Yahoo Tuning Groups Ultimate Backup tuning help with lattice diagrams, explanations?

O Tuning-List:

Here are some fixed-font simple diagrams [hopefully not wrapped on your screen], trying to find a way to define any note found in standard western music in terms of just intonation.

In C major, for instance, what is that Ab used as the minor ninth in a dominant ninth chord? Is it the one that is 36 cents sharper than its equal tempered equivalent, or the one that's 14 cents sharper (135/81), or the one that's 8 cents flatter (128/81)?

Not so much trying to answer that particular question (but please do!) as to frame a way to answer that KIND of question.

Comments, assistance, please. But the heavy-handed irony/mockery/flippancy that gets into this list from time to time, please not.

> > Pythagorean circle of pure fifths:
> > Gb * Db * Ab * Eb * Bb * F * C * G * D * A * E * B * F# * C#
> > Lattice with pure thirds above and below ("df"= double flat;+,*,- identify
the strands)

> > Edf+ Bdf+ Fb + Cb + Gb + Db + Ab + Eb + Bb + F + C + G + D + A + E > / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / > Cb * Gb * Db * Ab * Eb * Bb * F * C * G * D * A * E * B * F# * C# *
>/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
>Ab- Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E#
> > > Lattice with pure thirds above and below, twice removed:
> >(++) ddf adf edf bdf fb cb gb db ab eb bb f c > / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / > Edf+ Bdf+ Fb + Cb + Gb + Db + Ab + Eb + Bb + F + C + G + D + A + E
> / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
>* Cb * Gb * Db * Ab * Eb * Bb * F * C * G * D * A * E * B * F# * C# > / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / >Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E#
> \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
> c g d a e b f# c# g# d# a# e# b# fx cx
> > Same lattice as above, with approximate tuning in cents with respect to 12EDO > based on C: > >(++) ddf adf edf bdf fb cb gb db ab eb bb f c > +20 +22 +24 +26 +28 +30 +32 +34 +36 +38 +40 +42 +44
> / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
> Edf+ Bdf+ Fb + Cb + Gb + Db + Ab + Eb + Bb + F + C + G + D + A + E
> +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +24 +26 +28 +30
> / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
>* Cb * Gb * Db * Ab * Eb * Bb * F * C * G * D * A * E * B * F# * C# *
> -14 -12 -10 -8 -6 -4 -2 0 +2 +4 +6 +8 +10 +12 +14 > / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ >Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E#
> -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0
> \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
> c g d a e b f# c# g# d# a# e# b# fx cx
> -44 -42 -40 -38 -36 -34 -32 -30 -28 -26 -24 -22 -20 -18 -16 Thank you,

guglielmo

🔗Guglielmo <gugliel@guglielmomusic.com>

One of the ratios used for illustration in my post was incorrect:

Guglielmo wrote:
>In C major, for instance, what is that Ab used as the minor ninth in a
>dominant ninth chord? Is it the one that is 36 cents sharper than its
>equal tempered equivalent, or the one that's 14 cents sharper (135/81),
>or the one that's 8 cents flatter (128/81)?
>

CORRECTION:

the Ab should have compared the 8/5 Ab (4 pure fifths from C, 8 cents flatter than ET) to the 128/81 Ab (pure major third below C, 14 cents sharper than ET), distant by one comma (81/80) -- mistakenly I used the ration for an A (5/3) to get 135/81. Sorry to have confused the main question with a mistake in the little question!

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

On Fri, 09 Dec 2005, Guglielmo wrote:
>
> O Tuning-List:
>
> Here are some fixed-font simple diagrams [hopefully not wrapped on your
> screen], trying to find a way to define any note found in standard
> western music in terms of just intonation.
>
> In C major, for instance, what is that Ab used as the minor ninth in a
> dominant ninth chord? Is it the one that is 36 cents sharper than its
> equal tempered equivalent, or the one that's 14 cents sharper (135/81),
> or the one that's 8 cents flatter (128/81)?
>
> Not so much trying to answer that particular question (but please do!)
> as to frame a way to answer that KIND of question.
>
> Comments, assistance, please. But the heavy-handed
> irony/mockery/flippancy that gets into this list from time to time,
> please not.
>
> >
> > Pythagorean circle of pure fifths:
> >
> > Gb * Db * Ab * Eb * Bb * F * C * G * D * A * E * B * F# * C#
> >
> >
>
> Lattice with pure thirds above and below ("df"= double flat;+,*,- identify
> the strands)
>
> >
> > Edf+ Bdf+ Fb + Cb + Gb + Db + Ab + Eb + Bb + F + C + G + D + A +
E
> > / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
> > Cb * Gb * Db * Ab * Eb * Bb * F * C * G * D * A * E * B * F# * C#
*
> >/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\
> >Ab- Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# -
E#
> >
... etc

Gulglielmo,

I like the way you've laid the diagrams out using
such simple means.

