The Intersection of Music, Science, and Mathematics: Insights on Perception and Model-Fitting
Michael E McIntyre
Dept of Applied Mathematics & Theoretical Physics
University of Cambridge
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How did our ancestors survive?
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An idea to take seriously (with far-reaching implications):
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An idea to take seriously (with far-reaching implications):
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the perceived reality.
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An idea to take seriously (with far-reaching implications):
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the perceived reality.
Example: the “walking lights
”:
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Absolute Proof.)
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Absolute Proof.)
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:
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.
Another way to say it is:
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:
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m
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.
Another way to say it is:
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v
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–
and what is abstraction?
Answer:
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a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
M
o
d
e
l
s
a
n
d
m
o
d
e
l
-
f
i
t
t
i
n
g
r
e
q
u
i
r
e
m
a
t
h
e
m
a
t
i
c
s
(e.g. Euclidean geometry).
I
m
p
l
i
c
a
t
i
o
n
:
w
e
a
l
l
h
a
v
e
u
n
c
o
n
s
c
i
o
u
s
m
a
t
h
e
m
a
t
i
c
s
.
Another way to say it is:
W
e
a
l
l
h
a
v
e
a
n
u
n
c
o
n
s
c
i
o
u
s
p
o
w
e
r
o
f
a
b
s
t
r
a
c
t
i
o
n
–
and what is abstraction?
Answer:
t
h
e
a
b
i
l
i
t
y
t
o
h
a
n
d
l
e
m
a
n
y
p
o
s
s
i
b
i
l
i
t
i
e
s
a
t
o
n
c
e
(
e
v
e
n
a
n
i
n
f
i
n
i
t
e
n
u
m
b
e
r
o
f
p
o
s
s
i
b
i
l
i
t
i
e
s
)
.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
All this can be seen as a consequence of biological
n
a
t
u
r
a
l
s
e
l
e
c
t
i
o
n
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
All this can be seen as a consequence of biological
n
a
t
u
r
a
l
s
e
l
e
c
t
i
o
n
along with certain mathematical facts, especially
c
o
m
b
i
n
a
t
o
r
i
a
l
l
a
r
g
e
n
e
s
s
.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
All this can be seen as a consequence of biological
n
a
t
u
r
a
l
s
e
l
e
c
t
i
o
n
along with certain mathematical facts, especially
c
o
m
b
i
n
a
t
o
r
i
a
l
l
a
r
g
e
n
e
s
s
.
The unconscious brain must choose the
m
o
d
e
l
c
o
m
p
o
n
e
n
t
s
t
o
f
i
t
t
o
t
h
e
i
n
c
o
m
i
n
g
d
a
t
a
f
r
o
m
a
c
o
m
b
i
n
a
t
o
r
i
a
l
l
y
l
a
r
g
e
n
u
m
b
e
r
o
f
p
o
s
s
i
b
i
l
i
t
i
e
s
.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
“
P
l
a
t
o
n
i
c
”
v
e
r
s
u
s
“
c
o
n
s
t
r
u
c
t
i
v
i
s
t
”
–
another of our false dichotomies! Again see Lucidity and Science Part II, p. 296.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
So, internal-model components need to be
a
s
s
i
m
p
l
e
a
s
p
o
s
s
i
b
l
e
:
(Platonic objects – elegance – computational economy)
“
P
l
a
t
o
n
i
c
”
v
e
r
s
u
s
“
c
o
n
s
t
r
u
c
t
i
v
i
s
t
”
–
another of our false dichotomies! Again see Lucidity and Science Part II, p. 296.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
So, internal-model components need to be
a
s
s
i
m
p
l
e
a
s
p
o
s
s
i
b
l
e
:
(Platonic objects – elegance – computational economy)
“
P
l
a
t
o
n
i
c
”
v
e
r
s
u
s
“
c
o
n
s
t
r
u
c
t
i
v
i
s
t
”
–
another of our false dichotomies! Again see Lucidity and Science Part II, p. 296.
P
e
r
f
e
c
t
l
y
s
m
o
o
t
h
c
u
r
v
e
s
a
r
e
w
h
a
t
w
e
p
e
r
c
e
i
v
e
w
h
e
n
t
h
e
d
a
t
a
a
l
l
o
w
i
t
.
