in different types of chord progressions. Or, anyway,
this is the kind of ideas I am playing around with. :-)
--
Hans Aberg
Hans Aberg
<sp...@uce.gov> wrote:
> >What I am saying that if two players somehow need to match up their both
> >playing, they will avoid scalestretch, as that will sound poorly. But if
> >they play their own music line, they do not need to, so then they can
> >exploit such a feature. So it seems you have misunderstood what I am
> >saying here. And by "communicating the reference pitch", I do not mean
> >that it should be actually played, but that the musical interpretation is
> >such that the listener can figure around which pitch it is played. This
> >can happen either by suitable picthbends by the performer, but also if
> >both player and listener happens to be a part of the same music culture
> >where scales are known.
>
> So you think folks play in poor intonation with their own echoes when
> they're playing solo... my experience says otherwise.
No, I am saying that the intonation is freer, when playing solo.
Musicians that are trained in Western classical music, where such
intonations are likely to be in little use, may though perceive them as
poor intonations.
> But I will go
> along with claims that people go to such extremes as 19-limit just
> intonation for "intensity".
I have no idea what you are referring to here. These intonations take
place intuitively, and do not need a formal theory, nor are they likely to
be in musical performance.
And did you have a look at the page on gamelan tuning:
http://en.wikipedia.org/wiki/Gamelan#Tuning
Apparently, they customize what you call poor intonation in order to give
each gamelan orchestra a special musical color. :-)
And also in Western music, beats are not bad:
http://www.cs.cmu.edu/afs/cs/user/phoebe/accordion/accordion-tuning.html
--
Hans Aberg
Matthew Fields
><sp...@uce.gov> wrote:
>
>> >What I am saying that if two players somehow need to match up their both
>> >playing, they will avoid scalestretch, as that will sound poorly. But if
>> >they play their own music line, they do not need to, so then they can
>> >exploit such a feature. So it seems you have misunderstood what I am
>> >saying here. And by "communicating the reference pitch", I do not mean
>> >that it should be actually played, but that the musical interpretation is
>> >such that the listener can figure around which pitch it is played. This
>> >can happen either by suitable picthbends by the performer, but also if
>> >both player and listener happens to be a part of the same music culture
>> >where scales are known.
>>
>> So you think folks play in poor intonation with their own echoes when
>> they're playing solo... my experience says otherwise.
>
>No, I am saying that the intonation is freer, when playing solo.
>
>Musicians that are trained in Western classical music, where such
>intonations are likely to be in little use, may though perceive them as
>poor intonations.
Depends on what they're doing. Western-trained singers tend to know
how to glissando from a G# to an Ab, in the right context, though they
generally won't be able to tell you the circumstances in which that's
ascending and those in which it's descending.
>> But I will go
>> along with claims that people go to such extremes as 19-limit just
>> intonation for "intensity".
>
>I have no idea what you are referring to here. These intonations take
>place intuitively, and do not need a formal theory, nor are they likely to
>be in musical performance.
Indeed, these things are discovered by analyzing the performance practice
of string players and singers.
>And did you have a look at the page on gamelan tuning:
> http://en.wikipedia.org/wiki/Gamelan#Tuning
>Apparently, they customize what you call poor intonation in order to give
>each gamelan orchestra a special musical color. :-)
Not at all. That's all covered in Sethares.
>And also in Western music, beats are not bad:
> http://www.cs.cmu.edu/afs/cs/user/phoebe/accordion/accordion-tuning.html
The topic of beats good and bad wasn't what I was talking about. In
fact, the vibrant beats of equal-tempered major thirds are prized by
many pianists.
Hans Aberg
<sp...@uce.gov> wrote:
> >No, I am saying that the intonation is freer, when playing solo.
> >
> >Musicians that are trained in Western classical music, where such
> >intonations are likely to be in little use, may though perceive them as
> >poor intonations.
>
> Depends on what they're doing. Western-trained singers tend to know
> how to glissando from a G# to an Ab, in the right context, though they
> generally won't be able to tell you the circumstances in which that's
> ascending and those in which it's descending.
And the singers here:
http://www.balkanfolk.com/cd_185500
What do you your ears say? Are they right in intonation?
And, in another direction (and somewhat off-topic), what you do think of
the musical interpretation in
http://varna-sound.bulgarian-music.com/audio/621-Traichovo-horo.mp3
http://varna-sound.bulgarian-music.com/cd-621-Mitko_Marinov_-_Muto_-_Bulgarian_Folk_Music
> >> But I will go
> >> along with claims that people go to such extremes as 19-limit just
> >> intonation for "intensity".
> >
> >I have no idea what you are referring to here. These intonations take
> >place intuitively, and do not need a formal theory, nor are they likely to
> >be in musical performance.
