Re: 11TET

Paul H. Erlich wrote:

> Otherwise, 11-tET is
> highly irrational. In fact, Blackwood and I independently arrived at the
> conclusion that it is the best ET for random dissonance.

Interesting. In a doubtlessly flawed but interesting algorithm I posted the
results of last March, 13TET came out on "top" as a maximally dissonant ET.
Though I don't want to get into that whole complexity issue again, I graded
closeness to 13-limit intervals whose consonance was rated according to the
complexity of the ratio. In the range from 9 to 24TET, 13 was followed by
20, 14, 16, and 18 in terms of dissonance. 9, 19, and 12 were the most
consonant. I guess it goes to show how different algorithms can change
things. Only musical context would make these assignments meaningful, but
it was a fun exercise.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

On Thu, 29 Jul 1999, Bill Alves wrote:
> In a doubtlessly flawed but interesting algorithm I posted the
> results of last March, 13TET came out on "top" as a maximally dissonant ET.
> Though I don't want to get into that whole complexity issue again, I graded
> closeness to 13-limit intervals whose consonance was rated according to the
> complexity of the ratio. In the range from 9 to 24TET, 13 was followed by
> 20, 14, 16, and 18 in terms of dissonance. 9, 19, and 12 were the most
> consonant. I guess it goes to show how different algorithms can change
> things. Only musical context would make these assignments meaningful, but
> it was a fun exercise.

Just out of curiosity, where did 11 come out in the rankings from your
algorithm, Bill?

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Hey--do you think I need to lose some weight?"
-\-\-- o

>From: Paul Hahn <Paul-Hahn@library.wustl.edu>
>
>On Thu, 29 Jul 1999, Bill Alves wrote:
>> In a doubtlessly flawed but interesting algorithm I posted the
>> results of last March, 13TET came out on "top" as a maximally dissonant ET.
>> Though I don't want to get into that whole complexity issue again, I graded
>> closeness to 13-limit intervals whose consonance was rated according to the
>> complexity of the ratio. In the range from 9 to 24TET, 13 was followed by
>> 20, 14, 16, and 18 in terms of dissonance. 9, 19, and 12 were the most
>> consonant. I guess it goes to show how different algorithms can change
>> things. Only musical context would make these assignments meaningful, but
>> it was a fun exercise.
>
>Just out of curiosity, where did 11 come out in the rankings from your
>algorithm, Bill?

Hmm, pretty consonant, now that I look at it. Here are the results that
divide the octave evenly. The first item in each line is the number of
tones per octave, followed by the number of cents per step, followed by a
"score" on an arbitrary scale of average dissonance per interval in the
tuning system. The higher the number, the more dissonant.

9 133 23.392733
8 150 23.447059
19 63 24.398251
12 100 24.836317
24 50 25.886108
11 109 25.931373
22 55 26.046512
31 39 26.476374
29 41 26.657201
17 71 26.675579
26 46 26.732411
27 44 26.786492
21 57 26.817073
10 120 26.826625
25 48 26.883554
28 43 27.113369
15 80 27.156187
23 52 27.161438
30 40 27.259222
18 67 27.330252
16 75 27.925996
14 86 28.420479
20 60 28.514329
13 92 29.124706

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

I wrote,

>> Otherwise, 11-tET is
>> highly irrational. In fact, Blackwood and I independently arrived at the
>> conclusion that it is the best ET for random dissonance.

Bill Alves wrote,

>Interesting. In a doubtlessly flawed but interesting algorithm I posted the
>results of last March, 13TET came out on "top" as a maximally dissonant ET.
>Though I don't want to get into that whole complexity issue again, I graded
>closeness to 13-limit intervals whose consonance was rated according to the
>complexity of the ratio. In the range from 9 to 24TET, 13 was followed by
>20, 14, 16, and 18 in terms of dissonance. 9, 19, and 12 were the most
>consonant. I guess it goes to show how different algorithms can change
>things. Only musical context would make these assignments meaningful, but
>it was a fun exercise.

Just to clarify, we're talking 13-odd-limit intervals, right?