So, internal-model components need to be
a
s
s
i
m
p
l
e
a
s
p
o
s
s
i
b
l
e
:
(Platonic objects – elegance – computational economy)
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
“
P
l
a
t
o
n
i
c
”
v
e
r
s
u
s
“
c
o
n
s
t
r
u
c
t
i
v
i
s
t
”
–
another of our false dichotomies! Again see Lucidity and Science Part II, p. 296.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
So, internal-model components need to be
a
s
s
i
m
p
l
e
a
s
p
o
s
s
i
b
l
e
:
(Platonic objects – elegance – computational economy)
P
e
r
f
e
c
t
l
y
s
m
o
o
t
h
c
u
r
v
e
s
a
r
e
w
h
a
t
w
e
p
e
r
c
e
i
v
e
w
h
e
n
t
h
e
d
a
t
a
a
l
l
o
w
i
t
.
“
P
l
a
t
o
n
i
c
”
v
e
r
s
u
s
“
c
o
n
s
t
r
u
c
t
i
v
i
s
t
”
–
another of our false dichotomies! Again see Lucidity and Science Part II, p. 296.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
So, internal-model components need to be
a
s
s
i
m
p
l
e
a
s
p
o
s
s
i
b
l
e
:
(Platonic objects – elegance – computational economy)
P
e
r
f
e
c
t
l
y
s
m
o
o
t
h
c
u
r
v
e
s
a
r
e
w
h
a
t
w
e
p
e
r
c
e
i
v
e
w
h
e
n
t
h
e
d
a
t
a
a
l
l
o
w
i
t
.
A
p
e
r
f
e
c
t
l
y
s
t
r
a
i
g
h
t
l
i
n
e
i
s
a
n
o
t
h
e
r
P
l
a
t
o
n
i
c
o
b
j
e
c
t
:
“
P
l
a
t
o
n
i
c
”
v
e
r
s
u
s
“
c
o
n
s
t
r
u
c
t
i
v
i
s
t
”
–
another of our false dichotomies! Again see Lucidity and Science Part II, p. 296.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
So, internal-model components need to be
a
s
s
i
m
p
l
e
a
s
p
o
s
s
i
b
l
e
:
(Platonic objects – elegance – computational economy)
P
e
r
f
e
c
t
l
y
s
m
o
o
t
h
c
u
r
v
e
s
a
r
e
w
h
a
t
w
e
p
e
r
c
e
i
v
e
w
h
e
n
t
h
e
d
a
t
a
a
l
l
o
w
i
t
.
A
p
e
r
f
e
c
t
l
y
s
t
r
a
i
g
h
t
l
i
n
e
i
s
a
n
o
t
h
e
r
P
l
a
t
o
n
i
c
o
b
j
e
c
t
:
“
P
l
a
t
o
n
i
c
”
v
e
r
s
u
s
“
c
o
n
s
t
r
u
c
t
i
v
i
s
t
”
–
another of our false dichotomies! Again see Lucidity and Science Part II, p. 296.
T
h
e
r
e
a
r
e
m
a
n
y
m
o
r
e
P
l
a
t
o
n
i
c
o
b
j
e
c
t
s
.
E
.
g
.
a
n
i
n
f
i
n
i
t
e
l
y
l
a
r
g
e
n
u
m
b
e
r
o
f
s
m
o
o
t
h
c
u
r
v
e
s
.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
So, internal-model components need to be
a
s
s
i
m
p
l
e
a
s
p
o
s
s
i
b
l
e
:
(Platonic objects – elegance – computational economy)
“
P
l
a
t
o
n
i
c
”
v
e
r
s
u
s
“
c
o
n
s
t
r
u
c
t
i
v
i
s
t
”
–
another of our false dichotomies! Again see Lucidity and Science Part II, p. 296.
T
h
e
r
e
a
r
e
m
a
n
y
m
o
r
e
P
l
a
t
o
n
i
c
o
b
j
e
c
t
s
.
E
.
g
.
a
n
i
n
f
i
n
i
t
e
l
y
l
a
r
g
e
n
u
m
b
e
r
o
f
s
m
o
o
t
h
c
u
r
v
e
s
.
Here’s another smooth curve – can you see it?
