Some boring 12-tET stuff

First a boring 5-limit thing:

I'm sure this has been gone over, but I have always felt that
this relationship is important:

3^7/2^11 is 13.685006 cents sharp from 12-tET semitone
5/4 is 13.686286 flat from 12-tET major third

These numbers are very close. We can impose each as a limit
in some kind of tuning, where you never go beyond seven fifths
(chromatic half-step, adjusting 25/24 with the syntonic comma)
and a plain old major third. (Never needing 5^2/2^4, just use
8/5) Of course you cross 6 fifths by going to 7, but I think
this is acceptable due to chromaticism. (C becomes C#)

So the errors cancel in this situation C->C# (chromatic semitone)
and then C#->E# (just third) E# hits F with almost no error (-.00128
cents) I had a better example/construction at one time which I cannot
seem to remember right now. You can of course go C-E-E# instead.

And now a boring 12-tET 7-limit thing: "Hjelmstad's comma" (Oh I know
it's nothing new)

250,047/250,000

(1,000,188/1,000,000) = 1.000188

This is just (1.26)^3 /2 or (63/50)^3 /2

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> First a boring 5-limit thing:
>
> I'm sure this has been gone over, but I have always felt that
> this relationship is important:
>
> 3^7/2^11 is 13.685006 cents sharp from 12-tET semitone
> 5/4 is 13.686286 flat from 12-tET major third
>
> These numbers are very close. We can impose each as a limit
> in some kind of tuning, where you never go beyond seven fifths
> (chromatic half-step, adjusting 25/24 with the syntonic comma)
> and a plain old major third. (Never needing 5^2/2^4, just use
> 8/5) Of course you cross 6 fifths by going to 7, but I think
> this is acceptable due to chromaticism. (C becomes C#)

Duh. This comma is known of course, just (3^8)*5/2^15. Okay back to
my day job:)

> So the errors cancel in this situation C->C# (chromatic semitone)
> and then C#->E# (just third) E# hits F with almost no error (-
.00128
> cents) I had a better example/construction at one time which I
cannot
> seem to remember right now. You can of course go C-E-E# instead.
>
> And now a boring 12-tET 7-limit thing: "Hjelmstad's comma" (Oh I
know
> it's nothing new)
>
> 250,047/250,000
>
> (1,000,188/1,000,000) = 1.000188
>
> This is just (1.26)^3 /2 or (63/50)^3 /2
>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > First a boring 5-limit thing:
> >
> > I'm sure this has been gone over, but I have always felt that
> > this relationship is important:
> >
> > 3^7/2^11 is 13.685006 cents sharp from 12-tET semitone
> > 5/4 is 13.686286 flat from 12-tET major third
> >
> > These numbers are very close. We can impose each as a limit
> > in some kind of tuning, where you never go beyond seven fifths
> > (chromatic half-step, adjusting 25/24 with the syntonic comma)
> > and a plain old major third. (Never needing 5^2/2^4, just use
> > 8/5) Of course you cross 6 fifths by going to 7, but I think
> > this is acceptable due to chromaticism. (C becomes C#)
>
> Duh. This comma is known of course, just (3^8)*5/2^15. Okay back to
> my day job:)

But then again, this is against a just fourth, not a tempered one.
One point is that (5 * 3^7 * 2^-13)^12 =~ 2^5, or rather
5^12 * 3^84 * 2^-156 = 2^5, so 5^12 * 3^84 * 2^-161 is a comma,
which would be [-161, 84, 12] Atom of Kirchenberger?

