Black magic

I've already mentioned that if we take the N-et, and set r = N/ln(2),
and then calculate

r' = (r+G+1/8)/ln(r)

we get an adjusted tuning after setting N' = ln(2) r'. Here G
represents the nearest Gram point, which is round(g(r)), where

g(r) = r ln(r) - r - 1/8

and "round" rounds to the nearest integer. This strikes me as almost
black magic, it's so easy. Another piece of the same magic is this:
define a function

tend(N) = 180 (g(r) - round(g(r))),

where again r = N/ln(2), and the "180" makes tend read out in degrees
from -180 to 180. Tend gives the tendency of an et, being positive
for ets with a sharp tendency, and negative for flat ets. We have for
example:

N tend(N)

7 -23
10 8
12 13
15 42
19 -40
22 22
27 75
31 -22
34 40
41 -11
46 15
53 -3
58 67
72 -55
99 54

When using these to create MOS of M steps out of N, it is better that
the tendencies of M and N agree. Thus 19, 31, and 41 are reasonable
fits to the flat 72, while 22, 46 (and 21, where we have tend(21) =
14) are less apt, and 58 is downright awkward. On the other hand,
when adding two ets to get an et, then it is better if the tendencies
are opposite, where they tend to cancel. For instance both 22+31 and
19+34 lead to the neutral 53, whereas adding the slightly sharp 12 to
the distinctly sharp 15 leads to the very sharp 27.

Both the meantone and the 72 systems tend towards flatness, and it
might be interesting to look to the sharp systems (such as the 15 out
of 27 system I mentioned) for something a little different. 22 out of
46, or 27 out of 58, anyone?

I haven't had any zeta feedback--does any of this make sense?

--- In tuning-math@y..., genewardsmith@j... wrote:
> I've already mentioned that if we take the N-et, and set r = N/ln(2),
> and then calculate
>
> r' = (r+G+1/8)/ln(r)
>
> we get an adjusted tuning after setting N' = ln(2) r'. Here G
> represents the nearest Gram point, which is round(g(r)), where
>
> g(r) = r ln(r) - r - 1/8
>
> and "round" rounds to the nearest integer. This strikes me as almost
> black magic, it's so easy. Another piece of the same magic is this:
> define a function
>
> tend(N) = 180 (g(r) - round(g(r))),
>
> where again r = N/ln(2), and the "180" makes tend read out in degrees
> from -180 to 180. Tend gives the tendency of an et, being positive
> for ets with a sharp tendency, and negative for flat ets. We have for
> example:
>
> N tend(N)
>
> 7 -23
> 10 8
> 12 13
> 15 42
> 19 -40
> 22 22
> 27 75
> 31 -22
> 34 40
> 41 -11
> 46 15
> 53 -3
> 58 67
> 72 -55
> 99 54
>
> When using these to create MOS of M steps out of N, it is better that
> the tendencies of M and N agree. Thus 19, 31, and 41 are reasonable
> fits to the flat 72, while 22, 46 (and 21, where we have tend(21) =
> 14) are less apt, and 58 is downright awkward. On the other hand,
> when adding two ets to get an et, then it is better if the tendencies
> are opposite, where they tend to cancel. For instance both 22+31 and
> 19+34 lead to the neutral 53, whereas adding the slightly sharp 12 to
> the distinctly sharp 15 leads to the very sharp 27.
>
> Both the meantone and the 72 systems tend towards flatness, and it
> might be interesting to look to the sharp systems (such as the 15 out
> of 27 system I mentioned) for something a little different. 22 out of
> 46,

That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-out-of-26 (or 14-out-of-38) systems? Blackwood's 10-
out-of-15? The diminished scale (8-out-of-12)?
>
> I haven't had any zeta feedback--does any of this make sense?

Can you describe "tendency" a little more precisely for us ignoramuses?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> > Both the meantone and the 72 systems tend towards flatness, and
it
> > might be interesting to look to the sharp systems (such as the 15
out
> > of 27 system I mentioned) for something a little different. 22
out of
> > 46,

> That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-
out-of-26 (or 14-out-of-38) systems? Blackwood's 10-
> out-of-15? The diminished scale (8-out-of-12)?