>
> Indeed, these things are discovered by analyzing the performance practice
> of string players and singers.
>
> >And did you have a look at the page on gamelan tuning:
> > http://en.wikipedia.org/wiki/Gamelan#Tuning
> >Apparently, they customize what you call poor intonation in order to give
> >each gamelan orchestra a special musical color. :-)
>
> Not at all. That's all covered in Sethares.
The you should perhaps look at it again, because it is not known how the
gamelan orchestars are tuned. It says that some Westerners have jumped to
conclusions, but which are in fact wrong.
I could not find any gamelan tunings on these Sethares pages. Could you help me?
> >And also in Western music, beats are not bad:
> > http://www.cs.cmu.edu/afs/cs/user/phoebe/accordion/accordion-tuning.html
>
> The topic of beats good and bad wasn't what I was talking about. In
> fact, the vibrant beats of equal-tempered major thirds are prized by
> many pianists.
I just put it in to avoid the avoiding-beats is good. Of course, it is a
common technique on keyboards to strike adjacent chromatic keys to produce
a quarter-note with a strong vibrato produced by the beats. I use it all
the time.
--
Hans Aberg
Hans Aberg
In article <aWy_f.970$hi2...@news.itd.umich.edu>, "Matthew Fields"
<sp...@uce.gov> wrote:
> Western-trained singers tend to know
> how to glissando from a G# to an Ab, in the right context, though they
> generally won't be able to tell you the circumstances in which that's
> ascending and those in which it's descending.
There needs to be an agreement as to what pitches defines G# and Ab. I
just noticed, when playing chord in Scala, that if I start a session
playing the 12-TET dim, and then shifts to the just dim 24:28:33:45 (=
1:7/3*2:11/8:3*5/8) I created, then the latter at first sounds out of
tune. If I however, start using this just dim for awhile, so I can get
used to it, it sounds nicely minor-like. I can then shift between these
two chord, hearing them as being of different musical texture. Perhaps
this musical perception phenomenon is due to the fact that I am mostly
used to 12-TET.
--
Hans Aberg
Hans Aberg
hab...@math.su.se (Hans Aberg) wrote:
> ... Western-trained singers tend to know
> how to glissando from a G# to an Ab, in the right context, though they
> generally won't be able to tell you the circumstances in which that's
> ascending and those in which it's descending.
There needs to be an agreement as to what pitches defines G# and Ab. I
Hans Aberg
<sp...@uce.gov> wrote:
> >We are not speaking only about pianos here, but the use of scalestretch on
> >all instruments in use, and also how it may with the musical context.
> >If scalestretch is a phenomenon musically tied to inharmonicity, then on
> >musical instruments with negative inharmonicity, the scale should be
> >compressed instead.
>
> See the work of Sethares.
Stringed instruments have positive inharmonicity (I think), be it a piano
or a guitar. So if a compressed scale is used on a guitar, then that is
just an example of how scalestretch is used independently of
inharmonicity. Besides, the Overtone Tuning, if that is what you are
referring to (strange thing, not giving a more specific citation), does
not have any ties to inharmonicity at all, merely some partials. (One can
easily play around with such things in Scala.)
--
Hans Aberg
Hans Aberg
<sp...@uce.gov> wrote:
> ... Western-trained singers tend to know
> how to glissando from a G# to an Ab, in the right context, though they
> generally won't be able to tell you the circumstances in which that's
> ascending and those in which it's descending.
One can work intuitively in Scala with differences between accidentals by
pinning down them in 72-ET or 144-TET, also with the chance to listening
to intermediate pitches and get an idea of how big it is. In Scala "Play",
select notation system E72 or E144, and then a keyboard map for that. This
gives a keyboard for 72-ET or 144-TET. (Of course, in Scala, one can
choose any scale directly, like a Just scale containing only those
pitches; the idea here is that one should be able to slide along
the intermediate pitches, relating that to 12-TET.)
Then one needs to pin down the Just pitches in 72-TET or 144-TET. This is
not so difficult, because one already know where they are relative 12-TET,
and one just needs to compute the 72-TET or 144-TET offsets. 72-TET is
good for the partials 2, 3, 5, 7, 11 when not taking powers of them,
because the approximation is within 3 cents, an error hardly hearable
as absolute pitches. When using powers 5^2, 5^3, as in traditional Just
intonation, if one wants to have the same degree of accuracy, one needs to
go up to 144-TET. Working with 144-TET, one has (or I got :-)):
3/2 k7+1.96
5/4 k4_2+2.98
5^2/16 k8_3-2.37
5^3/128 k0_5+0.61
where "_" indicates a lowering in 144-TET tonestep, and the trailing
decimal indicates an offset in cents. This just says, relative 12-TET, for
the indicated partial, use the following 144-TET offsets:
2^k 0
3^k 0
5 -2
5^2 -3
5^3 -5
5^(-1) 2
5^(-2) 3
5^(-3) 5
Now, in (one form of) Just intonation, one uses:
Interval relative C relative frequency 144-TET offset
p5 G 3/2 0
+5 G# 25/16 -3
m6 Ab 8/5 2
M6 A 5/3 -2
The last column is fairly easy to fill in: For example, in G#, the
only offset is created by the factor 5^2, which has an 144-TET offset of
-3 by the table above.