My main problem with this analysis is that while 13-limit harmony is
certainly a reasonable yardstick, 13-limit dyads alone would not be enough
to unambiguously project the harmony (as we've discussed, ratios of 13, even
if tuned justly, tend to be confused with other ratios). One would want to
fit together three or more notes at a time to evoke clear 13-limit effects.
But in ETs, the best appoximation of one just ratio plus the best
approximation of another just ratio is not necessarily the best
approximation of the product of the ratios. This kind of "flaw" is what I
like to call inconsistency. The only ETs that are fully consistent within
the 13-limit are 26, 29, 41, 46, 58, 72, 80, 87, 94 . . . . For other ETs,
the type of analysis Bill describes would be best modified by choosing
(carefully!) a single template for an ET's approximation to the harmonic (or
subharmonic) series through 13, and evaluating all intervals according to
that template. Inconsistency would mean that some intervals would be forced
to have an error of more than half a step.

Herman Miller wrote,

>My example, on the other hand, highlights the sharp minor third (12 cents
>sharp of 6/5) and harmonic seventh (13 cents sharp of 7/4).

I would agree that in an octave-specific sense, the harmonic seventh is
clearly represented in 11-tET. However, it is inversionally equivalent to an
approximation of the harmonic ninth, which is not much more complex, so in
terms of interval classes, this is a potentially ambiguous one (though that
can be a good thing).

Clark wrote,

>The reason I like 22-tet? Aside from those consonant structures and good
>things, halve it and you get 11-tet!

Clark, do you like 22-tET a lot? Have you seen my paper on it?

> In the range from 9 to 24TET, 13 was followed by
> 20, 14, 16, and 18 in terms of dissonance. 9, 19, and 12 were the most
> consonant. I guess it goes to show how different algorithms can change
> things. Only musical context would make these assignments meaningful, but
> it was a fun exercise.

I've used both 11 and 13 TET in "serious" musical contexts, and they work...
as far as I'm concerned. I mean, like practically any ET, they have their
quirky characteristics, and are usable for some musical contexts.

> Just out of curiosity, where did 11 come out in the rankings from your
> algorithm, Bill?

Also out of curiousity, how does this compare with other methods? I'm most
familiar with Wendy Carlos method in the 1987 CMJ article "Tuning at the
Crossroads".

Rick

I suspect that the main reason my results were so different from Bill Alves'
is that I evaluated the quality of approximation to _all_ just intervals
within the given limit (which ranged from 3 to 11), while Bill restricted
his attention to those just intervals that came close to intervals of the
ET. Is that right, Bill?

>From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
>>> Otherwise, 11-tET is
>>> highly irrational. In fact, Blackwood and I independently arrived at the
>>> conclusion that it is the best ET for random dissonance.
>
I responded with the results of an algorithm which rated ETs according to
the closeness of their intervals to 13-limit JI intervals. My algorithm
showed 11TET as relatively consonant and 13TET as most dissonant. I've been
late in responding to Paul's further questions because I was out of town.
He asked:
>
>Just to clarify, we're talking 13-odd-limit intervals, right?
>
Yes, I looked at numerators from 4 to 14 and denominators from the square
root of the numerator to the numerator-1.

>My main problem with this analysis is that while 13-limit harmony is
>certainly a reasonable yardstick, 13-limit dyads alone would not be enough
>to unambiguously project the harmony (as we've discussed, ratios of 13, even
>if tuned justly, tend to be confused with other ratios).

I didn't mean for this algorithm to predict the usefulness of
interval-stacking harmonies through consistency or any similar measure.
Actually, at the time I was more interested in finding the most dissonant
ETs, which might be most useful for a kind of serialism that maximized
dissonance. That's why your post caught my eye in the first place.

>I suspect that the main reason my results were so different from Bill Alves'
>is that I evaluated the quality of approximation to _all_ just intervals
>within the given limit (which ranged from 3 to 11), while Bill restricted
>his attention to those just intervals that came close to intervals of the
>ET. Is that right, Bill?

What I did was to give the JI intervals (chosen within the half-octave
range as shown above) a ranking of complexity by simply adding the
numerators and denominators. Then in between these intervals, I drew a
curve down either side following the curve of the critical bandwidth.
Interval inversions were treated as identical with the non-inverted
interval. As I said, I don't think it is a perfect dissonance (or
"complexity") curve, but I just wanted to see what would happen with this
rough measure. I then averaged the dissonance from this table for each step
of the given ET.