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
So, internal-model components need to be
a
s
s
i
m
p
l
e
a
s
p
o
s
s
i
b
l
e
:
(Platonic objects – elegance – computational economy)
“
P
l
a
t
o
n
i
c
”
v
e
r
s
u
s
“
c
o
n
s
t
r
u
c
t
i
v
i
s
t
”
–
another of our false dichotomies! Again see Lucidity and Science Part II, p. 296.
T
h
e
r
e
a
r
e
m
a
n
y
m
o
r
e
P
l
a
t
o
n
i
c
o
b
j
e
c
t
s
.
E
.
g
.
a
n
i
n
f
i
n
i
t
e
l
y
l
a
r
g
e
n
u
m
b
e
r
o
f
s
m
o
o
t
h
c
u
r
v
e
s
.
Here’s another smooth curve – can you see it?
(unconscious
mathematics
again)
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
So, internal-model components need to be
a
s
s
i
m
p
l
e
a
s
p
o
s
s
i
b
l
e
:
(Platonic objects – elegance – computational economy)
“
P
l
a
t
o
n
i
c
”
v
e
r
s
u
s
“
c
o
n
s
t
r
u
c
t
i
v
i
s
t
”
–
another of our false dichotomies! Again see Lucidity and Science Part II, p. 296.
M
o
d
e
l
-
f
i
t
t
i
n
g
t
a
k
e
s
p
l
a
c
e
i
n
s
p
a
c
e
t
i
m
e
.
(
T
h
e
w
a
l
k
i
n
g
lights demonstrate it, as does any other movie.)
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
(unconscious
mathematics
again)
“
P
l
a
t
o
n
i
c
”
v
e
r
s
u
s
“
c
o
n
s
t
r
u
c
t
i
v
i
s
t
”
–
another of our false dichotomies! Again see Lucidity and Science Part II, p. 296.
M
o
d
e
l
-
f
i
t
t
i
n
g
t
a
k
e
s
p
l
a
c
e
i
n
s
p
a
c
e
t
i
m
e
.
(
T
h
e
w
a
l
k
i
n
g
lights demonstrate it, as does any other movie.)
W
e
c
a
n
p
e
r
c
e
i
v
e
c
o
n
t
i
n
u
o
u
s
m
o
t
i
o
n
d
e
s
p
i
t
e
a
c
t
u
a
l
d
i
s
c
o
n
t
i
n
u
i
t
i
e
s
.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
(unconscious
mathematics
again)
“
P
l
a
t
o
n
i
c
”
v
e
r
s
u
s
“
c
o
n
s
t
r
u
c
t
i
v
i
s
t
”
–
another of our false dichotomies! Again see Lucidity and Science Part II, p. 296.
M
o
d
e
l
-
f
i
t
t
i
n
g
t
a
k
e
s
p
l
a
c
e
i
n
s
p
a
c
e
t
i
m
e
.
(
T
h
e
w
a
l
k
i
n
g
lights demonstrate it, as does any other movie.)
W
e
c
a
n
p
e
r
c
e
i
v
e
c
o
n
t
i
n
u
o
u
s
m
o
t
i
o
n
d
e
s
p
i
t
e
a
c
t
u
a
l
d
i
s
c
o
n
t
i
n
u
i
t
i
e
s
.
Musical example: Mozart’s “flowing oil” – a feeling of
continuous motion evoked by
d
i
s
c
o
n
t
i
n
u
o
u
s
s
o
u
n
d
s
(
f
r
o
m
t
h
e
K
5
4
5
p
i
a
n
o
s
o
n
a
t
a
)
.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
(unconscious
mathematics
again)
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
M
u
s
i
c
t
a
k
e
s
u
s
d
e
e
p
e
r
s
t
i
l
l
.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
M
u
s
i
c
t
a
k
e
s
u
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”
Tui song recorded in
Invercargill, New Zealand:
Recording by Les McPherson
>
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”
Also basic to our perceptual and cognitive processes:
t
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”
Mathematics is full of changing patterns (of numbers, shapes,
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.
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g
s
s
t
a
y
t
h
e
s
a
m
e
:
“conservation theorems”... “invariance properties”…
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
M
u
s
i
c
c
o
n
s
i
s
t
s
o
f
o
r
g
a
n
i
c
a
l
l
y
-
c
h
a
n
g
i
n
g
s
o
u
n
d
p
a
t
t
e
r
n
s
.