But why, oh why, not use the standard notation
bb for double-flat?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.371 / Virus Database: 267.13.13/197 - Release Date: 9/12/05

🔗Mark Rankin <markrankin95511@yahoo.com>

Guglielmo,

This is the central heptad, the seven central tones
where the lattice or matrix begins.

The Major 6th and the Major 3rd are usually shown
above the Pythagorean line of 5ths, and the minor 6th
and the minor 3rd are usually shown below. It's not
*wrong* to invert them, one may present them however
one cares to, but this is the traditional layout.

A-----E
/ \ / \
/ \ / \
F-----C-----G
\ / \ /
\ / \ /
Ab-----Eb

By the way, this subject, the triaxial matrix of
thirds and fifths, has a history that goes back at
least 2500 years to the 'close-packed diagrams' of the
ancient Greeks, probably another 1500 years to the
more ancient Babylonians, and probably yet another
1500 years to the first humans with writing, the even
more ancient Sumerians.

Although you may not know what you're getting into,
this could turn out to be one hell of a fine ride!

As they say in Danish, Have it Good!

--Mark Rankin

--- Yahya Abdal-Aziz <yahya@melbpc.org.au> wrote:

>
> On Fri, 09 Dec 2005, Guglielmo wrote:
> >
> > O Tuning-List:
> >
> > Here are some fixed-font simple diagrams
> [hopefully not wrapped on your
> > screen], trying to find a way to define any note
> found in standard
> > western music in terms of just intonation.
> >
> > In C major, for instance, what is that Ab used as
> the minor ninth in a
> > dominant ninth chord? Is it the one that is 36
> cents sharper than its
> > equal tempered equivalent, or the one that's 14
> cents sharper (135/81),
> > or the one that's 8 cents flatter (128/81)?
> >
> > Not so much trying to answer that particular
> question (but please do!)
> > as to frame a way to answer that KIND of question.
> >
> > Comments, assistance, please. But the
> heavy-handed
> > irony/mockery/flippancy that gets into this list
> from time to time,
> > please not.
> >
> > >
> > > Pythagorean circle of pure fifths:
> > >
> > > Gb * Db * Ab * Eb * Bb * F * C * G * D * A
> * E * B * F# * C#
> > >
> > >
> >
> > Lattice with pure thirds above and below ("df"=
> double flat;+,*,- identify
> > the strands)
> >
> > >
> > > Edf+ Bdf+ Fb + Cb + Gb + Db + Ab + Eb + Bb + F
> + C + G + D + A +
> E
> > > / \ / \ / \ / \ / \ / \ / \ / \ / \ /
> \ / \ / \ / \ / \ /
> > > Cb * Gb * Db * Ab * Eb * Bb * F * C * G * D
> * A * E * B * F# * C#
> *
> > >/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \
> / \ / \ / \ / \ /
> \
> > >Ab- Eb - Bb - F - C - G - D - A - E - B -
> F# - C# - G# - D# - A# -
> E#
> > >
> ... etc
>
> Guglielmo,
>
> I like the way you've laid the diagrams out using
> such simple means.
>
> But why, oh why, not use the standard notation
> bb for double-flat?
>
> Regards,
> Yahya
>
> --
> No virus found in this outgoing message.
> Checked by AVG Free Edition.
> Version: 7.1.371 / Virus Database: 267.13.13/197 -
> Release Date: 9/12/05
>
>

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🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Mark Rankin <markrankin95511@y...> wrote:

>
> A-----E
> / \ / \
> / \ / \
> F-----C-----G
> \ / \ /
> \ / \ /
> Ab-----Eb
>
>
> By the way, this subject, the triaxial matrix of
> thirds and fifths, has a history that goes back at
> least 2500 years to the 'close-packed diagrams' of the
> ancient Greeks, probably another 1500 years to the
> more ancient Babylonians, and probably yet another
> 1500 years to the first humans with writing, the even
> more ancient Sumerians.

Do you have cites for this? I just finished writing a Wikipedia
article on "modulatory space" where the discussion carried the matter
much less farther back than that, but I'd love to work the Greeks in
if possible.