> > So the errors cancel in this situation C->C# (chromatic semitone)
> > and then C#->E# (just third) E# hits F with almost no error (-
> .00128
> > cents) I had a better example/construction at one time which I
> cannot
> > seem to remember right now. You can of course go C-E-E# instead.
> >
> > And now a boring 12-tET 7-limit thing: "Hjelmstad's comma" (Oh I
> know
> > it's nothing new)
> >
> > 250,047/250,000
> >
> > (1,000,188/1,000,000) = 1.000188
> >
> > This is just (1.26)^3 /2 or (63/50)^3 /2
> >
>

Hi Paul,

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > First a boring 5-limit thing:
> >
> > I'm sure this has been gone over, but I have always felt that
> > this relationship is important:
> >
> > 3^7/2^11 is 13.685006 cents sharp from 12-tET semitone
> > 5/4 is 13.686286 flat from 12-tET major third
> >
> > These numbers are very close. We can impose each as
> > a limit in some kind of tuning, where you never go
> > beyond seven fifths (chromatic half-step, adjusting
> > 25/24 with the syntonic comma) and a plain old
> > major third. (Never needing 5^2/2^4, just use 8/5)
> > Of course you cross 6 fifths by going to 7, but I
> > think this is acceptable due to chromaticism.
> > (C becomes C#)
>
> Duh. This comma is known of course, just (3^8)*5/2^15.
> Okay back to my day job:)

But that's not correct.

This interval is the skhisma. The difference between
the 12-edo 5th and the 3/2 ratio is a grad, which is
1/12 of a pythagorean-comma.

The difference between those two errors in your
illustration, is the difference between a skhisma and
a grad, shown here using vector addition with
rational 2,3,5-monzos (which give an exact value
with no rounding error):

[ -11 7 0 > = 3^7/2^11 = ~113.685006058 cents
- [ 1/12 0 0 > = 12-edo semitone = ~100.000000000 cents
-----------------
[-133/12 7 0 > = ~13.685006058 cents

[ 4/12 0 0 > = 12-edo major-3rd = ~400.000000000 cents
- [ -2 0 1 > = 5/4 ratio = ~386.313713865 cents
-----------------
[ 28/12 0 -1 > = ~13.686286135 cents

[ 28/12 0 -1 > = ~13.686286135 cents
- [-133/12 7 0 > = ~13.685006058 cents
-----------------
[161/12 -7 -1 > = ~0.001280077 cents

You can see exactly this result on my "grad" webpage:

http://tonalsoft.com/enc/g/grad.aspx

>> [ -19/12 1 0 ] grad
>> - [ -15 8 1 ] skhisma
>> ----------------------
>> [ 161/12 -7 -1 ] difference between grad and skhisma

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Paul,

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:

> But then again, this is against a just fourth, not
> a tempered one.

I just sent in a post explaining the correct discrepancy.

> One point is that (5 * 3^7 * 2^-13)^12 =~ 2^5, or rather
> 5^12 * 3^84 * 2^-156 = 2^5, so 5^12 * 3^84 * 2^-161 is
> a comma, which would be [-161, 84, 12] Atom of Kirchenberger?

You got his name wrong: it's "Kirnberger".

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Paul,
>
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <phjelmstad@> wrote:
> > >
> > > First a boring 5-limit thing:
> > >
> > > I'm sure this has been gone over, but I have always felt that
> > > this relationship is important:
> > >
> > > 3^7/2^11 is 13.685006 cents sharp from 12-tET semitone
> > > 5/4 is 13.686286 flat from 12-tET major third
> > >
> > > These numbers are very close. We can impose each as
> > > a limit in some kind of tuning, where you never go
> > > beyond seven fifths (chromatic half-step, adjusting
> > > 25/24 with the syntonic comma) and a plain old
> > > major third. (Never needing 5^2/2^4, just use 8/5)
> > > Of course you cross 6 fifths by going to 7, but I
> > > think this is acceptable due to chromaticism.
> > > (C becomes C#)
> >
> > Duh. This comma is known of course, just (3^8)*5/2^15.
> > Okay back to my day job:)
>
>
> But that's not correct.
>
> This interval is the skhisma. The difference between
> the 12-edo 5th and the 3/2 ratio is a grad, which is
> 1/12 of a pythagorean-comma.
>
> The difference between those two errors in your
> illustration, is the difference between a skhisma and
> a grad, shown here using vector addition with
> rational 2,3,5-monzos (which give an exact value
> with no rounding error):
>
>
> [ -11 7 0 > = 3^7/2^11 = ~113.685006058 cents
> - [ 1/12 0 0 > = 12-edo semitone = ~100.000000000 cents
> -----------------
> [-133/12 7 0 > = ~13.685006058 cents
>
>
> [ 4/12 0 0 > = 12-edo major-3rd = ~400.000000000 cents
> - [ -2 0 1 > = 5/4 ratio = ~386.313713865 cents
> -----------------
> [ 28/12 0 -1 > = ~13.686286135 cents
>
>
> [ 28/12 0 -1 > = ~13.686286135 cents
> - [-133/12 7 0 > = ~13.685006058 cents
> -----------------
> [161/12 -7 -1 > = ~0.001280077 cents
>
>
> You can see exactly this result on my "grad" webpage:
>
> http://tonalsoft.com/enc/g/grad.aspx
>
> >> [ -19/12 1 0 ] grad
> >> - [ -15 8 1 ] skhisma
> >> ----------------------
> >> [ 161/12 -7 -1 ] difference between grad and skhisma
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
Thanx Monz. As you see I realized the error in my error. I will look
at your pages. Last night, I had fun trying to combine
the "landscape" comma which we were chatting about on plain "tuning",
with the Atom of Kirnberger. One fun comma I got is this one:

<-16, 24, -24, 12| =~ 1.000752

Does that one ring any bells?

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> >
> > Hi Paul,
> >
> >
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <phjelmstad@> wrote:
> > >
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <phjelmstad@> wrote:
> > > >
> > > > First a boring 5-limit thing:
> > > >
> > > > I'm sure this has been gone over, but I have always felt that
> > > > this relationship is important:
> > > >
> > > > 3^7/2^11 is 13.685006 cents sharp from 12-tET semitone
> > > > 5/4 is 13.686286 flat from 12-tET major third
> > > >
> > > > These numbers are very close. We can impose each as
> > > > a limit in some kind of tuning, where you never go
> > > > beyond seven fifths (chromatic half-step, adjusting
> > > > 25/24 with the syntonic comma) and a plain old
> > > > major third. (Never needing 5^2/2^4, just use 8/5)
> > > > Of course you cross 6 fifths by going to 7, but I
> > > > think this is acceptable due to chromaticism.
> > > > (C becomes C#)
> > >
> > > Duh. This comma is known of course, just (3^8)*5/2^15.
> > > Okay back to my day job:)
> >
> >
> > But that's not correct.
> >
> > This interval is the skhisma. The difference between
> > the 12-edo 5th and the 3/2 ratio is a grad, which is
> > 1/12 of a pythagorean-comma.
> >
> > The difference between those two errors in your
> > illustration, is the difference between a skhisma and
> > a grad, shown here using vector addition with
> > rational 2,3,5-monzos (which give an exact value
> > with no rounding error):
> >
> >
> > [ -11 7 0 > = 3^7/2^11 = ~113.685006058 cents
> > - [ 1/12 0 0 > = 12-edo semitone = ~100.000000000 cents
> > -----------------
> > [-133/12 7 0 > = ~13.685006058 cents
> >
> >
> > [ 4/12 0 0 > = 12-edo major-3rd = ~400.000000000 cents
> > - [ -2 0 1 > = 5/4 ratio = ~386.313713865 cents
> > -----------------
> > [ 28/12 0 -1 > = ~13.686286135 cents
> >
> >
> > [ 28/12 0 -1 > = ~13.686286135 cents
> > - [-133/12 7 0 > = ~13.685006058 cents
> > -----------------
> > [161/12 -7 -1 > = ~0.001280077 cents
> >
> >
> > You can see exactly this result on my "grad" webpage:
> >
> > http://tonalsoft.com/enc/g/grad.aspx
> >
> > >> [ -19/12 1 0 ] grad
> > >> - [ -15 8 1 ] skhisma
> > >> ----------------------
> > >> [ 161/12 -7 -1 ] difference between grad and skhisma
> >
> >
> >
> > -monz
> > http://tonalsoft.com
> > Tonescape microtonal music software
> >
> Thanx Monz. As you see I realized the error in my error. I will
look
> at your pages. Last night, I had fun trying to combine
> the "landscape" comma which we were chatting about on
plain "tuning",
> with the Atom of Kirnberger. One fun comma I got is this one:
>
> <-16, 24, -24, 12| =~ 1.000752
>
> Does that one ring any bells?
>
> PGH

It's just four times the landscape comma, so there is no value
added here...monz, are you aware of any synergies in combining
these two commas? I am coming up empty...