As you've probably figured out by now, there's lots I'm not familiar
with--in fact, most of what I know is what I worked out for myself 20-
30 years ago, after reading Helmholtz and something called "Music, a
Science and an Art" by a JI advocate whose name I can't recall. It
all really started in grade school, where a music teacher came into
class and told us about white keys and black keys, and how there were
seven notes to the scale and twelve to the octave. When I
asked "Why?", she said "That's just the way it is." This sort of
answer never makes me happy.

However, let me guess:

Shrutar 2n mod 23, pattern 22222222223 * 2

Paul 10 out of 22, 2n mod 11, pattern 22223 * 2

(Similarly, 12 out of 22, 2n mod 11, pattern 222221 * 2)

Blackwood 10 out of 15, 2n mod 3, pattern 21 * 5

Etc. Is that right?

> Can you describe "tendency" a little more precisely for us
ignoramuses?

It measures the proportional distance to the nearest Gram point from
the N-et. What it actually does is give a high reading if 3, 5, 7
etc. pile up in one direction or another--tending to be sharp or
flat, and a low reading if they are less consistent, with some sharp
and some flat. In some sense all odd primes are considered, but
things are heavily weighted in favor of the first few.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-
out-of-26 (or 14-out-of-38) systems? Blackwood's 10-
> out-of-15? The diminished scale (8-out-of-12)?

Now that I have a sound card, I want to creat midi files, and have
just downloaded Scala. It seems rather formidable, but I noticed a
lot of Paul Erlich .sla files in a list of files there.

Is there a FAQ or something to lead a person through this? Most of
the FAQs seem to want to tell me how to do math!

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > Both the meantone and the 72 systems tend towards flatness, and
> it
> > > might be interesting to look to the sharp systems (such as the
15
> out
> > > of 27 system I mentioned) for something a little different. 22
> out of
> > > 46,
>
> > That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-
> out-of-26 (or 14-out-of-38) systems? Blackwood's 10-
> > out-of-15? The diminished scale (8-out-of-12)?
>
> As you've probably figured out by now, there's lots I'm not
familiar
> with--in fact, most of what I know is what I worked out for myself
20-
> 30 years ago, after reading Helmholtz and something called "Music,
a
> Science and an Art" by a JI advocate whose name I can't recall. It
> all really started in grade school, where a music teacher came into
> class and told us about white keys and black keys, and how there
were
> seven notes to the scale and twelve to the octave. When I
> asked "Why?", she said "That's just the way it is." This sort of
> answer never makes me happy.
>
> However, let me guess:
>
> Shrutar 2n mod 23, pattern 22222222223 * 2

Not quite . . . it's altered to get omnitetrachordality (the two
instances of '3' are placed a 3:2 apart).
>
> Paul 10 out of 22, 2n mod 11, pattern 22223 * 2

That's the symmetrical version . . . the omnitetrachodal version is
altered, with the two '3's a 3:2 apart.
>
> (Similarly, 12 out of 22, 2n mod 11, pattern 222221 * 2)

On my keyboard, I use an altered version, with the two '1's appearing
a 3:2 apart (between E and F and between B and C).
>
> Blackwood 10 out of 15, 2n mod 3, pattern 21 * 5

Right.
>
> Etc. Is that right?
>
> > Can you describe "tendency" a little more precisely for us
> ignoramuses?
>
> It measures the proportional distance to the nearest Gram point
from
> the N-et. What it actually does is give a high reading if 3, 5, 7
> etc. pile up in one direction or another--tending to be sharp or
> flat, and a low reading if they are less consistent, with some
sharp
> and some flat. In some sense all odd primes are considered, but
> things are heavily weighted in favor of the first few.

Very, very interesting! I'm familiar with Riemann's zeta function
from Schroeder's _Number Theory in Science and Communication_ but
that's about it.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-
> out-of-26 (or 14-out-of-38) systems? Blackwood's 10-
> > out-of-15? The diminished scale (8-out-of-12)?
>
> Now that I have a sound card, I want to creat midi files, and have
> just downloaded Scala. It seems rather formidable, but I noticed a
> lot of Paul Erlich .sla files in a list of files there.
>
> Is there a FAQ or something to lead a person through this? Most of
> the FAQs seem to want to tell me how to do math!

A FAQ for Scala? You can ask Manuel Op de Coul directly; he's on the
Tuning List and very helpful.