By the way, I found a slight bug in my code when reviewing it for this
post. While it doesn't greatly affect the ETs under discussion, it does
somewhat change the table I posted last week. Here are the new results (the
first column is the ET, followed by the cents per step, followed by a
"score" of relative dissonance of that ET's intervals -- larger numbers are
more dissonant). I've also included ETs 5 through 8:

6 200 19.834225
5 240 19.973856
7 171 21.825792
9 133 23.183391
8 150 23.447059
12 100 23.664962
19 63 23.884738
11 109 24.648459
24 50 25.312891
22 55 25.389193
10 120 25.408669
31 39 25.902604
26 46 26.05594
29 41 26.11359
21 57 26.159971
27 44 26.255991
25 48 26.333734
17 71 26.383244
23 52 26.433333
28 43 26.446524
30 40 26.706879
18 67 26.945378
15 80 26.961461
20 60 27.687783
14 86 27.751634
16 75 27.925996
13 92 28.047059

By the way, as I stated last March, I didn't limit this list to ETs that
evenly divided the octave. I looked at intervals only within the two octave
range, though, because I had to draw the line somewhere, and I figured that
most musical dyads would fall within that range. At the risk of greatly
adding to Dan Wolf's phone bill :) here is that list, with step sizes from
33 to 175 cents. The first number is the step size in cents and the second
number is the relative measure of dissonance as above:

171 21.825792
172 21.959276
173 22.794118
149 22.876471
133 23.183391
150 23.447059
169 23.470588
134 23.487889
151 23.639216
170 23.647059
100 23.664962
119 23.701238
63 23.884738
148 24.041177
152 24.143137
126 24.349673
174 24.429864
109 24.648459
132 24.653979
99 24.71867
175 25.19457
127 25.246732
131 25.266436
65 25.312092
50 25.312891
121 25.380805
55 25.389193
77 25.391177
120 25.408669
118 25.572755
135 25.589965
38 25.5963
168 25.626697
147 25.709804
37 25.715074
35 25.719291
89 25.742081
33 25.751634
101 25.800512
61 25.826625
53 25.838235
69 25.884083
54 25.890561
39 25.902604
36 25.922906
88 25.928733
51 26.053069
46 26.05594
41 26.11359
34 26.144118
42 26.159139
57 26.159971
114 26.201471
130 26.207613
103 26.244652
44 26.255991
47 26.308235
68 26.324394
48 26.333734
70 26.347594
95 26.367647
45 26.36991
108 26.372549
71 26.383244
146 26.398039
73 26.412684
79 26.418864
52 26.433333
43 26.446524
122 26.4613
56 26.481092
74 26.48482
125 26.493464
49 26.500613
76 26.526565
59 26.586029
91 26.625882
78 26.628431
62 26.664861
72 26.690257
40 26.706879
113 26.722059
155 26.745798
64 26.755167
83 26.839286
96 26.882353
153 26.913725
58 26.938971
67 26.945378
81 26.947262
80 26.961461
82 27.006303
66 27.036135
163 27.054622
110 27.084034
87 27.092593
85 27.35403
162 27.394958
105 27.40107
159 27.439076
90 27.466063
104 27.466578
167 27.531674
117 27.533824
138 27.534927
98 27.602302
139 27.610294
107 27.656863
97 27.665441
60 27.687783
112 27.694118
142 27.720588
86 27.751634
84 27.753151
106 27.756685
136 27.83045
154 27.84902
102 27.88491
160 27.92437
75 27.925996
141 27.928309
116 27.977941
94 27.982353
140 28.033088
92 28.047059
115 28.1
164 28.161765
158 28.336135
166 28.5181
123 28.56192
137 28.562284
143 28.568015
124 28.650327
128 28.683007
93 28.757647
161 28.773109
111 29.114846
156 29.397059
157 29.756302
144 29.88603
129 30.063726
145 30.238971
165 30.544118

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

[Paul H. Erlich:]
> Otherwise, 11-tET is highly irrational. In fact, Blackwood and I
independently arrived at the conclusion that it is the best ET for
random dissonance.

11e-out-of-23e (0 2 4 6 8 10 13 15 17 19 21 23 @:0 104 209 313 417 522
678 783 887 991 1096 1200) is a noticeably 'softened' (chromatic*)
eleven that I've used to satisfying (musical) ends. This recast eleven
could also be derived using a process (for creating sets and subsets)
that I outlined in an recent post (TD 277.15), when y=23, as 0 104 209
313 418 522 678 782 887 991 1096 1200 @:10s & 1L differs
insignificantly from the eleven equal in twenty-three equal.

Dan
____________
*Does anyone object to calling these (with chromatic step sizes at an
approximate 1.5 size differential) chromatic sets?