comma question

Well, I guess there is a little uncertainty in my mind here...

Also in response to Margo Schulter's interesting post and the idea of
using the ascii symbols "^" and "V" to indicate "commas" in a system,
after the Maneri symbols of arrows for the smallest interval 1/12
tone in 72-tET. Sorry, Monz... I understand you like to do it a
different way.

My question is... and I should know the answer for this:

is the 1/12 tone actually a comma in 72-tET?

Could someone please diagram this for me... Monz?? Anybody??

I would greatly appreciate it.

Thanks!

_______ _______ _______
Joseph Pehrson

I could have added an explanation for the size of the
septimal comma of 64/63 which is two steps or 1/6 tone in
72-tET as Margo said.
Take
H = best approximation to 7/4, in 72 this is 966.67 cents

64/63 = (2 * 2/1) / (2 * 3/2) / (7/4) =

2 * A - 2 * V - H = 2 * 1200 - 2 * 700 - 966.67 = 33.33 cents,
in steps: 2 * 72 - 2 * 42 - 58 = 2 steps.

What you mustn't do is take the value of the comma, and round
that to the nearest step. Often the value you get is the same,
but often not.

Manuel

--- In tuning@y..., jpehrson@r... wrote:
>
> My question is... and I should know the answer for this:
>
> is the 1/12 tone actually a comma in 72-tET?

The 1/12 tone acts as the _syntonic_ comma (81:80) in 72-tET.
Assuming that's what you meant.
>
> Could someone please diagram this for me... Monz?? Anybody??

What sort of diagram are you looking for? Here's one -- a standard 5-
limit lattice diagram:

Av
\
\
\
\
C-------G-------D-------A

The difference between the Av on the upper-left and the A on the
right is a syntonic comma, by definition. In 72-tET, this difference
comes out to 1/12 tone, thus the symbol "v".

--- In tuning@y..., <manuel.op.de.coul@e...> wrote:

/tuning/topicId_25422.html#25429

>
> I could have added an explanation for the size of the
> septimal comma of 64/63 which is two steps or 1/6 tone in
> 72-tET as Margo said.
> Take
> H = best approximation to 7/4, in 72 this is 966.67 cents
>
> 64/63 = (2 * 2/1) / (2 * 3/2) / (7/4) =
>
> 2 * A - 2 * V - H = 2 * 1200 - 2 * 700 - 966.67 = 33.33 cents,
> in steps: 2 * 72 - 2 * 42 - 58 = 2 steps.
>
> What you mustn't do is take the value of the comma, and round
> that to the nearest step. Often the value you get is the same,
> but often not.
>
> Manuel

Oh! I think perhaps I may be getting a "glimmer" of this.

So, this is why, let's say in 72-tET, when you lower the "E" by a
1/12 tone from 12-tET(in a simple C:E:G:Bb tetrad) you are getting
the "just E" 16.6 cents flatter than 12-tET where it has been
tempered out...

So that, obviously, is the syntonic comma "in action..."

And, similarly, when you lower the Bb in the same tetrad, you have to
lower it by 1/6 tone, or 33.3 cents, *two* steps of 72-tET. That way
the larger *septimal* comma, which has been tempered out in 12-tET
becomes the "just" Bb of 72-tET... the *septimal* comma "in
action..."

So that becomes a clear illustration of the two commas in 72-tET.

Correct??

Thanks!

_______ ______ _______
Joseph Pehrson

Joseph wrote:
>And, similarly, when you lower the Bb in the same tetrad, you have to
>lower it by 1/6 tone, or 33.3 cents, *two* steps of 72-tET. That way
>the larger *septimal* comma, which has been tempered out in 12-tET
>becomes the "just" Bb of 72-tET... the *septimal* comma "in
>action..."

>So that becomes a clear illustration of the two commas in 72-tET.

>Correct??

Yes, that's right. So when a comma is said to vanish in a certain
ET, it is represented by 0 steps. This is something you can check
with the command DIVIDE/CONSISTENT.

Manuel

--- In tuning@y..., jpehrson@r... wrote:

> Oh! I think perhaps I may be getting a "glimmer" of this.
>
> So, this is why, let's say in 72-tET, when you lower the "E" by a
> 1/12 tone from 12-tET(in a simple C:E:G:Bb tetrad) you are getting
> the "just E" 16.6 cents flatter than 12-tET where it has been
> tempered out...
>
> So that, obviously, is the syntonic comma "in action..."
>
> And, similarly, when you lower the Bb in the same tetrad, you have
to
> lower it by 1/6 tone, or 33.3 cents, *two* steps of 72-tET. That
way
> the larger *septimal* comma, which has been tempered out in 12-tET
> becomes the "just" Bb of 72-tET... the *septimal* comma "in
> action..."
>
> So that becomes a clear illustration of the two commas in 72-tET.
>
> Correct??

If you add the observation that 12-tET plays the role of Pythagorean
tuning in 72-tET, then yes, this is correct.