M
u
s
i
c
a
n
d
m
a
t
h
e
m
a
t
i
c
s
b
o
t
h
s
t
e
m
f
r
o
m
o
u
r
unconscious interest in abstract patterns for their own sake,
e
s
p
e
c
i
a
l
l
y
o
r
g
a
n
i
c
a
l
l
y
c
h
a
n
g
i
n
g
p
a
t
t
e
r
n
s
–
p
a
t
t
e
r
n
s
w
e
l
i
k
e
t
o
p
l
a
y
w
i
t
h
.
Mathematics is full of changing patterns (of numbers, shapes,
s
y
m
m
e
t
r
y
p
r
o
p
e
r
t
i
e
s
,
…
)
b
u
t
s
o
m
e
t
h
i
n
g
s
s
t
a
y
t
h
e
s
a
m
e
:
“conservation theorems”... “invariance properties”…
M
a
t
h
e
m
a
t
i
c
s
i
s
f
u
l
l
o
f
p
a
t
t
e
r
n
s
t
h
a
t
l
o
o
k
d
i
s
s
i
m
i
l
a
r
a
t
f
i
r
s
t
b
u
t
–
w
h
e
n
y
o
u
g
o
d
e
e
p
e
r
,
w
h
e
n
y
o
u
p
l
a
y
w
i
t
h
t
h
e
m
e
n
o
u
g
h
–
are found to be similar, or intimately related, perhaps
within a still larger set of possibilities:
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
M
u
s
i
c
c
o
n
s
i
s
t
s
o
f
o
r
g
a
n
i
c
a
l
l
y
-
c
h
a
n
g
i
n
g
s
o
u
n
d
p
a
t
t
e
r
n
s
.
M
u
s
i
c
a
n
d
m
a
t
h
e
m
a
t
i
c
s
b
o
t
h
s
t
e
m
f
r
o
m
o
u
r
unconscious interest in abstract patterns for their own sake,
e
s
p
e
c
i
a
l
l
y
o
r
g
a
n
i
c
a
l
l
y
c
h
a
n
g
i
n
g
p
a
t
t
e
r
n
s
–
p
a
t
t
e
r
n
s
w
e
l
i
k
e
t
o
p
l
a
y
w
i
t
h
.
Mathematics is full of changing patterns (of numbers, shapes,
s
y
m
m
e
t
r
y
p
r
o
p
e
r
t
i
e
s
,
…
)
b
u
t
s
o
m
e
t
h
i
n
g
s
s
t
a
y
t
h
e
s
a
m
e
:
“conservation theorems”... “invariance properties”…
M
a
t
h
e
m
a
t
i
c
s
i
s
f
u
l
l
o
f
p
a
t
t
e
r
n
s
t
h
a
t
l
o
o
k
d
i
s
s
i
m
i
l
a
r
a
t
f
i
r
s
t
b
u
t
–
w
h
e
n
y
o
u
g
o
d
e
e
p
e
r
,
w
h
e
n
y
o
u
p
l
a
y
w
i
t
h
t
h
e
m
e
n
o
u
g
h
–
are found to be similar, or intimately related, perhaps
within a still larger set of possibilities:
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
M
u
s
i
c
c
o
n
s
i
s
t
s
o
f
o
r
g
a
n
i
c
a
l
l
y
-
c
h
a
n
g
i
n
g
s
o
u
n
d
p
a
t
t
e
r
n
s
.
M
u
s
i
c
a
n
d
m
a
t
h
e
m
a
t
i
c
s
b
o
t
h
s
t
e
m
f
r
o
m
o
u
r
unconscious interest in abstract patterns for their own sake,
e
s
p
e
c
i
a
l
l
y
o
r
g
a
n
i
c
a
l
l
y
c
h
a
n
g
i
n
g
p
a
t
t
e
r
n
s
–
p
a
t
t
e
r
n
s
w
e
l
i
k
e
t
o
p
l
a
y
w
i
t
h
.
