Yahoo Tuning Groups Ultimate Backup tuning tiny interval: (s.c.-11 skh.) and (P.c.-12 skh.)

Has anyone before ever pointed out the tiny interval
2^161 * 3^-84 * 5^-12, which is only about 1/65 cent?

I found it by looking into the fact that the skhisma
is almost exactly 1/11 syntonic comma. This microscopic
interval is the difference between 11 skhismas and the
comma, and also the difference between 12 skhismas and
the Pythagorean comma. Using vector addition:

2^ 3^ 5^ ratio cents

syntonic comma [-4 4 - 1 ] 1.0125 ~21.5062896
11 skhismas - [-165 88 11 ] ~1.012491016 ~21.49092867
-----------------
[ 161 -84 -12 ] ~1.000008873 ~0.015360929

Pythagorean comma [ -19 12 0 ] ~1.013643265 ~23.46001038
12 skhismas - [-180 96 12 ] ~1.013634271 ~23.44464946
-----------------
[ 161 -84 -12 ] ~1.000008873 ~0.015360929

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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🔗genewardsmith@juno.com

🔗graham@microtonal.co.uk

🔗David Beardsley <davidbeardsley@biink.com>

Maybe you could express that as a ratio so us math impared
types can understand what you're talking about?

* David Beardsley
* http://biink.com
* http://mp3.com/davidbeardsley

----- Original Message -----
From: monz <joemonz@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Thursday, November 22, 2001 4:04 PM
Subject: [tuning] tiny interval: (s.c.-11 skh.) and (P.c.-12 skh.)

> Has anyone before ever pointed out the tiny interval
> 2^161 * 3^-84 * 5^-12, which is only about 1/65 cent?
>
> I found it by looking into the fact that the skhisma
> is almost exactly 1/11 syntonic comma. This microscopic
> interval is the difference between 11 skhismas and the
> comma, and also the difference between 12 skhismas and
> the Pythagorean comma. Using vector addition:
>
>
> 2^ 3^ 5^ ratio cents
>
> syntonic comma [-4 4 - 1 ] 1.0125 ~21.5062896
> 11 skhismas - [-165 88 11 ] ~1.012491016 ~21.49092867
> -----------------
> [ 161 -84 -12 ] ~1.000008873 ~0.015360929
>
>
>
> Pythagorean comma [ -19 12 0 ] ~1.013643265 ~23.46001038
> 12 skhismas - [-180 96 12 ] ~1.013634271 ~23.44464946
> -----------------
> [ 161 -84 -12 ] ~1.000008873 ~0.015360929
>
>
>
> love / peace / harmony ...
>
> -monz
> http://www.monz.org
> "All roads lead to n^0"
>
>
>
>
>
> _________________________________________________________
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🔗graham@microtonal.co.uk

🔗BobWendell@technet-inc.com

Without having attempted to think it through yet, Gene, I'm wondering
whether the perfect identity:

Syntonic comma - 11 schismas = Pythagorean comma - 12 schismas

is trivial, or significant. Oh, I think I just realized that the
schisma occurs at 8 perfect fifths down, the syntonic comma at 4
fifths up, which totals 12 fifths to generate the Pythagorean comma.
Just answered my own question. Pretty trivial I guess. Can't see how
it would lead to any major new discoveries. Thanks anyway, but would
welcome any additional observations or comments.

Sincerely,

Bob

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > Has anyone before ever pointed out the tiny interval
> > 2^161 * 3^-84 * 5^-12, which is only about 1/65 cent?
>
> This is one of the 5-limit commas I found when I went looking for
> them; I would have posted on it but I didn't know what to do with
> them once found. I could put up a list of such beasts if it would
be
> of interest, however.

🔗David Beardsley <davidbeardsley@biink.com>

Thanks!

----- Original Message -----
From: <graham@microtonal.co.uk>
To: <tuning@yahoogroups.com>
Sent: Friday, November 23, 2001 12:21 PM
Subject: [tuning] Re: tiny interval: (s.c.-11 skh.) and (P.c.-12 skh.)