Then it is easy to play these notes on the 144-TET keyboard: Just first
find the 12-TET key, and move the number of 144-TET keys. For example, to
play Just A, go to the 12-TET A, and move down two 144-TET tonesteps. To
play Just G#, go to 12-TET G# and move down three 144-TET tonesteps. And
so forth. This gives the opportunity to listen to the intermediate 144-TET
tones, thus giving an intuition on how big the difference is.
--
Hans Aberg
Margo Schulter
>
> Then I think that my suggestion of dim 24:28:33:45 sounds like a good
> minor dim, much more so than the 12-TET dim.
Hello, there, and having read your interesting dialogue with Graham, I
might just ask what basic interval structure we are going for here in a
four-note "diminished" chord.
The reason that I'm querying this is that 24:28:33:45 has two minor thirds
at 24:28 (7:6, about 267 cents, virtually just in 72-equal) and 28:33
(about 284 cents, approximated very closely as 283.333 cents); but an
upper interval of 33:45 or 11:15, about 537 cents (approximated as 533.33
cents) -- not a third, but a kind of "superfourth" just moving into the
territory of a very small diminished fifth or tritone.
Also, the outer seventh here is a major seventh at 24:45 or 8:15; the term
"diminished" chord, when applied to a four-voice sonority, suggests to me
either a diminished seventh (which can vary in size depending on the
tuning system and, here, on style or taste since we have so many choices)
or a minor seventh (a "half-diminished" chord).
One possible tuning of the type emphasizing mostly simple ratios of 5 and
7 for the variety with diminished seventh is this one proposed by
Helmholtz:
316 267 336
5:6 6:7 14:17
0 316 583 919
Just: 10: 12: 14: 17
72: 0 317 583 917
317 267 333
The just 10:12:14:17 tuning has pure minor thirds at 5:6 and 6:7, a
diminished fifth at 5:7, and a diminished seventh at 10:17 -- as well as
an upper "supraminor" third at 14:17, all of which 72-equal can
approximate within three cents, and some of which are accurate to around a
cent or better (compare 7:6, 266.871 cents, with 16/72 octave at
266.6666... cents).
For a "half-diminished" seventh chord with an outer minor seventh, we
could use, for example, 24:28:33:42 -- the same lower notes as in the
proposed 24:28:33:45, but here with a minor seventh at 24:42 or 4:7, and
an upper major third at 11:14.
267 284 418
7:6 33:28 14:11
0 267 551 969
Just: 24: 28: 33: 42
72: 0 267 550 967
267 283 417
Maybe I should mention one general approach to exploring a tuning like
this: looking for what are called by people such as Ervin Wilson and
George Secor isoharmonic chords, where the partials have the same
differences, or for example differ by either one or two. Thus:
2 2 2
3:5:7:9 0 884 1467 1902 72: 0 883 1467 1900
2 2 2
5:7:9:11 0 583 1018 1365 72: 0 583 1017 1367
1 2 2
6:7:9:11 0 267 702 1049 72: 0 267 700 1050
This next one isn't quite so accurate in 72, but still with all intervals
within 3.29 cents of just:
4 2 2
9:13:15:17 0 637 884 1101 72: 0 633 883 1100
Most appreciatively,
Margo Schulter
msch...@calweb.com
Hans Aberg
<msch...@web1.calweb.com> wrote:
> Hans Aberg <hab...@math.su.se> wrote:
> >
> > Then I think that my suggestion of dim 24:28:33:45 sounds like a good
> > minor dim, much more so than the 12-TET dim.
>
> Hello, there, and having read your interesting dialogue with Graham, I
> might just ask what basic interval structure we are going for here in a
> four-note "diminished" chord.
