Schisma vs. 12th root of 2

I have a stupid question that has always been at the back of my head: why can't you
derive 12tet by subtracting the schisma (1/12 pythagorean comma) from each of the
pythagorean fifths... obviously you can't because this would produce a whole number
ratio for the 12tet semitone... but how can you reconcile this with the 12th root of 2?
-Justin

>I have a stupid question that has always been at the back of my
>head: why can't you derive 12tet by subtracting the schisma (1/12
>pythagorean comma) from each of the pythagorean fifths...

You can. Following the recent thread, go to...

http://tinyurl.com/4xuu

...and hit "next" until you see "schismic/helmholtz/groven".
For whatever reason 12-tET isn't given as a sample ET... Paul,
why's that?

>obviously you can't because this would produce a whole number
>ratio for the 12tet semitone... but how can you reconcile this
>with the 12th root of 2?

2^1/12 is an irrational number near in size to the schisma,
which is exactly 32805/32768.

-Carl

I guess the question is, why is 12tet thought of as an irrational tuning if it can be
arrived at by tuning with the schisma? And why, then is not the schisma = 12-root-2?
-Justin

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I have a stupid question that has always been at the back of my
> >head: why can't you derive 12tet by subtracting the schisma (1/12
> >pythagorean comma) from each of the pythagorean fifths...
>
> You can. Following the recent thread, go to...
>
> http://tinyurl.com/4xuu
>
> ...and hit "next" until you see "schismic/helmholtz/groven".
> For whatever reason 12-tET isn't given as a sample ET... Paul,
> why's that?
>
> >obviously you can't because this would produce a whole number
> >ratio for the 12tet semitone... but how can you reconcile this
> >with the 12th root of 2?
>
> 2^1/12 is an irrational number near in size to the schisma,
> which is exactly 32805/32768.
>
> -Carl

Dear me, I wrote...

>> 2^1/12 is an irrational number near in size to the schisma,
>> which is exactly 32805/32768.

That's not right. The difference between a just fifth (3:2)
and a 12-tET fifth (700 cents) is exactly 1/12 of an pythag.
comma. And *that* interval is very near in size to a schisma.

>why is 12tet thought of as an irrational tuning if it can be
>arrived at by tuning with the schisma?

It can't be arrived at *exactly* flattening a 3:2 by a
schisma -- only *nearly*.

However, it can be arrived at exactly by flattening a 3:2 by
the 12th root of the pythag. comma, or by defining a semitone
to by the 12th root of 2. Either of these yield irrational
intervals.

-Carl

> That's not right. The difference between a just fifth (3:2)
> and a 12-tET fifth (700 cents) is exactly 1/12 of an pythag.
> comma. And *that* interval is very near in size to a schisma.

I thought the schisma was 1/12 the pythagorean comma.

> It can't be arrived at *exactly* flattening a 3:2 by a
> schisma -- only *nearly*.

Yes, I accept that but can't quite fathom the logic behind why you can't divide the
comma in 12 and simply subtract it off. -Justin

>> That's not right. The difference between a just fifth (3:2)
>> and a 12-tET fifth (700 cents) is exactly 1/12 of an pythag.
>> comma. And *that* interval is very near in size to a schisma.
>
>I thought the schisma was 1/12 the pythagorean comma.

The schisma is the difference between a major third and eight
fifths. It just happens to be very close to 1/12 a pythag.
comma.

>> It can't be arrived at *exactly* flattening a 3:2 by a
>> schisma -- only *nearly*.
>
>Yes, I accept that but can't quite fathom the logic behind why
>you can't divide the comma in 12 and simply subtract it off.

If you want equal-sized divisions, you need to take the 12th
root, not just put a 12 in the denominator. You're allowed
to divide log-freq. measures like cents by 12, but not ratios.

-Carl

hi Justin,

--- In tuning@yahoogroups.com, "Justin Weaver" <improvist@u...> wrote:

/tuning/topicId_46579.html#46585

> > [Carl Lumma:]
> > That's not right. The difference between a just fifth (3:2)
> > and a 12-tET fifth (700 cents) is exactly 1/12 of an pythag.
> > comma. And *that* interval is very near in size to a schisma.
>
> I thought the schisma was 1/12 the pythagorean comma.
>
> > It can't be arrived at *exactly* flattening a 3:2 by a
> > schisma -- only *nearly*.
>
> Yes, I accept that but can't quite fathom the logic behind
> why you can't divide the comma in 12 and simply subtract it
> off. -Justin

a logarithmic 1/12 of a Pythagorean comma is called a "grad":

http://sonic-arts.org/dict/grad.htm

the "skhisma" or "schisma" is the difference between, for example,
a 5/4 "major-3rd" and the Pythagorean "diminished-4th" found by
going 8 4ths upward or 8 5ths downward:

http://sonic-arts.org/dict/schisma.htm

on both of those Dictionary pages, i show the difference
between the two, which is indeed tiny:

