Zoom diagrams

I have been working with Paul Ehrlich's Zoom diagrams (The 2
dimensional diagrams that have temperaments as points and 5-limit
commas as lines). I am trying to derive a single formula that will
relate the slopes of the temperament family (comma) lines with their
factors (3^a*5^b.) I have found a formula for the first "hexant" (as
opposed to quadrant) using arctangent/60 degrees. Most of my results
come within 1 degree. However, taking meantone in the sixth "hexant" I
get about -14 degrees from measuring with a ruler, and calculate 12
degrees from 3^4/5^1. Now I know that going clockwise from 0 degrees to
negative 60 degrees brings you to 6/5 (minor third). Meantone is in
this hexant.

My question: Is there one formula for the whole diagram, or does each
60 degrees require different formulas? (Based on perfect fifth->major
third->major sixth->perfect fourth->minor sixth->minor third->perfect
fifth. Paul? Any thoughts?

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:
> I have been working with Paul Ehrlich's Zoom diagrams (The 2
> dimensional diagrams that have temperaments as points and 5-limit
> commas as lines). I am trying to derive a single formula that will
> relate the slopes of the temperament family (comma) lines with their
> factors (3^a*5^b.) I have found a formula for the first "hexant" (as
> opposed to quadrant) using arctangent/60 degrees. Most of my results
> come within 1 degree. However, taking meantone in the sixth "hexant"
I
> get about -14 degrees from measuring with a ruler, and calculate 12
> degrees from 3^4/5^1. Now I know that going clockwise from 0 degrees
to
> negative 60 degrees brings you to 6/5 (minor third). Meantone is in
> this hexant.
>
> My question: Is there one formula for the whole diagram, or does each
> 60 degrees require different formulas? (Based on perfect fifth->major
> third->major sixth->perfect fourth->minor sixth->minor third->perfect
> fifth. Paul? Any thoughts?

Note: I meant to say (arctangent(y/x)/60 degrees). I should mention I
calculated diaschizoid within 1 degree, in the sixth hexant.

Sorry -- I replied to this incorrectly before. Here's the correct
reply.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:
> I have been working with Paul Ehrlich's

That's Erlich :)

> Zoom diagrams (The 2
> dimensional diagrams that have temperaments as points and 5-limit
> commas as lines). I am trying to derive a single formula that will
> relate the slopes of the temperament family (comma) lines with their
> factors (3^a*5^b.) I have found a formula for the first "hexant" (as
> opposed to quadrant) using arctangent/60 degrees. Most of my results
> come within 1 degree. However, taking meantone in the sixth "hexant"
I
> get about -14 degrees from measuring with a ruler, and calculate 12
> degrees from 3^4/5^1. Now I know that going clockwise from 0 degrees
to
> negative 60 degrees brings you to 6/5 (minor third). Meantone is in
> this hexant.
>
> My question: Is there one formula for the whole diagram, or does
each
> 60 degrees require different formulas? (Based on perfect fifth-
>major
> third->major sixth->perfect fourth->minor sixth->minor third-
>perfect
> fifth. Paul? Any thoughts?

Hi Paul,

Good talking to you on the phone yesterday.

As I explained, I've used different sets of angles for different
diagrams.

I believe you're looking at the older diagrams where octave-
equivalence
is assumed and the three 5-odd-limit consonant interval classes are
put
on an equal footing.

Because of duality, you can simply calculate the angle that the
relevant comma is in the equilateral triangular lattice, and there's
your angle for the line on the "zoom" graph.

For example, the syntonic comma is found on the lattice as follows
(hit
reply to see this correctly on the web):

1/1-------3/2-------9/8------27/16
\
\
\
\
81/80

Thus it's 3.5 units to the right and sqrt(3)/2 = 0.86603 unit down
from the origin.

So x=+3.5, y=-0.86603.

Matlab has a two-argument arctangent function, ATAN2(Y,X)

atan2(-0.86603,3.5) = -0.24257 radians = -13.898 degrees.

That's pretty darn close to the -14 degrees you got from measuring
with a ruler!

Hopefully this is easy enough for you to extend to all the
other 'commas'.

Cool?

-P

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> Sorry -- I replied to this incorrectly before. Here's the correct
> reply.
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@m...> wrote:
> > I have been working with Paul Ehrlich's
>
> That's Erlich :)
>
> > Zoom diagrams (The 2
> > dimensional diagrams that have temperaments as points and 5-limit
> > commas as lines). I am trying to derive a single formula that will
> > relate the slopes of the temperament family (comma) lines with
their
> > factors (3^a*5^b.) I have found a formula for the first "hexant"
(as
> > opposed to quadrant) using arctangent/60 degrees. Most of my
results
> > come within 1 degree. However, taking meantone in the
sixth "hexant"
> I
> > get about -14 degrees from measuring with a ruler, and calculate
12
> > degrees from 3^4/5^1. Now I know that going clockwise from 0
degrees
> to
> > negative 60 degrees brings you to 6/5 (minor third). Meantone is
in
> > this hexant.
> >
> > My question: Is there one formula for the whole diagram, or does
> each
> > 60 degrees require different formulas? (Based on perfect fifth-
> >major
> > third->major sixth->perfect fourth->minor sixth->minor third-
> >perfect
> > fifth. Paul? Any thoughts?
>
> Hi Paul,
>
> Good talking to you on the phone yesterday.
>
> As I explained, I've used different sets of angles for different
> diagrams.
>
> I believe you're looking at the older diagrams where octave-
> equivalence
> is assumed and the three 5-odd-limit consonant interval classes are
> put
> on an equal footing.
>
> Because of duality, you can simply calculate the angle that the
> relevant comma is in the equilateral triangular lattice, and there's
> your angle for the line on the "zoom" graph.
>
> For example, the syntonic comma is found on the lattice as follows
> (hit
> reply to see this correctly on the web):
>
> 1/1-------3/2-------9/8------27/16
> \
> \
> \
> \
> 81/80
>
> Thus it's 3.5 units to the right and sqrt(3)/2 = 0.86603 unit down
> from the origin.
>
> So x=+3.5, y=-0.86603.
>
> Matlab has a two-argument arctangent function, ATAN2(Y,X)
>
> atan2(-0.86603,3.5) = -0.24257 radians = -13.898 degrees.
>
> That's pretty darn close to the -14 degrees you got from measuring
> with a ruler!
>
> Hopefully this is easy enough for you to extend to all the
> other 'commas'.
>
> Cool?
>
> -P

Way cool! Thanks man!