At the time, it was a purely theoretical construction based on the idea
that the diminished chord is an alteration of the minor chord. Then, for
the construction of major chords, I used the partials 2, 3, 5, and for the
minor chords, the partials 2, 3, 7, or rather the notes of them that might
be associated with some yet hypothetical major and minor scales. The
partial 2 will be used to move a relative frequency into the fundamental
octave, and the partial 3, at not too high powers, will be used to move it
along the semitones. As the focus is on compatibility with 72-TET, the
partials 5, 7, having duodecitone offsets relative 12-TET, -2, powers
other that +1, -1, as well as mixtures of them, will be avoided, as then
they are indistinguishable from other rational intervals with the same
partials, preferring the power +1, with the idea that partials will mix
better. This way, I arrived at the chords:
M7 1:5/4:3/2:7/4 = 4:5:6:7
Mmaj7 1:5/4:3/2:3*5/8 = 8:10:12:15
m7 1:7/3*2:3/2:7/4 = 12:14:18:21
Then, the diminished chord
dim 1:7/3*2:11/8:5/3 = 24:28:33:40
first gets the minor third 7/3*2 and the major sixth 5/3 (there probably
was a typo in my other post on this last one, writing wrongly the major
seventh 3*5/8). So, so far, I have created a partial diminished chord,
without the third note of it. So then, it just happens that the partial
11, when reduced to the fundamental octave, is a good candidate. So I just
put it in. This adds a pleasing symmetry, both as a Just chord, as the
three partials 5, 7, 11 are used once, but also, I get a use for the
partial 11 in 72-TET, where it has a full quartertone offset relative
12-TET.
At the time, I did not have access to Scala, or any other program, getting
me chance to listen to it. So it was purely theoretical construction,
based on the idea to develop a 11-limit for use with 72-TET. In addition,
chords should be used together with scales and chord progressions, and I
deliberately skipped that issue in this very first construction. The
intent was just doing some first steps onto this path.
--
Hans Aberg
Margo Schulter
> Then, the diminished chord
> ? dim? ? ?1:7/3*2:11/8:5/3 =?24:28:33:40
> first gets the minor third?7/3*2 and the major sixth?5/3 (there probably
> was a typo in my other post on this last one, writing wrongly the major
> seventh?3*5/8).
Thank you for clarifying what I was querying: that the ratio for a major
sixth at 24:40 or 3:5 was intended. An interesting feature of this
diminished seventh is that, like the 10:12:14:17 of Helmholtz, it has an
upper "supraminor" third, here at 33:40 or about 333 cents -- about three
cents smaller than the 14:17 of Helmholtz.
There are other possibilities, for example 18:21:25:30, also with a 3:5
major sixth for the diminished seventh (about 0-267-568-884 cents), or in
72-equal (or 144-equal) 0-267-567-883.
Of course, it is possible either to choose one form of each relevant
chord, or deliberately to mix different intonational shadings of a given
form in a piece. With a tuning system as large as this, there are lots of
possibilities with either approach.
Hans Aberg
<msch...@web1.calweb.com> wrote:
There are clearly several possibilities. In one tune, in minor, the tonic
minor triad is alternated with a diminished chord. If the tonic is at D
(2/4), it starts (1/8) A A A A | G# G# G# G# | A F (1/4)D | ... So this is
an example of the chord m altered with a dim with the same root. If the
7/6 is used for the minor third in the hypothetical associated minor
sclae, it seems that the diminished chord I made might be suitable in this
context.
But one can also think of, in the major scale, switching from the dominant
M7 to a diminished chord. Then, one should probably use other rational
intervals for this diminished chord variation.
This is just attempts to create some empirical methods to construct and
use these rational (Just) chords.
If I should attempt to produce some musical motivations, listening in in
SCala, it appears that the common denominator of these chords, i.e., the
root or fundamental if the other intervals are given integers,
is responsible for creating partials that brighten the chord, though I am
not entirely sure about it as a general principle. Then, choosing such a
common denominator small helps to thin (make less bright) the chord. Or,
that its at least one idea I am plying around with. One can, for example,
create a major triad or seventh using only the partials 2, 3, in which
case they will end up pretty close to 12-TET:
M 1:81/64:3/2 = 64:81:96
M7 1:81/64:3/2:16/9 = 576:729:864:1024
but the 12-TET equivalents are then brighter still. If one mixes in the
partial 5, one gets
M 1:5/4:3/2 = 4:5:6
M7 1:5/4:3/2:9/5 = 20:25:30:36
Here, one can see that mixing in the partial 5, reduces the common
denominators of the chords above 64 -> 4, 576 -> 20, also producing
musically less bright chords.
So perhaps this has been one historical motivation: To produce less bright
chords.
In this context, it appears that Western orchestral musical instruments
have been developed into become brighter: A person measured up the
partials of the Kaval, a Bulgarian flute that is open at the side blown,
and blown at the edge, and compared with the partials of the standard
traverse flute, and found that the latter had more partials.
So one though that comes across my mind, is that when switching to 12-TET,
then the chords brighten up, and the musical instruments playing the
musical lines need to compensate for that, in order to be able to keep up
the competition with the background carpet. This could have some
importance in musical performance, because if one is using archaic
instruments with less partials in them, they might not work well as solo
or musical line instruments against a 12-TET carpet, becoming too weak in
musical color.
--
Hans Aberg