Here are two patterns that are intimately related:
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
Here are two patterns that are intimately related:
Back to music:
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
E
.
g
.
t
h
e
w
a
y
h
a
r
m
o
n
y
w
o
r
k
s
(
w
e
b
s
e
a
r
c
h
“
m
u
s
i
c
a
l
h
y
p
e
r
s
p
a
c
e
”
)
:
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
E
.
g
.
t
h
e
w
a
y
h
a
r
m
o
n
y
w
o
r
k
s
(
w
e
b
s
e
a
r
c
h
“
m
u
s
i
c
a
l
h
y
p
e
r
s
p
a
c
e
”
)
:
P
o
w
e
r
f
u
l
,
c
o
n
t
i
n
u
o
u
s
h
a
r
m
o
n
i
c
m
o
t
i
o
n
u
s
e
s
o
r
g
a
n
i
c
c
h
a
n
g
e
– some things changing slightly while others stay the same –
w
h
e
r
e
“
s
l
i
g
h
t
l
y
”
n
o
w
h
a
s
t
w
o
a
l
t
e
r
n
a
t
i
v
e
m
e
a
n
i
n
g
s
:
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
E
.
g
.
t
h
e
w
a
y
h
a
r
m
o
n
y
w
o
r
k
s
(
w
e
b
s
e
a
r
c
h
“
m
u
s
i
c
a
l
h
y
p
e
r
s
p
a
c
e
”
)
:
P
o
w
e
r
f
u
l
,
c
o
n
t
i
n
u
o
u
s
h
a
r
m
o
n
i
c
m
o
t
i
o
n
u
s
e
s
o
r
g
a
n
i
c
c
h
a
n
g
e
– some things changing slightly while others stay the same –
w
h
e
r
e
“
s
l
i
g
h
t
l
y
”
n
o
w
h
a
s
t
w
o
a
l
t
e
r
n
a
t
i
v
e
m
e
a
n
i
n
g
s
:
”harmonically close”
:
”melodically close”
:
(
A
n
d
p
o
w
e
r
f
u
l
c
h
o
r
d
s
a
r
e
m
a
d
e
o
f
h
a
r
m
o
n
i
c
-
s
e
r
i
e
s
s
u
b
s
e
t
s
.
D
e
b
u
s
s
y
w
a
s
t
h
e
first great composer to recognize all this – see
Debussy and the Harmonic Series
by Sir Peter Platt.)
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
E
.
g
.
t
h
e
w
a
y
h
a
r
m
o
n
y
w
o
r
k
s
(
w
e
b
s
e
a
r
c
h
“
m
u
s
i
c
a
l
h
y
p
e
r
s
p
a
c
e
”
)
:
P
o
w
e
r
f
u
l
,
c
o
n
t
i
n
u
o
u
s
h
a
r
m
o
n
i
c
m
o
t
i
o
n
u
s
e
s
o
r
g
a
n
i
c
c
h
a
n
g
e
– some things changing slightly while others stay the same –
w
h
e
r
e
“
s
l
i
g
h
t
l
y
”
n
o
w
h
a
s
t
w
o
a
l
t
e
r
n
a
t
i
v
e
m
e
a
n
i
n
g
s
:
”harmonically close”
:
”melodically close”
:
(
A
n
d
p
o
w
e
r
f
u
l
c
h
o
r
d
s
a
r
e
m
a
d
e
o
f
h
a
r
m
o
n
i
c
-
s
e
r
i
e
s
s
u
b
s
e
t
s
.
D
e
b
u
s
s
y
w
a
s
t
h
e
first great composer to recognize all this – see
Debussy and the Harmonic Series
by Sir Peter Platt.)
(
W
e
s
t
e
r
n
m
u
s
i
c
h
a
s
t
w
o
m
a
i
n
p
i
t
c
h
-
t
e
m
p
l
a
t
e
s
,
i
n
s
l
i
g
h
t
t
e
n
s
i
o
n
–
a
r
i
c
h
a
r
t
i
s
t
i
c
r
e
s
o
u
r
c
e
!
!