> In-Reply-To: <008d01c17442$159f27a0$5dc5fd18@union1.nj.home.com>
> David Beardsley asked:
>
> > Maybe you could express that as a ratio so us math impared
> > types can understand what you're talking about?
>
> 2923003274661805836407369665432566039311865085952
> -------------------------------------------------
> 2922977339492680612451840826835216578535400390625
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
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>
>
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>
>
>

🔗BobWendell@technet-inc.com

Bob had said this implies:

Syntonic comma - 11 schismas = Pythagorean comma - 12 schismas
>

Bob now adds:
Correction. I transposed without realizing it the right side of this
equation. Let P = Pythagorean comma, S = Syntonic comma, and Z =
schisma, so:

S - 11Z = 12Z - P

This may not be so trivial after all. It implies that:

P + S = 23Z

This is true to the hundredth of a cent.

Also P + S is very nearly 45 cents

and

80(P+S) = 3 8vas - 2.7 cents

Isn't there anything we can do with this, Gene? I notice that using
(P + S) as a generator, we can closely graze all the harmonics up
through 19 a la Secor, but not quite so accurately.

--- In tuning@y..., BobWendell@t... wrote:
> Without having attempted to think it through yet, Gene, I'm
wondering
> whether the perfect identity:
>
> Syntonic comma - 11 schismas = Pythagorean comma - 12 schismas
>
> is trivial, or significant. Oh, I think I just realized that the
> schisma occurs at 8 perfect fifths down, the syntonic comma at 4
> fifths up, which totals 12 fifths to generate the Pythagorean
comma.
> Just answered my own question. Pretty trivial I guess. Can't see
how
> it would lead to any major new discoveries. Thanks anyway, but
would
> welcome any additional observations or comments.
>
> Sincerely,
>
> Bob
>
>
> --- In tuning@y..., genewardsmith@j... wrote:
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > > Has anyone before ever pointed out the tiny interval
> > > 2^161 * 3^-84 * 5^-12, which is only about 1/65 cent?
> >
> > This is one of the 5-limit commas I found when I went looking for
> > them; I would have posted on it but I didn't know what to do with
> > them once found. I could put up a list of such beasts if it would
> be
> > of interest, however.

🔗BobWendell@technet-inc.com

Addendum to last post in this thread:
Pythagorean comma minus Syntonic comma equals schisma, or that is:

P - S = Z

This is true to 0.00000004 cents, which may just be a rounding error
in my HP scientific calculator.

Together with the equation below (P + S = 23Z), this directly implies:

12S = 11P

or conversely

S/11 = P/12

which explains why 12-tET is equivalent to 11th-comma meantone.

--- In tuning@y..., BobWendell@t... wrote:
> Bob had said this implies:
>
> Syntonic comma - 11 schismas = Pythagorean comma - 12 schismas
> >
>
> Bob now adds:
> Correction. I transposed without realizing it the right side of
this
> equation. Let P = Pythagorean comma, S = Syntonic comma, and Z =
> schisma, so:
>
> S - 11Z = 12Z - P
>
> This may not be so trivial after all. It implies that:
>
> P + S = 23Z
>
>
> This is true to the hundredth of a cent.
>
> Also P + S is very nearly 45 cents
>
> and
>
> 80(P+S) = 3 8vas - 2.7 cents
>
> Isn't there anything we can do with this, Gene? I notice that using
> (P + S) as a generator, we can closely graze all the harmonics up
> through 19 a la Secor, but not quite so accurately.
>
>
>
>
> --- In tuning@y..., BobWendell@t... wrote:
> > Without having attempted to think it through yet, Gene, I'm
> wondering
> > whether the perfect identity:
> >
> > Syntonic comma - 11 schismas = Pythagorean comma - 12 schismas
> >
> > is trivial, or significant. Oh, I think I just realized that the
> > schisma occurs at 8 perfect fifths down, the syntonic comma at 4
> > fifths up, which totals 12 fifths to generate the Pythagorean
> comma.
> > Just answered my own question. Pretty trivial I guess. Can't see
> how
> > it would lead to any major new discoveries. Thanks anyway, but
> would
> > welcome any additional observations or comments.
> >
> > Sincerely,
> >
> > Bob
> >
> >
> > --- In tuning@y..., genewardsmith@j... wrote:
> > > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > >
> > > > Has anyone before ever pointed out the tiny interval
> > > > 2^161 * 3^-84 * 5^-12, which is only about 1/65 cent?
> > >
> > > This is one of the 5-limit commas I found when I went looking
for
> > > them; I would have posted on it but I didn't know what to do
with
> > > them once found. I could put up a list of such beasts if it
would
> > be
> > > of interest, however.