Why do you call it a matrix?

http://en.wikipedia.org/wiki/Modulatory_space

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, Guglielmo <gugliel@g...> wrote:
>
> O Tuning-List:
>
> Here are some fixed-font simple diagrams [hopefully not wrapped on
your
> screen],

They got wrapped. Could you upload an ASCII file?

> trying to find a way to define any note found in standard
> western music in terms of just intonation.
>
> In C major, for instance, what is that Ab used as the minor ninth
in a
> dominant ninth chord? Is it the one that is 36 cents sharper than
its
> equal tempered equivalent, or the one that's 14 cents sharper
(135/81),
> or the one that's 8 cents flatter (128/81)?

In my opinion, it may be several of these as well as other ratios.
Part of the answer depends on what you 'mean' by ratios -- an
immediate perception of a simultaneously-sounding sonority as part of
a harmonic series, or an extended calculation based on a sequence of
such sonorities through common tones to arrive at a ratio, or . . .)
In my experience, most Western music can be rendered in adaptive JI,
but most cannot be rendered in strict JI, at least not in anything
like an unambiguous way.

> Comments, assistance, please. But the heavy-handed
> irony/mockery/flippancy that gets into this list from time to time,
> please not.

I apologize if I may appear heavy-handed sometimes. I'm not the
greatest with ASCII communication.

Anyway, as best as I can gather from the wrapped lattices, our
lattices seem to be the same, just flipped top-to-bottom. You might
want to look at Figure 6 here (and the rest of the paper) and tell me
what you think:

http://lumma.org/tuning/erlich/erlich-tFoT.pdf

Cheers!

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, Mark Rankin <markrankin95511@y...>
wrote:
>
> Guglielmo,
>
> This is the central heptad, the seven central tones
> where the lattice or matrix begins.

Harry Partch refers to this as the 5-limit Tonality Diamond, and Erv
Wilson diagrams it as you do below, as I did before I was aware of
his writings . . . It wouldn't surprise me at all if some Babylonian
did so as well!

> The Major 6th and the Major 3rd are usually shown
> above the Pythagorean line of 5ths, and the minor 6th
> and the minor 3rd are usually shown below. It's not
> *wrong* to invert them, one may present them however
> one cares to, but this is the traditional layout.
>
>
>
> A-----E
> / \ / \
> / \ / \
> F-----C-----G
> \ / \ /
> \ / \ /
> Ab-----Eb
>
>
> By the way, this subject, the triaxial matrix of
> thirds and fifths, has a history that goes back at
> least 2500 years to the 'close-packed diagrams' of the
> ancient Greeks, probably another 1500 years to the
> more ancient Babylonians, and probably yet another
> 1500 years to the first humans with writing, the even
> more ancient Sumerians.

I'm sure Monz and Daniel Wolf would love to learn more about this.
Heretofore we've been told that Tanaka invented the triangular
lattice of musical thirds and fifths!

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Mark Rankin <markrankin95511@yahoo.com>

Dear Paul (Polyester Walrus?),

You're right. The earliest tri-axial matrix that I've
ever seen
is Tanaka's, from 1892 A.D. I've searched for years
to find concrete evidence of earlier examples, but to
no avail so far.
Also, I might point out the Tanaka's lattice includes
the NE-SW axes, / /, in black ink, and the NW-SE axes
\ \, in black ink, but for some reason unknown to me,
it shows only a white void where the horizontal axes
lie!

There are only two or three ways to cover a plane with
regular tiles: Squares, Hexagons, and Triangles
(which are biased). The "closest packing" of the
ancients must, therefore be tri-axial.

--- wallyesterpaulrus <wallyesterpaulrus@yahoo.com>
wrote:

> --- In tuning@yahoogroups.com, Mark Rankin
> <markrankin95511@y...>
> wrote:
> >
> > Guglielmo,
> >
> > This is the central heptad, the seven central
> tones
> > where the lattice or matrix begins.
>
> Harry Partch refers to this as the 5-limit Tonality
> Diamond, and Erv
> Wilson diagrams it as you do below, as I did before
> I was aware of
> his writings . . . It wouldn't surprise me at all if
> some Babylonian
> did so as well!
>
> > The Major 6th and the Major 3rd are usually shown
> > above the Pythagorean line of 5ths, and the minor
> 6th
> > and the minor 3rd are usually shown below. It's
> not
> > *wrong* to invert them, one may present them
> however
> > one cares to, but this is the traditional layout.
> >
> >
> >
> > A-----E
> > / \ / \
> > / \ / \
> > F-----C-----G
> > \ / \ /
> > \ / \ /
> > Ab-----Eb
> >
> >
> > By the way, this subject, the triaxial matrix of
> > thirds and fifths, has a history that goes back at
> > least 2500 years to the 'close-packed diagrams' of
> the
> > ancient Greeks, probably another 1500 years to the
> > more ancient Babylonians, and probably yet another
> > 1500 years to the first humans with writing, the
> even
> > more ancient Sumerians.
>
> I'm sure Monz and Daniel Wolf would love to learn
> more about this.
> Heretofore we've been told that Tanaka invented the
> triangular
> lattice of musical thirds and fifths!
>
>
>
>

__________________________________________________
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🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, Mark Rankin <markrankin95511@y...> wrote:
>
> Dear Paul (Polyester Walrus?),
>
> You're right. The earliest tri-axial matrix that I've
> ever seen is Tanaka's, from 1892 A.D. I've searched
> for years to find concrete evidence of earlier examples,
> but to no avail so far.
> Also, I might point out the Tanaka's lattice includes
> the NE-SW axes, / /, in black ink, and the NW-SE axes
> \ \, in black ink, but for some reason unknown to me,
> it shows only a white void where the horizontal axes
> lie!
>
> There are only two or three ways to cover a plane with
> regular tiles: Squares, Hexagons, and Triangles
> (which are biased). The "closest packing" of the
> ancients must, therefore be tri-axial.

Riemann's _Tonnetz_ may have been earlier:

http://tonalsoft.com/enc/t/tonnetz.aspx

The references to it which i've seen always say that
he published it in 1915 -- but he was developing his
theory much earlier than that, and i haven't read
the German originals, so i'm not exactly sure when
he originally used it.

However, another book i've never seen is Oettingen's:

Oettingen, Arthur von. 1866.
_Harmoniesystem in Dualer Entwicklung:
.. Studien zur Theorie der Musik_.
Leipzig: W. Gl ser.

But everyone who discusses the _Tonnetz_ gives him
the credit for inventing it (something which i need
to put into my webpage).

And Oettingen's work was a development of Euler's
music-theory, which he published first in 1739, and
then with a follow-up in 1773:

http://tonalsoft.com/monzo/euler/euler-en.htm

BTW, the Wikipedia article on "pitch space"

http://en.wikipedia.org/wiki/Pitch_space

is wrong about Riemann's _Tonnetz_ -- it is shown
there as biaxial, but it was indeed triaxial, as can
be seen on my _Tonnetz_ page.

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

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > is wrong about Riemann's _Tonnetz_ -- it is shown
> > there as biaxial, but it was indeed triaxial, as can
> > be seen on my _Tonnetz_ page.
>
> Lehrdahl says it is biaxial. Do you have Riemann to consult?

I don't have any of Riemann's books, but an article in a
recent _Journal of Music Theory_ gives a reproduction of
the _Tonnetz_ from Riemann's book, and it is exactly like
the one i reproduced in ASCII on my webpage, which is
clearly triaxial.

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

🔗threesixesinarow <CACCOLA@NET1PLUS.COM>

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@N...> wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> >
> > Do you have Riemann to consult?
> >
> http://www.8ung.at/fzmw/1999/1999_1.htm

Excellent! Thanks, Clark.

There's the triaxial _Tonnetz_ on p 23 ("SEITE: 23" in German).

This book was indeed published in 1915, so unless Riemann
used this type of diagram earlier, Tanaka did it before him.

But Oettingen was earlier still.
Clark, can you find Oettingen's book?

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

🔗threesixesinarow <CACCOLA@NET1PLUS.COM>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> can you find Oettingen's book?
>

No, just the old post cached here http://tinyurl.com/844o5

You might find Poole's pedalboard interesting, though. "Further
capabilities in this key-board will appear as it is studied." These
schematics follow the diagrams:

...mi...
do....sol
(ex1)

se......
...mi...
do....sol
(ex2)

Poole, HW "On Perfect Harmony in Music". The Americal Journal of
Science and Arts, 2nd series vol xliv nr.130 July 1867 p.17

Clark

🔗Guglielmo <gugliel@guglielmomusic.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@N...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > can you find Oettingen's book?
> >
>
> No, just the old post cached here http://tinyurl.com/844o5

Thanks. I noticed this in that post too:

>> "Jonathan Glasier, the director of Sonic Arts, extends
>> an invitation to other experimental musicians who may
>> be coming to San Diego. He offers performance opportunities
>> and possibly also short-term lodging at the Gallery if
>> he is given a few weeks notice to work out scheduling,
>> etc. Sound sculptors, performance artists, etc. are also
>> welcome. His phone number is (619) 231-3673. The Sonic Arts
>> Gallery is located at 2961 Beech Street, San Diego, CA 92102.

