Re: double negative and other periodicity tricks...

For 5-limit representation in an EDO, the period of 12 seems
to build converging solutions (until it stops). This produces
the familys

12edo : 0 12 24 36 48 etc...
double negative ? 2 14 26 38 50 ...
schismic : 5 17 29 41 53 65 77 ...
meantone : 7 19 31 43 55 67 ...
diachismic : 10 22 34 46 58 ...

any particular names and/or qualities for...

1 13 25 37 49 ...
3 15 27 39 51 ...
4 16 28 40 52 ...
6 18 30 42 54 ...
8 20 32 44 56 ...
9 21 33 45 57 ...
11 23 35 47 59 ...

Any known periods for other limits, or combinations of prime factors
(for instance a {3,7} system that gets the "5"s opporunistically). Any
easy easy easy way to come up with the period for an arbitrary pair
of numbers (besides brute force trail and error which is very easy in
excel for instance).

Bob Valentine

In-Reply-To: <200201300728.JAA110492@ius578.iil.intel.com>
Robert C Valentine wrote:

> 1 13 25 37 49 ...
Quintuple positive

> 3 15 27 39 51 ...
Triple positive

> 4 16 28 40 52 ...
Quadruple negative

> 6 18 30 42 54 ...
Sextuple positive/negative

> 8 20 32 44 56 ...
Quadruple positive

> 9 21 33 45 57 ...
Triple negative

> 11 23 35 47 59 ...
Quintuple negative

> Any known periods for other limits, or combinations of prime factors
> (for instance a {3,7} system that gets the "5"s opporunistically). Any
> easy easy easy way to come up with the period for an arbitrary pair
> of numbers (besides brute force trail and error which is very easy in
> excel for instance).

Don't know what you mean by "period". You can get the linear temperament
consistent with a pair of equal temperaments from
<http://microtonal.co.uk/temper/>. For the relationship between that and
nthly positive, see <http://x31eq.com/notakey.htm>.

Graham

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:
>
> For 5-limit representation in an EDO, the period of 12 seems
> to build converging solutions (until it stops). This produces
> the familys
>
> 12edo : 0 12 24 36 48 etc...
> double negative ? 2 14 26 38 50 ...
> schismic : 5 17 29 41 53 65 77 ...
> meantone : 7 19 31 43 55 67 ...
> diachismic : 10 22 34 46 58 ...

Hmm . . . these are different types of families. For example, 50 is a
meantone, but it is also doubly negative. So you'll have to more
clearly define what you're after.

> From: graham@microtonal.co.uk
> Subject: Re: double negative and other periodicity tricks...
>

<snip catalog of positives and negatives>

thanks Graham.

>
>
> > Any known periods for other limits, or combinations of prime factors
> > (for instance a {3,7} system that gets the "5"s opporunistically). Any
> > easy easy easy way to come up with the period for an arbitrary pair
> > of numbers (besides brute force trail and error which is very easy in
> > excel for instance).
>
> Don't know what you mean by "period".

By period I meant that these were all based on constructing scales
by k+12n (12 is the period). This doesn't work forever (as Paul points
out, 50 is in the double-negative column but is also a meantone).

It I try to search for something that looks like a similar scale
construction formula for a {3,7} system, I get a similar behavior
for k+5n.

5, 10, 15, 20, 25... sits on a fixed error (as multiples of 12
do {3,5}.

6, 11, 16, 21, 26, 31, 36, 41 converges most rapidly to a low error
on 41 before it starts wanderring. This
list seems to "own" {3,7} for low valued
EDOS

7, 12, 17, 22, 27, ... goes all the way out to 77 before the error
starts to get worse.

etc...

So I know exactly how to brute force these things in a spreadsheet
but I'm not sure if there is a simpler method, an obvious method,
something obviously wrong with this method, etc... Note that this
ignores other issues like consistency etc.

Bob Valentine

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:

> any particular names and/or qualities for...

This is defined by the wedgie, which in this case is just a vector product which can be interpreted as a comma, the defining comma of the corresponding 5-limit linear temperament. Thw wedgie can be either with 12 or (most of the time) with one member of this list with another.

> 1 13 25 37 49 ...

262144/253125

> 3 15 27 39 51 ...

128/125

> 4 16 28 40 52 ...

648/625
> 6 18 30 42 54 ...

Like 2 14 26 38, this is not really a 5-limit system

> 8 20 32 44 56 ...

Same comment

> 9 21 33 45 57 ...

128/125 again

> 11 23 35 47 59 ...

6561/6250

In-Reply-To: <200201310900.LAA56810@ius578.iil.intel.com>
Robert C Valentine wrote:

> By period I meant that these were all based on constructing scales
> by k+12n (12 is the period). This doesn't work forever (as Paul points
> out, 50 is in the double-negative column but is also a meantone).

