MOS/DE positions

Hi,

given a generator and a period how can I calculate where the moments
of symmetry/ditributionally even scales occur?

Is there a reasonably simple formula for this or is it easier to just
go through all scale sizes and check if there are at most two step
sizes?

I want to make a computer program in QBASIC for this purpose.

Kalle

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> Hi,
>
> given a generator and a period how can I calculate where the moments
> of symmetry/ditributionally even scales occur?

You can do it by finding the semiconvergents (aka secondary
convergents) of g/n, where g is the number of steps in a generator and
n is the number of steps in a period. This ought to be perfect for a
QBASIC program. The Wikipedia article on continued fractions does not
mention semiconvergents, however, so I suppose I'll have to add that.

For instance, suppose your generator is 13 and your period is 41, or
in other words, that you are taking steps of 13 out of an octave of
41. Then the semiconvergents for 13/41 go 1/3, 1/4, 2/7, 3/10, 4/13
... and the MOS are therefore 3, 4, 7, 10, 13 ... If the generator is
25 and the period is 68, we would get 2, 3, 5, 8, 11, 19, 30, 49, 68.
These of course are MOS for the temperament with generator 25/68 of an
octave, but we can multiply them by 9 and then they will be generators
for MOS where the period is 25/612 of a octave, and the period is 1/9
of an octave, or 68/612 octave. Multiplying by 9, we now get MOS/DE of
size 18, 27, 45, 72, 99, 171, 270, 441 and 612; each of these repeats
the same pattern nine times in an octave.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...>
wrote:
> > Hi,
> >
> > given a generator and a period how can I calculate where the
moments
> > of symmetry/ditributionally even scales occur?
>
> You can do it by finding the semiconvergents (aka secondary
> convergents) of g/n, where g is the number of steps in a generator
and
> n is the number of steps in a period. This ought to be perfect for
a
> QBASIC program. The Wikipedia article on continued fractions does
not
> mention semiconvergents, however, so I suppose I'll have to add
that.
>
> For instance, suppose your generator is 13 and your period is 41,
or
> in other words, that you are taking steps of 13 out of an octave of
> 41. Then the semiconvergents for 13/41 go 1/3, 1/4, 2/7, 3/10, 4/13
> ... and the MOS are therefore 3, 4, 7, 10, 13 ... If the generator
is
> 25 and the period is 68, we would get 2, 3, 5, 8, 11, 19, 30, 49,
68.
> These of course are MOS for the temperament with generator 25/68
of an
> octave, but we can multiply them by 9 and then they will be
generators
> for MOS where the period is 25/612 of a octave, and the period is
1/9
> of an octave, or 68/612 octave. Multiplying by 9, we now get
MOS/DE of
> size 18, 27, 45, 72, 99, 171, 270, 441 and 612; each of these
repeats
> the same pattern nine times in an octave.

Thank you, Gene!

You can find my own boneheaded take on the subject here:

< http://web.syr.edu/~rsholmes/music/xen/scale_mos.html >

- Rich Holmes

Kalle Aho wrote:
> Hi, > > given a generator and a period how can I calculate where the moments > of symmetry/ditributionally even scales occur?
> > Is there a reasonably simple formula for this or is it easier to just > go through all scale sizes and check if there are at most two step > sizes?
> > I want to make a computer program in QBASIC for this purpose. For each successive MOS/DE scale, each large step is divided into two smaller steps; one the size of the original small steps and another size of step that can be either larger or smaller. So you also need to keep track of the size of the intervals. Then the number of intervals of the same size as the old small step is the sum of the old large and small steps, and the number of the new intervals is the number of the old large steps.

I use a spreadsheet for this. Put the period in cell A1 and the generator in B1. The step sizes of the next MOS are then B1 and A1-B1. Put the larger step on the left side and the smaller step on the right:

A2 =MAX(A1-B1;B1)
B2 =MIN(A1-B1;B1)

Then I use column C for the number of large steps in a period and column D for the number of small steps. For the first row, just put 1 in C1 and 0 in D1. Then there's a little bit of a trick for the following rows:

C2 =(A2+B1-A1)/(A2-B2)*(C1+D1)+((A1-B1)-B2)/(A2-B2)*C1
D2 =((A1-B1)-B2)/(A2-B2)*(C1+D1)+(A2+B1-A1)/(A2-B2)*C1

But it's easier if you're writing a program: here's a bit of pseudocode that should work.