>> Note that the skhisma is nearly the same size as the grad,
>> the difference between them being only ~0.001280077 (= ~1/781)
>> cent:
>>
>> 2^x 3^y 5^z
>>
>> [ -19/12 1 0 ] grad
>> - [ -15 8 1 ] skhisma
>> ----------------------
>> [ 161/12 -7 -1 ] difference between grad and skhisma

does that answer your question once and for all?

-monz

>
> If you want equal-sized divisions, you need to take the 12th
> root, not just put a 12 in the denominator. You're allowed
> to divide log-freq. measures like cents by 12, but not ratios.

Of course you'd be multiplying the ratio by 1/12, but that would still yield a whole
number ratio, not an irrational number.

>
> >> Note that the skhisma is nearly the same size as the grad,
> >> the difference between them being only ~0.001280077 (= ~1/781)
> >> cent:
> >>
> >> 2^x 3^y 5^z
> >>
> >> [ -19/12 1 0 ] grad
> >> - [ -15 8 1 ] skhisma
> >> ----------------------
> >> [ 161/12 -7 -1 ] difference between grad and skhisma
>
>
> does that answer your question once and for all?
>
>

Yes, but now my question is: why isn't 1/12 the pythagorean comma = 1 grad vs. the
12th root? I.e., why can't you subtract (i.e. divide in ratios) the 1/12-comma to close
the loop of fifths. -Justin

>> If you want equal-sized divisions, you need to take the 12th
>> root, not just put a 12 in the denominator. You're allowed
>> to divide log-freq. measures like cents by 12, but not ratios.
>
>Of course you'd be multiplying the ratio by 1/12,

No, you wouldn't. Hearing is logarithmic. If you want equal
divisions of some ratio, you must take a root. To get twelve
equal divisions of 2:1, you take the 12th root of 2. You seemed
to understand this. To get 12 equal divisions of the pythag.
comma, you must take the 12th root of the pythag. comma.

-Carl

--- In tuning@yahoogroups.com, "Justin Weaver" <improvist@u...> wrote:
> >
> > >> Note that the skhisma is nearly the same size as the grad,
> > >> the difference between them being only ~0.001280077 (= ~1/781)
> > >> cent:
> > >>
> > >> 2^x 3^y 5^z
> > >>
> > >> [ -19/12 1 0 ] grad
> > >> - [ -15 8 1 ] skhisma
> > >> ----------------------
> > >> [ 161/12 -7 -1 ] difference between grad and skhisma
> >
> >
> > does that answer your question once and for all?
> >
> >
>
> Yes, but now my question is: why isn't 1/12 the
> pythagorean comma = 1 grad vs. the 12th root?

i don't really understand that question.

> I.e., why can't you subtract (i.e. divide in ratios)
> the 1/12-comma to close the loop of fifths. -Justin

the grad (logarithmic 1/12 of a Pythagorean comma) is an
irrational number. mathematically it's [(2^-19)*(3^12)]^(1/12).

the skhisma is a ratio. the two are incommensurable.

if you tuned up a scale of 12 Pythagorean 5ths each tempered
(narrowed) by a skhisma, you'd get this (reduced to one 8ve):

5th ~cents
0 0
1 700.0012801
2 200.0025602
3 900.0038402
4 400.0051203
5 1100.0064
6 600.0076805
7 100.0089605
8 800.0102406
9 300.0115207
10 1000.012801
11 500.0140809
12 0.015360929

it's extremely doubtful that anyone could hear a difference
between that and regular 12edo, but you can see mathematically
that they are not exactly the same.

-monz

--- In tuning@yahoogroups.com, "Justin Weaver" <improvist@u...> wrote:

> I have a stupid question that has always been at the back of my
head: why can't you
> derive 12tet by subtracting the schisma (1/12 pythagorean comma)
from each of the
> pythagorean fifths... obviously you can't because this would
produce a whole number
> ratio for the 12tet semitone... but how can you reconcile this with
the 12th root of 2?
> -Justin

hi justin, this is not a stupid question, it comes up here about once
a year. one person who noticed this near-coincidence was the famous
kirnberger, who proposed tuning 12-equal exactly in this way --
subtracting a schisma from each of the pythagorean fifths. the
schisma is not exactly 1/12 pythagorean comma, which would be
irrational -- rather, the schisma is the just ratio 32805:32768,
which can be derived from a chain of eight just fifths and a just
major third. the result of doing as kirnberger suggested will yield a
12-tone chain that falls short of closing on itself by the ratio