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@...>
wrote:

> It's just four times the landscape comma, so there is no value
> added here...monz, are you aware of any synergies in combining
> these two commas? I am coming up empty...

If you combine them, you come up with the 7-limit 612&1848 temperament.
Both 612 and 1848 are more notable in the 11-limit than the 7-limit,
and it really makes the most sense to consider this in an 11-limit
framework. This temperament is what I've called "atomic temperament":

/tuning-math/message/11565

I've suggested it might make a good framework for "olympian level"
Sagittal notation.

Graham suggested the temperament might be due to Hans Aberg:

/tuning-math/message/11566

I don't know if we ever figured out whether the temperament should be
called "atomic" or "aberg".

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@>
> wrote:
>
> > It's just four times the landscape comma, so there is no value
> > added here...monz, are you aware of any synergies in combining
> > these two commas? I am coming up empty...
>
> If you combine them, you come up with the 7-limit 612&1848
temperament.
> Both 612 and 1848 are more notable in the 11-limit than the 7-
limit,
> and it really makes the most sense to consider this in an 11-limit
> framework. This temperament is what I've called "atomic
temperament":
>
> /tuning-math/message/11565
>
> I've suggested it might make a good framework for "olympian level"
> Sagittal notation.
>
> Graham suggested the temperament might be due to Hans Aberg:
>
> /tuning-math/message/11566
>
> I don't know if we ever figured out whether the temperament should
be
> called "atomic" or "aberg".

Interesting. Here is a another more elementary way to combine them,
I call it "splitting the atom"

As monz said, the skisma to the grad is 1/12 the Atom of Kirnberger.
(Before clearing denominators).

This is kind of fun:

126/100 is the ratio used in the landscape comma
(3^5 * 5^3)/2^12 * 7) is a cool ratio, almost exactly 2^1/12, a
tempered semitone. Comparing this with 2^1/12, and then taking 12/7
of this, is my "synthetic comma"

((125/126)^1/7 * 3)^12/2^19

(126/100(2^1/3) is 1/3 of landscape

So I guess Landscape^4 / Synth^7 = the Atom

and Landscape^1/3 /Synth^7/12 = Splitting the Atom (Atom^1/12)

Forgive any bad calculations, just my humble way of comparing
Landscape with the Atom.

Does 3^5 * 5^3 / 2^12 * 7 (2^1/12) have a name?

PGH

>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@...>
wrote:

I'm not sure what the heck you are doing, but

(landscape/atom)^(1/3) = |-55 30 2 1> =
36030948116563575/36028797018963968

Aside from 612 and 1848, this is tempered out by 3125, which is a
notable 7-limit system. However, it isn't a super-strong comma as suxch
things go; for example 78125000/78121827. which 3125 also tempers out,
is smaller.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@>
> wrote:
>
> I'm not sure what the heck you are doing, but
>
> (landscape/atom)^(1/3) = |-55 30 2 1> =
> 36030948116563575/36028797018963968

One thing this accomplishes is tnis: giving a comma basis for atomic
temperament in the 7-limit. The atom and the landscape comma don't do
that, but the landscape comma and the above comma, which might be
what you wanted to call the synthetic comma, are the TM reduced comma
basis for 7-limit atomic.

An 11-limit comma basis is {9801/9800, 151263/151250,
184549376/184528125}. From 612+1848=2460, we can see that atomic is
excellent up to the 19-limit, BTW, so this could continue.

What do you think of the idea of basing a notation system on twelve
equally spaced nominals, as Johnny Reinhard does? Atomic leads to a
slicker version of that idea.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@>
> wrote:
>
> I'm not sure what the heck you are doing, but
>
> (landscape/atom)^(1/3) = |-55 30 2 1> =
> 36030948116563575/36028797018963968

Thanks

Some of what I am doing:

Just combining Landscape and Atom of K.