large_interval_size = large_interval_size - small_interval_size
num_small_steps = num_small_steps + num_large_steps
if (small_interval_size > large_interval_size) then
swap small_interval_size, large_interval_size
swap num_small_steps, num_large_steps
end if

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Kalle Aho wrote:
> > Hi,
> >
> > given a generator and a period how can I calculate where the
moments
> > of symmetry/ditributionally even scales occur?
> >
> > Is there a reasonably simple formula for this or is it easier to
just
> > go through all scale sizes and check if there are at most two
step
> > sizes?
> >
> > I want to make a computer program in QBASIC for this purpose.
>
> For each successive MOS/DE scale, each large step is divided into
two
> smaller steps; one the size of the original small steps and
another size
> of step that can be either larger or smaller. So you also need to
keep
> track of the size of the intervals. Then the number of intervals
of the
> same size as the old small step is the sum of the old large and
small
> steps, and the number of the new intervals is the number of the
old
> large steps.
>
> I use a spreadsheet for this. Put the period in cell A1 and the
> generator in B1. The step sizes of the next MOS are then B1 and A1-
B1.
> Put the larger step on the left side and the smaller step on the
right:
>
> A2 =MAX(A1-B1;B1)
> B2 =MIN(A1-B1;B1)
>
> Then I use column C for the number of large steps in a period and
column
> D for the number of small steps. For the first row, just put 1 in
C1 and
> 0 in D1. Then there's a little bit of a trick for the following
rows:
>
> C2 =(A2+B1-A1)/(A2-B2)*(C1+D1)+((A1-B1)-B2)/(A2-B2)*C1
> D2 =((A1-B1)-B2)/(A2-B2)*(C1+D1)+(A2+B1-A1)/(A2-B2)*C1
>
> But it's easier if you're writing a program: here's a bit of
pseudocode
> that should work.
>
> large_interval_size = large_interval_size - small_interval_size
> num_small_steps = num_small_steps + num_large_steps
> if (small_interval_size > large_interval_size) then
> swap small_interval_size, large_interval_size
> swap num_small_steps, num_large_steps
> end if

Very nice, Herman. Thanks!

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
> You can find my own boneheaded take on the subject here:
>
> < http://web.syr.edu/~rsholmes/music/xen/scale_mos.html >
>
> - Rich Holmes

Thanks!

Here

http://web.syr.edu/~rsholmes/music/xen/scale_nonlinear.html

you tell us:

"One thing I initially found bothersome about Erlich's scales is
their close relationship with 22-equal tuning. The diatonic scale is
not as closely bound to an equal temperament. Reasonably useful
versions of diatonic scales are found in a fairly wide range of
tunings including Pythagorean, 12-equal, 19-equal, and 31-equal. In
all of these four fifths make a major third, so the pattern of a
major triad on the circle of fifths is the same; and no use is made
of wraparound, so the temperament does not influence the number of
notes needed to make three major triads."

This is interesting because I have exactly opposite feelings. What I
like about decatonic scales is precisely the fact that they so
strongly suggest 22 if you want to use equal temperament.

Kalle

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> --- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> http://web.syr.edu/~rsholmes/music/xen/scale_nonlinear.html
>
> you tell us:

Here's another interesting quote from the same page:

"Note carefully he is saying 22-equal gives consonant harmony but out
of tune scales -- the important distinction between harmonic and
melodic usefulness of a tuning. Three observations: first, the
perceived intonation of the 22-equal diatonic scale may have little
relevance for new scales. Second, I'm not sure to what degree the
perceived tuning deficiencies of the 22-equal diatonic scale is
learned response due to familiarity with 12-equal scales (as opposed
to consonance of triads, which presumably relates to beats and is less
subjective). Third, it's probably too much to ask (though it might be
nice) for a not-too-large scale constructed for 7-limit music to
include an excellent 5-limit diatonic scale as a subset."

I've found that people often react to 7-limit just intonation in the
same way, finding it consonant but "out of tune". I think the melodic
"in tune" and "out of tune" has a very large element of learned
response in it. It is interesting, however, that going in the flat
direction does not seem to cause as much difficulty. Generally
reactions to meantone are positive.