I find the scale tree more useful for classifying scales. The meantones
are (expand messages, etc)

.7 12
. 19
. 26 31
. 33 45 50 43

You can call this branch 7&12. The singly negative scales are an arc
going 7-19-31-43 and getting ever closer to 12. Doubly negative scales on
this diagram are the combination of two singly positive scale, but
otherwise there's nothing special about them. All these scales will work
with the same 7 or 12 rank keyboard mapping. The 12&19 sub-branch uses a
consistent 7-limit mapping.

You could call "singly negative" a loose synonym for 12&7. But others
object to that, which is no problem because you can always call them 12&7.

The "double negative" temperaments are roughly 2&12, which is a branch
from a different scale tree because everything's divisible by 2.

.2 12
. 14
. 16 26
. 18 30 40 38

50 would be on the next row, as 12+38. So 50 has the properties of both a
2&12 and a 7&12 scale.

> It I try to search for something that looks like a similar scale
> construction formula for a {3,7} system, I get a similar behavior
> for k+5n.

What does {3, 7} mean here?

> 5, 10, 15, 20, 25... sits on a fixed error (as multiples of 12
> do {3,5}.
>
> 6, 11, 16, 21, 26, 31, 36, 41 converges most rapidly to a low error
> on 41 before it starts wanderring. This
> list seems to "own" {3,7} for low valued
> EDOS
>
> 7, 12, 17, 22, 27, ... goes all the way out to 77 before the error
> starts to get worse.

In Erv Wilson's landmark paper "On Linear Notations..." archived at
<http://www.anaphoria.com/xen3a.PDF>, these last ones are described as
"quintally negative systems". The line before could be "quintally double
positive". I don't actually know what the 0+5n systems are called.

> etc...
>
> So I know exactly how to brute force these things in a spreadsheet
> but I'm not sure if there is a simpler method, an obvious method,
> something obviously wrong with this method, etc... Note that this
> ignores other issues like consistency etc.

If you want to generalise the "nthly m-positive" system to other bases
than 5, 7 and 12 or other generators than a fifth, it gets messy. It's
easier to say "1+5n" if you mean that pattern of scale. But usually what
you really want to know about is the "6&5" pattern.

So far, this is all melodic. You tend to find new harmonic mappings apply
to new branches of the scale tree. You can define any linear temperament
as a combination of two equal temperaments. That's what my online script
does. The shorthand for this is, eg h5&h6, which means you take the
closest approximation to the prime-number ratios from 5- and 6-equal.

Graham

> From: graham@microtonal.co.uk
> Subject: Re: double negative and other periodicity tricks...
>
> Robert C Valentine wrote:
>
> > By period I meant that these were all based on constructing scales
> > by k+12n (12 is the period). This doesn't work forever (as Paul points
> > out, 50 is in the double-negative column but is also a meantone).
>
> I find the scale tree more useful for classifying scales. The meantones
> are (expand messages, etc)
>
> .7 12
> . 19
> . 26 31
> . 33 45 50 43
>

Cool, this finally clicked with me. By looking for a k+pn I was
establishing part of what is represented here, basically my spreadsheet
looked something like...

k p n T
7 12 0 7
1 19
2 31

<good stuff snipped, thanks>

>
> > It I try to search for something that looks like a similar scale
> > construction formula for a {3,7} system, I get a similar behavior
> > for k+5n.
>
> What does {3, 7} mean here?
>

{3, 7} is 7-limit, without 5, so

1/1 8/7 7/6 4/3 3/2 12/7 7/4

would be the basic construct. I was looking for a period that
I could use the same way as 12 in {3,5}. Looking at things
"your way", I might choose two ETs with low errors and
generate a scale tree from them. For instance, 5 seems to
have a low error vs 3:2 and 7:4, with 9 the next lowest below 10.

9 5
14
23 19
32 47 33 24
41 55 56 70 52 38 43 29

...but this doesn't look right.

>
> In Erv Wilson's landmark paper "On Linear Notations..." archived at
> <http://www.anaphoria.com/xen3a.PDF>, these last ones are described as
> "quintally negative systems". The line before could be "quintally double
> positive". I don't actually know what the 0+5n systems are called.
>

<thanks, snip>

> So far, this is all melodic. You tend to find new harmonic mappings apply
> to new branches of the scale tree. You can define any linear temperament
> as a combination of two equal temperaments. That's what my online script
> does. The shorthand for this is, eg h5&h6, which means you take the
> closest approximation to the prime-number ratios from 5- and 6-equal.
>

so was that right or wrong above?

Bob

>
> Graham