2923003274661805836407369665432566039311865085952
-------------------------------------------------
2922977339492680612451840826835216578535400390625

or, more usefully, 2^161 * 3^-84 * 5^-12, or [161 -84 -12] for short.

if, as carl suggested, you went to

http://tinyurl.com/4xuu

but then clicked on "last" and scrolled down to the bottom, you would
see that i've called this the "atom of kirnberger" and it's only
0.015361 cent.

the atom of kirnberger is not only the difference between 12 schismas
and a pythagorean comma, it's also the difference between 11 schismas
and a syntonic comma. since it's so small, it's typically ignored,
and so it's said that these 11-schisma and 12-schisma constructions
are exact. they're not.

Thanks, this is perfectly clear now. -Justin

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning@yahoogroups.com, "Justin Weaver" <improvist@u...> wrote:
> > >
> > > >> Note that the skhisma is nearly the same size as the grad,
> > > >> the difference between them being only ~0.001280077 (= ~1/781)
> > > >> cent:
> > > >>
> > > >> 2^x 3^y 5^z
> > > >>
> > > >> [ -19/12 1 0 ] grad
> > > >> - [ -15 8 1 ] skhisma
> > > >> ----------------------
> > > >> [ 161/12 -7 -1 ] difference between grad and skhisma
> > >
> > >
> > > does that answer your question once and for all?
> > >
> > >
> >
> > Yes, but now my question is: why isn't 1/12 the
> > pythagorean comma = 1 grad vs. the 12th root?
>
>
> i don't really understand that question.
>
>
> > I.e., why can't you subtract (i.e. divide in ratios)
> > the 1/12-comma to close the loop of fifths. -Justin
>
>
> the grad (logarithmic 1/12 of a Pythagorean comma) is an
> irrational number. mathematically it's [(2^-19)*(3^12)]^(1/12).
>
> the skhisma is a ratio. the two are incommensurable.
>
> if you tuned up a scale of 12 Pythagorean 5ths each tempered
> (narrowed) by a skhisma, you'd get this (reduced to one 8ve):
>
> 5th ~cents
> 0 0
> 1 700.0012801
> 2 200.0025602
> 3 900.0038402
> 4 400.0051203
> 5 1100.0064
> 6 600.0076805
> 7 100.0089605
> 8 800.0102406
> 9 300.0115207
> 10 1000.012801
> 11 500.0140809
> 12 0.015360929
>
>
> it's extremely doubtful that anyone could hear a difference
> between that and regular 12edo, but you can see mathematically
> that they are not exactly the same.
>
>
>
> -monz

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

/tuning/topicId_46579.html#46601

> --- In tuning@yahoogroups.com, "Justin Weaver" <improvist@u...>
wrote:
>
> > I have a stupid question that has always been at the back of my
> head: why can't you
> > derive 12tet by subtracting the schisma (1/12 pythagorean comma)
> from each of the
> > pythagorean fifths... obviously you can't because this would
> produce a whole number
> > ratio for the 12tet semitone... but how can you reconcile this
with
> the 12th root of 2?
> > -Justin
>
> hi justin, this is not a stupid question, it comes up here about
once
> a year. one person who noticed this near-coincidence was the famous
> kirnberger, who proposed tuning 12-equal exactly in this way --
> subtracting a schisma from each of the pythagorean fifths. the
> schisma is not exactly 1/12 pythagorean comma, which would be
> irrational -- rather, the schisma is the just ratio 32805:32768,
> which can be derived from a chain of eight just fifths and a just
> major third. the result of doing as kirnberger suggested will yield
a
> 12-tone chain that falls short of closing on itself by the ratio
>
> 2923003274661805836407369665432566039311865085952
> -------------------------------------------------
> 2922977339492680612451840826835216578535400390625
>
> or, more usefully, 2^161 * 3^-84 * 5^-12, or [161 -84 -12] for
short.
>
> if, as carl suggested, you went to
>
> http://tinyurl.com/4xuu
>
> but then clicked on "last" and scrolled down to the bottom, you
would
> see that i've called this the "atom of kirnberger" and it's only
> 0.015361 cent.
>
> the atom of kirnberger is not only the difference between 12
schismas
> and a pythagorean comma, it's also the difference between 11
schismas
> and a syntonic comma. since it's so small, it's typically ignored,
> and so it's said that these 11-schisma and 12-schisma constructions
> are exact. they're not.

***This is a great post. I love that fraction! :)

J. Pehrson