Synthetic comma:

|-145/7, 60/7, 36/7, -12/7>

Little Synthetic comma:

|-145/84, 5/7, 3/7, -1/7>

Baby Landscape:

|-4/3, 2, -2, 1>

Proton of K:

|-161/12, 7, 12/7>

Unnamed comma:

|-145/12, 5, 3, -1>

Up by 12:

|-145, 60, 36, -12>

As an semitone approx:

|-12, 5, 3, -1> (What is this?) Thanks

PGH
> Aside from 612 and 1848, this is tempered out by 3125, which is a
> notable 7-limit system. However, it isn't a super-strong comma as
suxch
> things go; for example 78125000/78121827. which 3125 also tempers
out,
> is smaller.
>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@>
> > wrote:
> >
> > I'm not sure what the heck you are doing, but
> >
> > (landscape/atom)^(1/3) = |-55 30 2 1> =
> > 36030948116563575/36028797018963968
>
> One thing this accomplishes is tnis: giving a comma basis for
atomic
> temperament in the 7-limit. The atom and the landscape comma don't
do
> that, but the landscape comma and the above comma, which might be
> what you wanted to call the synthetic comma, are the TM reduced
comma
> basis for 7-limit atomic.

I'll have to run some calculations. I'm glad they (together) form
some kind of comma basis, somewhere. Especially in the 7-limit.
>
> An 11-limit comma basis is {9801/9800, 151263/151250,
> 184549376/184528125}. From 612+1848=2460, we can see that atomic is
> excellent up to the 19-limit, BTW, so this could continue.

Neat

> What do you think of the idea of basing a notation system on twelve
> equally spaced nominals, as Johnny Reinhard does? Atomic leads to a
> slicker version of that idea.

As Mr. Twelve, I like it.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@>
> > wrote:
> >
> > I'm not sure what the heck you are doing, but
> >
> > (landscape/atom)^(1/3) = |-55 30 2 1> =
> > 36030948116563575/36028797018963968
>
> Thanks
>
> Some of what I am doing:
>
> Just combining Landscape and Atom of K.
>
> Synthetic comma:
>
> |-145/7, 60/7, 36/7, -12/7>
>
> Little Synthetic comma:
>
> |-145/84, 5/7, 3/7, -1/7>
>
> Baby Landscape:
>
> |-4/3, 2, -2, 1>
>
> Proton of K:
>
> |-161/12, 7, 12/7>
>
> Unnamed comma:
>
> |-145/12, 5, 3, -1>
>
> Up by 12:
>
> |-145, 60, 36, -12>
>
> As an semitone approx:
>
> |-12, 5, 3, -1> (What is this?) Thanks

* * * * * * Therefore,

Synth^7 * Landscape^4

|-145, 60, 36, -12> and |-16, 24, -24, 12> produce the atom

Synth^7/12 * Landscape^1/3

|-145/12, 5, 3, -1> and |-4/3, 2, -2, 1> produce the proton

I haven't reconciled with your landscape/atom yet, but I am probably
just too tired.

The basis of what I was doing is merely (One of these is Synth)

((125/126)^1/7 * 3)^12
-----------------------
2^19

Not super imaginative

Also

((126/125)^1/7 * 2^(19/12))^12
------------------------------
3^12

the same thing backwards.