🔗Paul Erlich <paul@stretch-music.com>

🔗BobWendell@technet-inc.com

🔗monz <joemonz@yahoo.com>

Hi Bob,

> From: <BobWendell@technet-inc.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, November 26, 2001 10:48 AM
> Subject: [tuning] Re: tiny interval: (s.c.-11 skh.) and Correction
>
>
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> >
> > --- In tuning@y..., BobWendell@t... wrote:
> > >
> > > Addendum to last post in this thread:
> > > Pythagorean comma minus Syntonic comma equals schisma, or that is:
> > >
> > > P - S = Z
> > >
> > > This is true to 0.00000004 cents, which may just be a
> > > rounding error
> > > in my HP scientific calculator.

That definitely *is* a rounding error.

> > [Paul Erlich's response:]
> >
> > Bob, the Pythagorean comma minus the syntonic comma equals the
> > schisma exactly:
> >
> >
> > 3s 5s
> > ... ...
> > Pyth comma 12 0
> > - Synt comma 4 1
> > ---------------------------
> > = Schishma 8 -1
> >
> >
> > Get it?
>
> Yes, I posted something long ago to this effect. I just got thrown
> off temperarily by my calulator error. But don't you have your
> columns labeled backwards (for those who might be confused by this)?
> You're referring to P5s and M3s in that order, left to right, correct?

No. By "3s" and "5s", Paul means the *prime-factors* 3 and 5.
In other words:

P = 2^-19 * 3^12 * 5^0
- S = 2^-4 * 3^4 * 5^-1
---------------------------
2^-15 * 3^8 * 5^1 = skhisma

One of the most unfortunate things about tuning math
and its application to Eurocentric diatonic music-theory
is the appearance of the "false cognates", whereby
prime-factor 3 represents the interval called a "5th" and
prime-factor 5 represents the interval called a "major 3rd".

Now a clarification about your "equivalences":

> From: <BobWendell@technet-inc.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, November 26, 2001 7:45 AM
> Subject: [tuning] Re: tiny interval: (s.c.-11 skh.) and Correction
>
>
> Together with the equation below (P + S = 23Z), this directly implies:
>
> 12S = 11P
>
> or conversely
>
> S/11 = P/12
>
> which explains why 12-tET is equivalent to 11th-comma meantone.
>

As for your first "equation":

> 12S = 11P

12S = 2^(-4*12) * 3^(4*12) * 5^(-1*12) = 2^-48 * 3^48 * 5^-12
- 11P = 2^(-19*11) * 3^(12*11) * 5^(0*11) = 2^-209 * 3^132 * 5^0
-----------------------
2^161 * 3^-84 * 5^-12
= ~0.015360929 (= ~1/65) cent

This is precisely the same interval which started this thread:

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, November 22, 2001 1:04 PM
> Subject: [tuning] tiny interval: (s.c.-11 skh.) and (P.c.-12 skh.)
>
>
> Has anyone before ever pointed out the tiny interval
> 2^161 * 3^-84 * 5^-12, which is only about 1/65 cent?
>
> I found it by looking into the fact that the skhisma
> is almost exactly 1/11 syntonic comma. This microscopic
> interval is the difference between 11 skhismas and the
> comma, and also the difference between 12 skhismas and
> the Pythagorean comma.