)
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
E
.
g
.
t
h
e
w
a
y
h
a
r
m
o
n
y
w
o
r
k
s
(
w
e
b
s
e
a
r
c
h
“
m
u
s
i
c
a
l
h
y
p
e
r
s
p
a
c
e
”
)
:
P
o
w
e
r
f
u
l
,
c
o
n
t
i
n
u
o
u
s
h
a
r
m
o
n
i
c
m
o
t
i
o
n
u
s
e
s
o
r
g
a
n
i
c
c
h
a
n
g
e
– some things changing slightly while others stay the same –
w
h
e
r
e
“
s
l
i
g
h
t
l
y
”
n
o
w
h
a
s
t
w
o
a
l
t
e
r
n
a
t
i
v
e
m
e
a
n
i
n
g
s
:
”harmonically close”
:
”melodically close”
:
O
n
e
l
a
s
t
p
o
i
n
t
:
m
u
s
i
c
r
e
m
i
n
d
s
u
s
t
h
a
t
t
h
e
b
r
a
i
n
’
s
unconscious model-fitting is done
i
n
t
i
m
e
a
s
w
e
l
l
a
s
s
p
a
c
e
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
O
n
e
l
a
s
t
p
o
i
n
t
:
m
u
s
i
c
r
e
m
i
n
d
s
u
s
t
h
a
t
t
h
e
b
r
a
i
n
’
s
unconscious model-fitting is done
i
n
t
i
m
e
a
s
w
e
l
l
a
s
s
p
a
c
e
(or rather, in this case, musical hyperspace).
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
O
n
e
l
a
s
t
p
o
i
n
t
:
m
u
s
i
c
r
e
m
i
n
d
s
u
s
t
h
a
t
t
h
e
b
r
a
i
n
’
s
unconscious model-fitting is done
i
n
t
i
m
e
a
s
w
e
l
l
a
s
s
p
a
c
e
(or rather, in this case, musical hyperspace).
Therefore,
s
u
b
j
e
c
t
i
v
e
t
i
m
e
c
a
n
d
i
f
f
e
r
f
r
o
m
o
b
j
e
c
t
i
v
e
(
o
u
t
s
i
d
e
-
w
o
r
l
d
)
t
i
m
e
because
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
s
u
b
j
e
c
t
i
v
e
,
i
.
e
.
,
p
e
r
c
e
i
v
e
d
,
t
i
m
e
i
s
a
n
i
n
t
e
r
n
a
l
-
m
o
d
e
l
p
r
o
p
e
r
t
y
.
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
s
u
b
j
e
c
t
i
v
e
,
i
.
e
.
,
p
e
r
c
e
i
v
e
d
,
t
i
m
e
i
s
a
n
i
n
t
e
r
n
a
l
-
m
o
d
e
l
p
r
o
p
e
r
t
y
.
(So it needn’t obey the usual laws of physics regarding causality.)
W
e
b
s
e
a
r
c
h
e
-
b
o
o
k
”
l
u
c
i
d
i
t
y
p
r
i
n
c
i
p
l
e
s
”
s
u
b
j
e
c
t
i
v
e
,
i
.
e
.
,
p
e
r
c
e
i
v
e
d
,
t
i
m
e
i
s
a
n
i
n
t
e
r
n
a
l
-
m
o
d
e
l
p
r
o
p
e
r
t
y
.
(So it needn’t obey the usual laws of physics regarding causality.)
E
x
a
m
p
l
e
:
a
c
a
u
s
a
l
i
t
y
i
l
l
u
s
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Exploring the relationship between music, science, and mathematics reveals profound connections in understanding perception through model-fitting. An intriguing idea suggests that the unconscious brain fits internal models to sensory data, shaping our perceived reality. This concept has far-reaching implications and can be illustrated by phenomena like the walking lights.