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> > <genewardsmith@> wrote:
> > >
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <phjelmstad@>
> > > wrote:
> > >
> > > I'm not sure what the heck you are doing, but
> > >
> > > (landscape/atom)^(1/3) = |-55 30 2 1> =
> > > 36030948116563575/36028797018963968
> >
> > Thanks
> >
> > Some of what I am doing:
> >
> > Just combining Landscape and Atom of K.
> >
> > Synthetic comma:
> >
> > |-145/7, 60/7, 36/7, -12/7>
> >
> > Little Synthetic comma:
> >
> > |-145/84, 5/7, 3/7, -1/7>
> >
> > Baby Landscape:
> >
> > |-4/3, 2, -2, 1>
> >
> > Proton of K:
> >
> > |-161/12, 7, 12/7>
> >
> > Unnamed comma:
> >
> > |-145/12, 5, 3, -1>
> >
> > Up by 12:
> >
> > |-145, 60, 36, -12>
> >
> > As an semitone approx:
> >
> > |-12, 5, 3, -1> (What is this?) Thanks
>
> * * * * * * Therefore,
>
> Synth^7 * Landscape^4
>
> |-145, 60, 36, -12> and |-16, 24, -24, 12> produce the atom
>
> Synth^7/12 * Landscape^1/3
>
> |-145/12, 5, 3, -1> and |-4/3, 2, -2, 1> produce the proton
>
> I haven't reconciled with your landscape/atom yet, but I am probably
> just too tired.
>
> The basis of what I was doing is merely (One of these is Synth)
>
> ((125/126)^1/7 * 3)^12
> -----------------------
> 2^19
>
> Not super imaginative
>
> Also
>
> ((126/125)^1/7 * 2^(19/12))^12
> ------------------------------
> 3^12
>
> the same thing backwards.
>
>
> PGH

I can tell everybody hates it. It doesn't improve on the grad,
I guess this will just go into the bin of "novelty commas"

In cents:

grad 1.955000865
skhisma 1.953720788
landscape .325441411
landscape^4 1.301765643
inner synthetic 1.97068094 (little big, this is (125/126)^1/7
synthetic .188160939
synthetic^7 1.317126572
synthetic^7/12 |-145/12, 5, 3, -1> .109760548
synthetic^1/12 .015680078

atom .015360929
proton .001280077

"We have a lot on our plates, so need to leverage new synergies"

- Joe Manager

Re: Synthetic comma:

Here's what I can salvage from this:

Major third based on Landscape: 3^2 * 7 / 5^2 * 2
Perfect fourth due to Atom: 3^7 * 5 /2^13
Synthetic half-step (almost 2^1/12): 3^5 * 5^3/ 7 * 2^12

Using these building blocks, and a 31-symbol system, centered on "D",
you can build an almost perfect 12t-ET scale: (Ignore powers of 2)

{0, 0, 0>
|5, 3, -1>
|-14, -2, 0>
|-9, 1, -1>
|2, -2, 1>
|7, 1, 0>
{12, 4, -1>

etc.

For example, you never need more than ##(x)or bb. However, assuming
the scale is C to C, 5^4 would be D## (okay), but 3^-14 would be Cbb
(yuck)and from D to D, 5^4 is E## (yuck) but 3^-14 is Dbb (okay).
So a solution is to take the scale from G to G, giving A## and Gbb
(both fine). I'm not creating single chains, even though one could.
The longest chain is 24 fifths, the last one in the list)

In cents: (Six places, truncated)

0.000000
99.890239
200.002560
299.892799
400.108480
499.998719
599.888959

etc.

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:

(Clipped)

> Hi Paul,
>
> This interval is the skhisma. The difference between
> the 12-edo 5th and the 3/2 ratio is a grad, which is
> 1/12 of a pythagorean-comma.
>
> The difference between those two errors in your
> illustration, is the difference between a skhisma and
> a grad, shown here using vector addition with
> rational 2,3,5-monzos (which give an exact value
> with no rounding error):
>
>
> [ -11 7 0 > = 3^7/2^11 = ~113.685006058 cents
> - [ 1/12 0 0 > = 12-edo semitone = ~100.000000000 cents
> -----------------
> [-133/12 7 0 > = ~13.685006058 cents
>
>
> [ 4/12 0 0 > = 12-edo major-3rd = ~400.000000000 cents
> - [ -2 0 1 > = 5/4 ratio = ~386.313713865 cents
> -----------------
> [ 28/12 0 -1 > = ~13.686286135 cents
>
>
> [ 28/12 0 -1 > = ~13.686286135 cents
> - [-133/12 7 0 > = ~13.685006058 cents
> -----------------
> [161/12 -7 -1 > = ~0.001280077 cents
>
>
> You can see exactly this result on my "grad" webpage:
>
> http://tonalsoft.com/enc/g/grad.aspx
>
> >> [ -19/12 1 0 ] grad
> >> - [ -15 8 1 ] skhisma
> >> ----------------------
> >> [ 161/12 -7 -1 ] difference between grad and skhisma
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software