As for your second "equation":

> S/11 = P/12

S/11 = 2^-4/11 * 3^4/11 * 5^-1/11
P/12 = 2^-19/12 * 3^12/12 * 5^0/12

= [must create a common denominator:]

2^(-4*12)/(11*12) * 3^(4*12)/(11*12) * 5^(-1*12)/(11*12)
2^(-19*11)/(12*11) * 3^(12*11)/(12*11) * 5^(0*11)/(12*11)

= 2^-48/132 * 3^48/132 * 5^-12/132
- 2^-209/132 * 3^132/132 * 5^0/132
------------------------------------
2^161/132 * 3^-84/132 * 5^-12/132

= ~0.000116371 (=~1/8593) cent

So in both cases there is a tiny remainder after the
subtraction is performed. Admittedly, the 1/65-cent
difference would be inaudible under probably every circumstance,
and the 1/8593-cent difference is certainly always inaudible.

But still, I feel that it's important to be aware of
*when* mathematical operations work out *exactly* and when
they don't, especially if the word "exactly" is being
bandied about.

That's one of the major reasons why I like to use prime-factor
notation. As I show above, it elucidates minor discrepancies
like this which can become lost in the rounding errors which
are built into our software and electronic calculators.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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Get your free @yahoo.com address at http://mail.yahoo.com

🔗BobWendell@technet-inc.com

Hey, Joe! Long time no talk. Thanks for the thorough elucidation.
Yes, I did realize the origin of the tiny interval. I also should
have realized what Paul meant by the 3s and 5s. I'm very used to
thinking that way, too, but somehow slipped a cog on this one. Not
used to that particular way of notating it. (I like the use of the
term "false cognate" for this mistake, by the way. I've never heard
it used in this context before, but it works nicely here.)

What I still don't quite get, though, (still got a slipping cog?) is
this:

If the difference (call it dCz for delta comma-schisma) between 12
schismas and the Pythag comma is essentally the same as the
difference between the Syntonic comma and 11 schismas, that means

12Z- P = S - 11Z = dCz

where Z is a schisma, P is the Pyth comma and S is the Synt comma.

This implies that P + S = 23Z,
or very nearly so.
This doesn't seem as exact as it should be, though, considering the
closeness (almost the same to many decimal places, isn't it?) of the
two differences.

🔗jpehrson@rcn.com

🔗monz <joemonz@yahoo.com>

Hi Bob,

> From: <BobWendell@technet-inc.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, November 26, 2001 2:05 PM
> Subject: [tuning] Re: tiny interval: (s.c.-11 skh.) and Correction
>
>
> What I still don't quite get, though, (still got a slipping cog?) is
> this:
>
> If the difference (call it dCz for delta comma-schisma) between 12
> schismas and the Pythag comma is essentally the same as the
> difference between the Syntonic comma and 11 schismas, that means
>
> 12Z- P = S - 11Z = dCz
>
> where Z is a schisma, P is the Pyth comma and S is the Synt comma.
>
> This implies that P + S = 23Z, or very nearly so.
> This doesn't seem as exact as it should be, though, considering the
> closeness (almost the same to many decimal places, isn't it?) of the
> two differences.

You had it right before you made your "correction"; i.e.,
the Pythagorean comma is *larger* than 12 skhismas, so that
your further transposition of the terms was the real error,
which is what caused this to not work out right.