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What can music tell us about science and mathematics? Michael E McIntyre Dept of Applied Mathematics & Theoretical Physics University of Cambridge Websearch e-book lucidity principles
What can music tell us about science and mathematics? What indeed is music? What is science? What is mathematics? Websearch e-book lucidity principles
What can music tell us about science and mathematics? What indeed is music? What is science? What is mathematics? How did our ancestors survive? Websearch e-book lucidity principles
What can music tell us about science and mathematics? What indeed is music? What is science? What is mathematics? How did our ancestors survive? How does perception work? Websearch e-book lucidity principles
An idea to take seriously (with far-reaching implications): perception works by model-fitting. Websearch e-book lucidity principles
An idea to take seriously (with far-reaching implications): perception works by model-fitting. The unconscious brain actively fits an internal model to the sensory data coming in from the outside world. Websearch e-book lucidity principles
An idea to take seriously (with far-reaching implications): perception works by model-fitting. The unconscious brain actively fits an internal model to the sensory data coming in from the outside world. If the fit is good enough, the internal model becomes the perceived reality. Websearch e-book lucidity principles
An idea to take seriously (with far-reaching implications): perception works by model-fitting. The unconscious brain actively fits an internal model to the sensory data coming in from the outside world. If the fit is good enough, the internal model becomes the perceived reality. Example: the walking lights : Websearch e-book lucidity principles
And what is a model? Websearch e-book lucidity principles
And what is a model? (in the sense used here a very general sense) Websearch e-book lucidity principles
And what is a model? (in the sense used here a very general sense) Answer: a partial and approximate representation of reality (e.g. of a real person really walking) Websearch e-book lucidity principles
And what is a model? (in the sense used here a very general sense) Answer: a partial and approximate representation of reality (e.g. of a real person really walking) NB: Science works the same way. Websearch e-book lucidity principles
And what is a model? (in the sense used here a very general sense) Answer: a partial and approximate representation of reality (e.g. of a real person really walking) NB: Science works the same way. (e.g. Einstein s theory of gravity) Websearch e-book lucidity principles
And what is a model? (in the sense used here a very general sense) Answer: a partial and approximate representation of reality (e.g. of a real person really walking) NB: Science works the same way. (e.g. Einstein s theory of gravity) (Implication: science is not about Absolute Truth or Absolute Proof.) Websearch e-book lucidity principles
And what is a model? (in the sense used here a very general sense) Answer: a partial and approximate representation of reality (e.g. of a real person really walking) NB: Science works the same way. (e.g. Einstein s theory of gravity) (Implication: science is not about Absolute Truth or Absolute Proof.) (For one thing, we must assumethat there s an outside world ) Websearch e-book lucidity principles
Models and model-fitting require mathematics Websearch e-book lucidity principles
Models and model-fitting require mathematics (e.g. Euclidean geometry). Websearch e-book lucidity principles
Models and model-fitting require mathematics (e.g. Euclidean geometry). Implication: we all have unconscious mathematics. Websearch e-book lucidity principles
Models and model-fitting require mathematics (e.g. Euclidean geometry). Implication: we all have unconscious mathematics. Another way to say it is: We all have an unconscious power of abstraction Websearch e-book lucidity principles
Models and model-fitting require mathematics (e.g. Euclidean geometry). Implication: we all have unconscious mathematics. Another way to say it is: We all have an unconscious power of abstraction and what is abstraction? Answer: the ability to handle many possibilities at once Websearch e-book lucidity principles
Models and model-fitting require mathematics (e.g. Euclidean geometry). Implication: we all have unconscious mathematics. Another way to say it is: We all have an unconscious power of abstraction and what is abstraction? Answer: the ability to handle many possibilities at once (even an infinite number of possibilities). Websearch e-book lucidity principles
All this can be seen as a consequence of biological natural selection Websearch e-book lucidity principles
All this can be seen as a consequence of biological natural selection along with certain mathematical facts, especially combinatorial largeness. Websearch e-book lucidity principles
All this can be seen as a consequence of biological natural selection along with certain mathematical facts, especially combinatorial largeness. The unconscious brain must choose the model components to fit to the incoming data from a combinatorially large number of possibilities. Websearch e-book lucidity principles
So, internal-model components need to be as simple as possible: (Platonic objects elegance computational economy) Websearch e-book lucidity principles
So, internal-model components need to be as simple as possible: (Platonic objects elegance computational economy) Websearch e-book lucidity principles
So, internal-model components need to be as simple as possible: (Platonic objects elegance computational economy) Perfectly smooth curves are what we perceive when the data allow it. Websearch e-book lucidity principles