I thought I would revisit this "bit of Landscape" and I came
up with a better comma than my rather useless "synthetic comma"

Let's take 126/125, which is used in the landscape comma

(126/125 * 5/4)^3

And take it's seventh root, and compare to the grad:

[ -19/12 1 0 0] grad
[ 1/7 2/7 -3/7 1/7] new comma
________________________
[ 145/84 -5/7 -3/7 1/7]

Both the grad and the new comma are about 2 cents,
and the result is 1.000009057209369065465996554943 or

0.015680078240483982111367261356183 cents

126/125 = 13.794766605395312086186699991152 cents
Just to tempered semitone ~13.685006058 cents
Just to tempered third ~13.686286135 cents

Now the ultimate minicomma would have to be the atom^1/84th,
which is:

.000182868142857 cents, which would arrive by comparing the grad,
(okay as is) and the tempered-over-just third ratio, split in seven
pieces.

Of course, this is close to 126/125 (1.26/1.25) split into seven
pieces, which I gave above, and which I call a "bit of landscape".

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
>
> (Clipped)
>
> > Hi Paul,
> >
> > This interval is the skhisma. The difference between
> > the 12-edo 5th and the 3/2 ratio is a grad, which is
> > 1/12 of a pythagorean-comma.
> >
> > The difference between those two errors in your
> > illustration, is the difference between a skhisma and
> > a grad, shown here using vector addition with
> > rational 2,3,5-monzos (which give an exact value
> > with no rounding error):
> >
> >
> > [ -11 7 0 > = 3^7/2^11 = ~113.685006058 cents
> > - [ 1/12 0 0 > = 12-edo semitone = ~100.000000000 cents
> > -----------------
> > [-133/12 7 0 > = ~13.685006058 cents
> >
> >
> > [ 4/12 0 0 > = 12-edo major-3rd = ~400.000000000 cents
> > - [ -2 0 1 > = 5/4 ratio = ~386.313713865 cents
> > -----------------
> > [ 28/12 0 -1 > = ~13.686286135 cents
> >
> >
> > [ 28/12 0 -1 > = ~13.686286135 cents
> > - [-133/12 7 0 > = ~13.685006058 cents
> > -----------------
> > [161/12 -7 -1 > = ~0.001280077 cents
> >
> >
> > You can see exactly this result on my "grad" webpage:
> >
> > http://tonalsoft.com/enc/g/grad.aspx
> >
> > >> [ -19/12 1 0 ] grad
> > >> - [ -15 8 1 ] skhisma
> > >> ----------------------
> > >> [ 161/12 -7 -1 ] difference between grad and skhisma
> >
> >
> >
> > -monz
> > http://tonalsoft.com
> > Tonescape microtonal music software
>
>
> I thought I would revisit this "bit of Landscape" and I came
> up with a better comma than my rather useless "synthetic comma"
>
> Let's take 126/125, which is used in the landscape comma
>
> (126/125 * 5/4)^3
>
> And take it's seventh root, and compare to the grad:
>
> [ -19/12 1 0 0] grad
> [ 1/7 2/7 -3/7 1/7] new comma
> ________________________
> [ 145/84 -5/7 -3/7 1/7]
>
> Both the grad and the new comma are about 2 cents,
> and the result is 1.000009057209369065465996554943 or
>
> 0.015680078240483982111367261356183 cents
>
> 126/125 = 13.794766605395312086186699991152 cents
> Just to tempered semitone ~13.685006058 cents
> Just to tempered third ~13.686286135 cents
>
> Now the ultimate minicomma would have to be the atom^1/84th,
> which is:
>
> .000182868142857 cents, which would arrive by comparing the grad,
> (okay as is) and the tempered-over-just third ratio, split in seven
> pieces.
>
> Of course, this is close to 126/125 (1.26/1.25) split into seven
> pieces, which I gave above, and which I call a "bit of landscape".