Therefore, as shown in my original post on this thread
(</tuning/topicId_30506.html#30506>):

dCz
= P - 12Z
= S - 11Z
= 2^161 * 3^-84 * 5^-12
= ~0.015360929 cent

Solving the equation exposes all the correct relationships:

P - 12Z = S - 11Z
P = S + Z
P - S = Z

Proof of the above algebra, in numbers:

S = 2^-4 * 3^4 * 5^-1
+ Z = 2^-15 * 3^8 * 5^1
-------------------
P = 2^-19 * 3^12 * 5^0

and

P = 2^-19 * 3^12 * 5^0
- S = 2^-4 * 3^4 * 5^-1
-------------------
Z = 2^-15 * 3^8 * 5^1

Now let's take a look at the numbers in "P + S = 23Z, or very nearly so":

P = 2^-19 * 3^12
+ S = 2^-4 * 3^4 * 5^-1
--------------------
2^-23 * 3^16 * 5^-1 = ~44.96629998 cents

2^-15 * 3^8 * 5^1
* 23
---------------------
23Z = 2^-345 * 3^184 * 5^23 = ~44.93557812 cents

So yes, they are very close, but not exactly the same.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗monz <joemonz@yahoo.com>

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, November 26, 2001 8:01 AM
> Subject: [tuning] Re: tiny interval: (s.c.-11 skh.) and Correction
>
>
> Bob, the Pythagorean comma minus the syntonic comma equals the
> schisma exactly:
>
>
> 3s 5s
> ... ...
> Pyth comma 12 0
> - Synt comma 4 1
> ---------------------------
> = Schishma 8 -1

I've already commented on this, but only now realized that
Paul made a mistake here. The "8ve"-equivalent factoring
for the Syntonic Comma is 3^4 * 5^-1, which makes the skhisma
3^8 * 5^1.

Some comments on the erroneous numbers given by Paul:

3^8 * 5^-1 is actually a very wide "major 3rd" (~429 cents)
which is a syntonic comma wider than the Pythagorean 81:64,
and only a skhisma (~2 cents) wider than the 32:25 JI
"diminished 4th".

3^4 * 5^1 is actually the flavor of "augmented 5th" (~794 cents)
which falls between the JI (~773 cents) and Pythagorean (~816 cents)
versions, being a syntonic comma wider than the former and a
syntonic comma narrower than the latter. Ellis's name for
it was "extreme sharp 5th", and the ratio is 405:256.

BTW, Rameau called 25:16 the "augmented 5th"; Ellis's name
for it is "grave superfluous 5th". This exhibits an inconsistency
in Ellis's application of terminology for rational intervals, because
he generally used "superfluous" to indicate a widening by 25:24,
"grave" to indicate a narrowing by 81:80, and "acute" to indicate
a widening by 81:80.

By this reckoning, since 3:2 is named "5th" without any other
qualification, Ellis should have called 25:16 simply "superfluous 5th",
and our ~794-cent 405:256 should have been "acute superfluous 5th".
If he had used these terms, it would also have been consistent with
the name "acute 5th" which he used for 3^5 * 5^-1 = 243:160 (~723 cents).

Be careful with operations on those plus and minus signs!

-monz

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🔗BobWendell@technet-inc.com

Yes, thanks, Monz! Sorry! I had realized last night that I had it
right the first time and saw that the discrepancy between the sum of
the commas and 23 schismas would therefore be two of your teensy
weensy intervals. So the approximation is about 2/65 cents off.

It strikes me as very interesting that the difference between the
Pythagorean comma and 12 schismas would be so close to the difference
between the Syntonic comma and ll schismas. I feel there must be
something useful about this relationship if we could just dig it out.
What do you think?

