<s>
In	O
musical	O
tuning	O
,	O
a	O
lattice	B-Application
"	O
is	O
a	O
way	O
of	O
modeling	O
the	O
tuning	O
relationships	O
of	O
a	O
just	O
intonation	O
system	O
.	O
</s>
<s>
It	O
is	O
an	O
array	B-Architecture
of	O
points	O
in	O
a	O
periodic	O
multidimensional	O
pattern	O
.	O
</s>
<s>
Each	O
point	O
on	O
the	O
lattice	B-Application
corresponds	O
to	O
a	O
ratio	O
(	O
i.e.	O
,	O
a	O
pitch	O
,	O
or	O
an	O
interval	O
with	O
respect	O
to	O
some	O
other	O
point	O
on	O
the	O
lattice	B-Application
)	O
.	O
</s>
<s>
The	O
lattice	B-Application
can	O
be	O
two-	O
,	O
three-	O
,	O
or	O
n-dimensional	O
,	O
with	O
each	O
dimension	O
corresponding	O
to	O
a	O
different	O
prime-number	O
partial	O
.	O
"	O
</s>
<s>
When	O
listed	O
in	O
a	O
spreadsheet	B-Application
a	O
lattice	B-Application
may	O
be	O
referred	O
to	O
as	O
a	O
tuning	B-Application
table	I-Application
.	O
</s>
<s>
The	O
points	O
in	O
a	O
lattice	B-Application
represent	O
pitch	O
classes	O
(	O
or	O
pitches	O
if	O
octaves	O
are	O
represented	O
)	O
,	O
and	O
the	O
connectors	O
in	O
a	O
lattice	B-Application
represent	O
the	O
intervals	O
between	O
them	O
.	O
</s>
<s>
The	O
connecting	O
lines	O
in	O
a	O
lattice	B-Application
display	O
intervals	O
as	O
vectors	O
,	O
so	O
that	O
a	O
line	O
of	O
the	O
same	O
length	O
and	O
angle	O
always	O
has	O
the	O
same	O
intervalic	O
relationship	O
between	O
the	O
points	O
it	O
connects	O
,	O
no	O
matter	O
where	O
it	O
occurs	O
in	O
the	O
lattice	B-Application
.	O
</s>
<s>
Lattices	B-Application
in	O
just	O
intonation	O
(	O
limited	O
to	O
intervals	O
comprising	O
primes	O
,	O
their	O
powers	O
,	O
and	O
their	O
products	O
)	O
are	O
theoretically	O
infinite	O
(	O
because	O
no	O
power	O
of	O
any	O
prime	O
equals	O
any	O
power	O
of	O
another	O
prime	O
)	O
.	O
</s>
<s>
However	O
,	O
lattices	B-Application
are	O
sometimes	O
also	O
used	O
to	O
notate	O
limited	O
subsets	O
that	O
are	O
particularly	O
interesting	O
(	O
such	O
as	O
an	O
Eikosany	O
illustrated	O
further	O
below	O
or	O
the	O
various	O
ways	O
to	O
extract	O
particular	O
scale	O
shapes	O
from	O
a	O
larger	O
lattice	B-Application
)	O
.	O
</s>
<s>
Examples	O
of	O
musical	O
lattices	B-Application
include	O
the	O
Tonnetz	O
of	O
Euler	B-Language
(	O
1739	O
)	O
and	O
Hugo	O
Riemann	O
and	O
the	O
tuning	O
systems	O
of	O
composer-theorists	O
Ben	O
Johnston	O
and	O
James	O
Tenney	O
.	O
</s>
<s>
Thus	O
Pythagorean	O
tuning	O
,	O
which	O
uses	O
only	O
the	O
perfect	O
fifth	O
(	O
3/2	O
)	O
and	O
octave	O
(	O
2/1	O
)	O
and	O
their	O
multiples	O
(	O
powers	O
of	O
2	O
and	O
3	O
)	O
,	O
is	O
represented	O
through	O
a	O
two-dimensional	O
lattice	B-Application
(	O
or	O
,	O
given	O
octave	O
equivalence	O
,	O
a	O
single	O
dimension	O
)	O
,	O
while	O
standard	O
(	O
5-limit	O
)	O
just	O
intonation	O
,	O
which	O
adds	O
the	O
use	O
of	O
the	O
just	O
major	O
third	O
(	O
5/4	O
)	O
,	O
may	O
be	O
represented	O
through	O
a	O
three-dimensional	O
lattice	B-Application
though	O
"	O
a	O
twelve-note	O
'	O
chromatic	O
 '	O
scale	O
may	O
be	O
represented	O
as	O
a	O
two-dimensional	O
(	O
3	O
,	O
5	O
)	O
projection	O
plane	O
within	O
the	O
three-dimensional	O
(	O
2	O
,	O
3	O
,	O
5	O
)	O
space	O
needed	O
to	O
map	O
the	O
scale	O
.	O
</s>
<s>
Erv	O
Wilson	O
has	O
made	O
significant	O
headway	O
with	O
developing	O
lattices	B-Application
than	O
can	O
represent	O
higher	O
limit	O
harmonics	O
,	O
meaning	O
more	O
than	O
2	O
dimensions	O
,	O
while	O
displaying	O
them	O
in	O
2	O
dimensions	O
.	O
</s>
<s>
Here	O
is	O
a	O
template	O
he	O
used	O
to	O
generate	O
what	O
he	O
called	O
an	O
"	O
Euler	B-Language
"	O
lattice	B-Application
after	O
where	O
he	O
drew	O
his	O
inspiration	O
.	O
</s>
<s>
Each	O
prime	O
harmonic	O
(	O
each	O
vector	O
representing	O
a	O
ratio	O
of	O
1/n	O
or	O
n/1	O
where	O
n	O
is	O
a	O
prime	O
)	O
has	O
a	O
unique	O
spacing	O
,	O
avoiding	O
clashes	O
even	O
when	O
generating	O
lattices	B-Application
of	O
multidimensional	O
,	O
harmonically	O
based	O
structure	O
.	O
</s>