Left out that landscape^(1/3) = 1.26 / 2^(1/3) if my calculations are
correct, which can also be split into 7 pieces, and closes the loop
here, I think:-)

> PGH
>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> >
> > (Clipped)
> >
> > > Hi Paul,
> > >
> > > This interval is the skhisma. The difference between
> > > the 12-edo 5th and the 3/2 ratio is a grad, which is
> > > 1/12 of a pythagorean-comma.
> > >
> > > The difference between those two errors in your
> > > illustration, is the difference between a skhisma and
> > > a grad, shown here using vector addition with
> > > rational 2,3,5-monzos (which give an exact value
> > > with no rounding error):
> > >
> > >
> > > [ -11 7 0 > = 3^7/2^11 = ~113.685006058 cents
> > > - [ 1/12 0 0 > = 12-edo semitone = ~100.000000000 cents
> > > -----------------
> > > [-133/12 7 0 > = ~13.685006058 cents
> > >
> > >
> > > [ 4/12 0 0 > = 12-edo major-3rd = ~400.000000000 cents
> > > - [ -2 0 1 > = 5/4 ratio = ~386.313713865 cents
> > > -----------------
> > > [ 28/12 0 -1 > = ~13.686286135 cents
> > >
> > >
> > > [ 28/12 0 -1 > = ~13.686286135 cents
> > > - [-133/12 7 0 > = ~13.685006058 cents
> > > -----------------
> > > [161/12 -7 -1 > = ~0.001280077 cents
> > >
> > >
> > > You can see exactly this result on my "grad" webpage:
> > >
> > > http://tonalsoft.com/enc/g/grad.aspx
> > >
> > > >> [ -19/12 1 0 ] grad
> > > >> - [ -15 8 1 ] skhisma
> > > >> ----------------------
> > > >> [ 161/12 -7 -1 ] difference between grad and skhisma
> > >
> > >
> > >
> > > -monz
> > > http://tonalsoft.com
> > > Tonescape microtonal music software
> >
> >
> > I thought I would revisit this "bit of Landscape" and I came
> > up with a better comma than my rather useless "synthetic comma"
> >
> > Let's take 126/125, which is used in the landscape comma
> >
> > (126/125 * 5/4)^3
> >
> > And take it's seventh root, and compare to the grad:
> >
> > [ -19/12 1 0 0] grad
> > [ 1/7 2/7 -3/7 1/7] new comma
> > ________________________
> > [ 145/84 -5/7 -3/7 1/7]
> >
> > Both the grad and the new comma are about 2 cents,
> > and the result is 1.000009057209369065465996554943 or
> >
> > 0.015680078240483982111367261356183 cents
> >
> > 126/125 = 13.794766605395312086186699991152 cents
> > Just to tempered semitone ~13.685006058 cents
> > Just to tempered third ~13.686286135 cents
> >
> > Now the ultimate minicomma would have to be the atom^1/84th,
> > which is:
> >
> > .000182868142857 cents, which would arrive by comparing the grad,
> > (okay as is) and the tempered-over-just third ratio, split in
seven
> > pieces.
> >
> > Of course, this is close to 126/125 (1.26/1.25) split into seven
> > pieces, which I gave above, and which I call a "bit of landscape".
>
> Left out that landscape^(1/3) = 1.26 / 2^(1/3) if my calculations
are
> correct, which can also be split into 7 pieces, and closes the loop
> here, I think:-)

> > PGH

Of course I meant compare each of these to the grad, or to seven
times the grad, if you don't want to split into seven pieces, in
these last two situations also:

grad^7/((2^1/3)/(5/4))= atom^1/12
grad^7/(126/125) new comma
grad^7/(Landscape)^1/3 = grad^7/(1.26/2^(1/3))

Three parts of the same triangle. Must seem rather sophomoric.

~0.001280077 cents
~0.109760547 cents
~13.57652558 cents

which of course when added back together with the correct signs is
just grad^7. Landscape^1/3 is

~0.1084804702 cents, close to newcomma, and also atom^7
~0.107526468 cents, although typically (landscape/atom)^1/3 is a
comma, not this.

HGP