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> Hi Bob,
>
>
> > From: <BobWendell@t...>
> > To: <tuning@y...>
> > Sent: Monday, November 26, 2001 2:05 PM
> > Subject: [tuning] Re: tiny interval: (s.c.-11 skh.) and Correction
> >
> >
> > What I still don't quite get, though, (still got a slipping cog?)
is
> > this:
> >
> > If the difference (call it dCz for delta comma-schisma) between
12
> > schismas and the Pythag comma is essentally the same as the
> > difference between the Syntonic comma and 11 schismas, that means
> >
> > 12Z- P = S - 11Z = dCz
> >
> > where Z is a schisma, P is the Pyth comma and S is the Synt comma.
> >
> > This implies that P + S = 23Z, or very nearly so.
> > This doesn't seem as exact as it should be, though, considering
the
> > closeness (almost the same to many decimal places, isn't it?) of
the
> > two differences.
>
>
> You had it right before you made your "correction"; i.e.,
> the Pythagorean comma is *larger* than 12 skhismas, so that
> your further transposition of the terms was the real error,
> which is what caused this to not work out right.
>
> Therefore, as shown in my original post on this thread
> (</tuning/topicId_30506.html#30506>):
>
> dCz
> = P - 12Z
> = S - 11Z
> = 2^161 * 3^-84 * 5^-12
> = ~0.015360929 cent
>
>
> Solving the equation exposes all the correct relationships:
>
> P - 12Z = S - 11Z
> P = S + Z
> P - S = Z
>
>
> Proof of the above algebra, in numbers:
>
>
> S = 2^-4 * 3^4 * 5^-1
> + Z = 2^-15 * 3^8 * 5^1
> -------------------
> P = 2^-19 * 3^12 * 5^0
>
> and
>
> P = 2^-19 * 3^12 * 5^0
> - S = 2^-4 * 3^4 * 5^-1
> -------------------
> Z = 2^-15 * 3^8 * 5^1
>
>
>
> Now let's take a look at the numbers in "P + S = 23Z, or very
nearly so":
>
>
> P = 2^-19 * 3^12
> + S = 2^-4 * 3^4 * 5^-1
> --------------------
> 2^-23 * 3^16 * 5^-1 = ~44.96629998 cents
>
>
>
>
> 2^-15 * 3^8 * 5^1
> * 23
> ---------------------
> 23Z = 2^-345 * 3^184 * 5^23 = ~44.93557812 cents
>
>
> So yes, they are very close, but not exactly the same.
>
>
>
>
> love / peace / harmony ...
>
> -monz
> http://www.monz.org
> "All roads lead to n^0"
>
>
>
>
>
> _________________________________________________________
> Do You Yahoo!?
> Get your free @yahoo.com address at http://mail.yahoo.com

🔗BobWendell@technet-inc.com

🔗Paul Erlich <paul@stretch-music.com>

I wrote,

> --- In tuning@y..., BobWendell@t... wrote:
>
> > It strikes me as very interesting that the difference between the
> > Pythagorean comma and 12 schismas would be so close to the
> difference
> > between the Syntonic comma and ll schismas. I feel there must be
> > something useful about this relationship if we could just dig it
> out.
> > What do you think?
>
> The difference between these differences is therefore an extremely
> tiny unison vector, which could therefore define an astoundingly
> accurate linear temperament for the 5-limit.

Whoops! The difference between these differences is zero, so can't be
used to define any temperament.

However, each of the differences itself is very small to begin with,
so can define temperaments. The most obvious example is the ET in
which the syntonic comma is 11 steps and the Pythagorean comma is 12
steps: 612-tET. This is the "scale of schismas" in the same sense
that 53-tET is the "scale of commas".

🔗BobWendell@technet-inc.com

Paul (complete text below):
Whoops! The difference between these differences is zero, so can't be
used to define any temperament.

Bob:
So that would seem to me only to increase its significance. Does
anyone understand why P - 12Z = S - 11Z?....EXACTLY!!!???

It doesn't seem reasonable to assume this is merely some kind of
happy mathematical happenstance. The odds against that kind of
precision occurring by chance are obviously extremely high.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> I wrote,
>
> > --- In tuning@y..., BobWendell@t... wrote:
> >
> > > It strikes me as very interesting that the difference between
the
> > > Pythagorean comma and 12 schismas would be so close to the
> > difference
> > > between the Syntonic comma and ll schismas. I feel there must
be
> > > something useful about this relationship if we could just dig
it
> > out.
> > > What do you think?
> >
> > The difference between these differences is therefore an
extremely
> > tiny unison vector, which could therefore define an astoundingly
> > accurate linear temperament for the 5-limit.
>
> Whoops! The difference between these differences is zero, so can't
be
> used to define any temperament.
>
> However, each of the differences itself is very small to begin
with,
> so can define temperaments. The most obvious example is the ET in
> which the syntonic comma is 11 steps and the Pythagorean comma is
12
> steps: 612-tET. This is the "scale of schismas" in the same sense
> that 53-tET is the "scale of commas".