So, internal-model components need to be as simple as possible: (Platonic objects elegance computational economy) Perfectly smooth curves are what we perceive when the data allow it. Websearch e-book lucidity principles
So, internal-model components need to be as simple as possible: (Platonic objects elegance computational economy) Perfectly smooth curves are what we perceive when the data allow it. A perfectly straight line is another Platonic object: Websearch e-book lucidity principles
So, internal-model components need to be as simple as possible: (Platonic objects elegance computational economy) Perfectly smooth curves are what we perceive when the data allow it. A perfectly straight line is another Platonic object: Websearch e-book lucidity principles
So, internal-model components need to be as simple as possible: (Platonic objects elegance computational economy) There are many more Platonic objects. E.g. an infinitely large number of smooth curves. Websearch e-book lucidity principles
So, internal-model components need to be as simple as possible: (Platonic objects elegance computational economy) There are many more Platonic objects. E.g. an infinitely large number of smooth curves. Here s another smooth curve can you see it? Websearch e-book lucidity principles
So, internal-model components need to be as simple as possible: (Platonic objects elegance computational economy) There are many more Platonic objects. E.g. an infinitely large number of smooth curves. Here s another smooth curve can you see it? (unconscious mathematics again) Websearch e-book lucidity principles
Model-fitting takes place in spacetime. (The walking lights demonstrate it, as does any other movie.) (unconscious mathematics again) Websearch e-book lucidity principles
Model-fitting takes place in spacetime. (The walking lights demonstrate it, as does any other movie.) We can perceive continuous motion despite actual discontinuities. (unconscious mathematics again) Websearch e-book lucidity principles
Model-fitting takes place in spacetime. (The walking lights demonstrate it, as does any other movie.) We can perceive continuous motion despite actual discontinuities. Musical example: Mozart s flowing oil a feeling of continuous motion evoked by discontinuous sounds(from the K545 piano sonata). (unconscious mathematics again) Websearch e-book lucidity principles
Music takes us deeper still. Websearch e-book lucidity principles
Music takes us deeper still. Natural selection again: auditory scene analysis. Websearch e-book lucidity principles
Music takes us deeper still. Natural selection again: auditory scene analysis. An implication is that the harmonic series is another Platonic object, part of the brain s model-building repertoire Websearch e-book lucidity principles
Music takes us deeper still. Natural selection again: auditory scene analysis. An implication is that the harmonic series is another Platonic object, part of the brain s model-building repertoire Websearch e-book lucidity principles
Music takes us deeper still. Natural selection again: auditory scene analysis. An implication is that the harmonic series is another Platonic object, part of the brain s model-building repertoire (cross-cultural indeed cross-species) Websearch e-book lucidity principles
Music takes us deeper still. Natural selection again: auditory scene analysis. An implication is that the harmonic series is another Platonic object, part of the brain s model-building repertoire (cross-cultural indeed cross-species) Websearch e-book lucidity principles
Music takes us deeper still. Natural selection again: auditory scene analysis. An implication is that the harmonic series is another Platonic object, part of the brain s model-building repertoire (cross-cultural indeed cross-species) Websearch e-book lucidity principles
Music takes us deeper still. Natural selection again: auditory scene analysis. An implication is that the harmonic series is another Platonic object, part of the brain s model-building repertoire (cross-cultural indeed cross-species) Websearch e-book lucidity principles
Music takes us deeper still. Natural selection again: auditory scene analysis. An implication is that the harmonic series is another Platonic object, part of the brain s model-building repertoire (cross-cultural indeed cross-species) Websearch e-book lucidity principles
Music takes us deeper still. Natural selection again: auditory scene analysis. An implication is that the harmonic series is another Platonic object, part of the brain s model-building repertoire (cross-cultural indeed cross-species) Websearch e-book lucidity principles
Music takes us deeper still. Natural selection again: auditory scene analysis. An implication is that the harmonic series is another Platonic object, part of the brain s model-building repertoire (cross-cultural indeed cross-species) Websearch e-book lucidity principles
Music takes us deeper still. Natural selection again: auditory scene analysis. An implication is that the harmonic series is another Platonic object, part of the brain s model-building repertoire (cross-cultural indeed cross-species) Websearch e-book lucidity principles
Tui song recorded in Invercargill, New Zealand: > Recording by Les McPherson Websearch e-book lucidity principles
Also basic to our perceptual and cognitive processes: the organic-change principle: The unconscious brain is interested in and likes to play with patterns in which some things change slightly, while others stay the same. (Another abstract idea.) And it takes us from juvenile play toward some of the deepest connections between music and mathematics. Websearch e-book lucidity principles