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

/tuning/topicId_25422.html#25468

> --- In tuning@y..., jpehrson@r... wrote:
>
> > Oh! I think perhaps I may be getting a "glimmer" of this.
> >
> > So, this is why, let's say in 72-tET, when you lower the "E" by a
> > 1/12 tone from 12-tET(in a simple C:E:G:Bb tetrad) you are
getting
> > the "just E" 16.6 cents flatter than 12-tET where it has been
> > tempered out...
> >
> > So that, obviously, is the syntonic comma "in action..."
> >
> > And, similarly, when you lower the Bb in the same tetrad, you
have
> to
> > lower it by 1/6 tone, or 33.3 cents, *two* steps of 72-tET. That
> way
> > the larger *septimal* comma, which has been tempered out in 12-
tET
> > becomes the "just" Bb of 72-tET... the *septimal* comma "in
> > action..."
> >
> > So that becomes a clear illustration of the two commas in 72-tET.
> >
> > Correct??
>
> If you add the observation that 12-tET plays the role of
Pythagorean tuning in 72-tET, then yes, this is correct.

Hi Paul...

Why this *particular* clarification again??

Thanks!

Joseph

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_25422.html#25468
>
> > --- In tuning@y..., jpehrson@r... wrote:
> >
> > > Oh! I think perhaps I may be getting a "glimmer" of this.
> > >
> > > So, this is why, let's say in 72-tET, when you lower the "E" by
a
> > > 1/12 tone from 12-tET(in a simple C:E:G:Bb tetrad) you are
> getting
> > > the "just E" 16.6 cents flatter than 12-tET where it has been
> > > tempered out...
> > >
> > > So that, obviously, is the syntonic comma "in action..."
> > >
> > > And, similarly, when you lower the Bb in the same tetrad, you
> have
> > to
> > > lower it by 1/6 tone, or 33.3 cents, *two* steps of 72-tET.
That
> > way
> > > the larger *septimal* comma, which has been tempered out in 12-
> tET
> > > becomes the "just" Bb of 72-tET... the *septimal* comma "in
> > > action..."
> > >
> > > So that becomes a clear illustration of the two commas in 72-
tET.
> > >
> > > Correct??
> >
> > If you add the observation that 12-tET plays the role of
> Pythagorean tuning in 72-tET, then yes, this is correct.
>
> Hi Paul...
>
> Why this *particular* clarification again??
>
> Thanks!

Because the commas are defined as the difference between different
pitches in JI, _not_ as the difference between pitches in JI and
pitches in 12-tET.

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

/tuning/topicId_25422.html#25531
>
> Because the commas are defined as the difference between different
> pitches in JI, _not_ as the difference between pitches in JI and
> pitches in 12-tET.

Hi Paul...

So you're saying that when we lower an "E" by 1/12 tone from the "E"
in 12-tET the TRUE syntonic comma would be a lowering from
PYTHAGOREAN to the just "E," instead??

How does the arithmetic work on that again?? We're going from a
Pythagorean third of 408 cents to a just major third of 386 cents,
correct?? That's 22 cents or the 81:80, correct??

So the lowering in 72-tET is from 400 cents to 386 or 14 cents...
well, actually it's to 383 cents because of the error.... so 16.66
cents...

So the syntonic comma is "off" by 22-16 = 6 cents??

________ ______ _______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> Hi Paul...
>
> So you're saying that when we lower an "E" by 1/12 tone from the "E"
> in 12-tET the TRUE syntonic comma would be a lowering from
> PYTHAGOREAN to the just "E," instead??
>
> How does the arithmetic work on that again?? We're going from a
> Pythagorean third of 408 cents to a just major third of 386 cents,
> correct?? That's 22 cents or the 81:80, correct??

Yup.
>
> So the lowering in 72-tET is from 400 cents to 386 or 14 cents...
> well, actually it's to 383 cents because of the error.... so 16.66
> cents...

Yup.
>
> So the syntonic comma is "off" by 22-16 = 6 cents??

Yeah, but remember, a lot of times (e.g. for Western common practice music) we need the
syntonic comma to vanish entirely . . . it's only the consonant intervals that cause "pain" when
they're "off" by 6 cents or x cents, not intervals like the syntonic comma (IMHO).

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

/tuning/topicId_25422.html#25564

> >
> > So the syntonic comma is "off" by 22-16 = 6 cents??
>
> Yeah, but remember, a lot of times (e.g. for Western common
practice music) we need the
> syntonic comma to vanish entirely . . . it's only the consonant
intervals that cause "pain" when
> they're "off" by 6 cents or x cents, not intervals like the
syntonic comma (IMHO).

So might one say that the syntonic comma is a STRUCTURAL, rather than
AUDIBLE characteristic of certain scales??

__________ _________ _______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> So might one say that the syntonic comma is a STRUCTURAL, rather
than
> AUDIBLE characteristic of certain scales??

It's definitely structural. It can _become_ audible in certain chord
progressions, but sometimes (as in much, but not all, of Pachelbel's
canon), it's not audible at all, unless you count the inequality of
the 9:8 and 10:9 melodic whole steps.