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., BobWendell@t... wrote:
> Paul (complete text below):
> Whoops! The difference between these differences is zero, so can't
be
> used to define any temperament.
>
> Bob:
> So that would seem to me only to increase its significance. Does
> anyone understand why P - 12Z = S - 11Z?....EXACTLY!!!???

As Monz showed:

P = S + Z (by definition

-12Z = -12Z

---------------------------

P - 12Z = S - 11Z

Hope that's not too mathematical for this list :)

> It doesn't seem reasonable to assume this is merely some kind of
> happy mathematical happenstance. The odds against that kind of
> precision occurring by chance are obviously extremely high.

1-to-0, in fact.

🔗genewardsmith@juno.com

🔗BobWendell@technet-inc.com

Yes, I feel silly. I should have seen that. Thank you both, Gene and
Paul. I do math manipulations much more sophisticated than that and
then turn around and miss something that trivial. I guess my math
brain is oxidizing or something.

You had mentioned, Gene, at one point that you were tempted to
recommend a book or two on tuning-related math like abstract algebra
and number theory so that some of us less mathematically learned
folks could get the background to understand better some of your and
Paul's posts. I would really appreciate that if you would go ahead
and do it (if I haven't already disqualified myself in your mind,
chuckle).

Gratefully,

Bob

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> > So that would seem to me only to increase its significance. Does
> > anyone understand why P - 12Z = S - 11Z?....EXACTLY!!!???
>
> You don't need to know what P, S and Z are to see that
> linear equation P - 12Z = S - 11Z is equivalent to P - S = Z. This
> tells us that the reason for it all is that this is how Z is
defined.

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗robert_wendell <BobWendell@technet-inc.com>

Hi, Paul! Never responded to this post of yours below. Too busy with
Christmas concerts, the second and last of which was yesterday
afternoon. Very successful, but tired puppy here!

I very much treasure the availability of wonderful information here.
I think it would be more productive if I were to study the archives
more thoroughly before I subject you to my queries, many of which I'm
sure would disappear after having done so.

Greetings to all. I'm still here. Just busy. I probably won't be
online here with the tuning group much for awhile, but I intend to be
here for years, most likely in perpetuity.

Happy holidays to all! (And a much better new year.)

Respectfully,

Bob

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
>
> > You had mentioned, Gene, at one point that you were tempted to
> > recommend a book or two on tuning-related math like abstract
> algebra
> > and number theory so that some of us less mathematically learned
> > folks could get the background to understand better some of your
> and
> > Paul's posts.
>
> For Gene's posts, yes, that would be essential. For my posts, I
think
> I could explain everything right here from square one. Any
questions?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <genewardsmith@juno.com>

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> > --- On Thu Nov 22, 2001 4:04 pm, "monz" <joemonz@y...> wrote:
> >
> > > Has anyone before ever pointed out the tiny interval
> > > 2^161 * 3^-84 * 5^-12, which is only about 1/65 cent?
> >
> > monz, we just encountered this again on tuning-math, as a sort of
> > stopping point in our survey of 5-limit commas and linear
> > temperaments.
>
> It's popped up from time to time before as well.

It is the leader of a gang of three connected with the 4296-et, to one of which Paul gave the name "pirate". They are

pirate [-90, -15, 49] ets: 730, 1783, 2513, 4296, 6809

raider [71, -99, 37] ets: 7, 1171, 4296

viking (atom) [161, -84, -12] ets: 12, 612, 4296, 16572, 20868, 25164

Clearly we are now ready to name these suckers, an idea which did not go over before. I think names which tie them together would be nice.

Incidentally, if these *still* are not far-out enough for someone there is always the 78005 family to be considered. :)