Some things never change. :)

> You might find Poole's pedalboard interesting, though.
> "Further capabilities in this key-board will appear as
> it is studied." These schematics follow the diagrams:

>
> ...mi...
> do....sol
> (ex1)
>
> se......
> ...mi...
> do....sol
> (ex2)
>
> Poole, HW "On Perfect Harmony in Music". The Americal Journal of
> Science and Arts, 2nd series vol xliv nr.130 July 1867 p.17

I started working on a webpage about Poole a long time ago
and never finished it. Here's what i have:

http://sonic-arts.org/monzo/poole/poole.htm

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

🔗Mark Rankin <markrankin95511@yahoo.com>

The library closed last night at 9:00 PM, cutting my
reply short. Now I'm back with 22 minutes before the
library closes again (I should come earlier!).

There are only 3 ways to fully cover a plane with
tiles: with squares, with hexagons, and with
triangles (which are biased, with one kind pointing up
and a second group pointing down).

What I was trying to say last night was that there is
a quote from one of the ancient Greek philosophers
(Aristoxenos?) who mentions certain philosophers "and
their close-packed diagrams". Since the triangular or
(tri-axial) grid is the "close-packed" one, as
compared to the square one or the full hexagon one, I
tend to consider this as evidence that the tri-axial
matrix was known in ancient Greece.

Ernest McClain mentions "Sand Tables" which may have
been used for making matrices.

-- Mark

--- Mark Rankin <markrankin95511@yahoo.com> wrote:

> Dear Paul (Polyester Walrus?),
>
> You're right. The earliest tri-axial matrix that
> I've ever seen is Tanaka's, from 1891 or 1892 A.D.
I've > searched for years trying to find
concrete evidence of earlier > examples, but to so far
to no avail.
> Also, I might point out the Tanaka's lattice
> includes the NE-SW axes, / /, in black ink, and the
NW-SE
> axes \ \, in black ink, but for some reason unknown
to
> me, it shows only a white void where the horizontal
axes
> lie!
>
> There are only two or three ways to cover a plane
> with regular tiles: Squares, Hexagons, and
Triangles
> (which are biased). The "closest packing" of the
> ancients must, therefore be tri-axial.
>
>
>
>
> --- wallyesterpaulrus <wallyesterpaulrus@yahoo.com>
> wrote:
>
> > --- In tuning@yahoogroups.com, Mark Rankin
> > <markrankin95511@y...>
> > wrote:
> > >
> > > Guglielmo,
> > >
> > > This is the central heptad, the seven central
> > tones
> > > where the lattice or matrix begins.
> >
> > Harry Partch refers to this as the 5-limit
> Tonality
> > Diamond, and Erv
> > Wilson diagrams it as you do below, as I did
> before
> > I was aware of
> > his writings . . . It wouldn't surprise me at all
> if
> > some Babylonian
> > did so as well!
> >
> > > The Major 6th and the Major 3rd are usually
> shown
> > > above the Pythagorean line of 5ths, and the
> minor
> > 6th
> > > and the minor 3rd are usually shown below. It's
> > not
> > > *wrong* to invert them, one may present them
> > however
> > > one cares to, but this is the traditional
> layout.
> > >
> > >
> > >
> > > A-----E
> > > / \ / \
> > > / \ / \
> > > F-----C-----G
> > > \ / \ /
> > > \ / \ /
> > > Ab-----Eb
> > >
> > >
> > > By the way, this subject, the triaxial matrix of
> > > thirds and fifths, has a history that goes back
> at
> > > least 2500 years to the 'close-packed diagrams'
> of
> > the
> > > ancient Greeks, probably another 1500 years to
> the
> > > more ancient Babylonians, and probably yet
> another
> > > 1500 years to the first humans with writing, the
> > even
> > > more ancient Sumerians.
> >
> > I'm sure Monz and Daniel Wolf would love to learn
> > more about this.
> > Heretofore we've been told that Tanaka invented
> the
> > triangular
> > lattice of musical thirds and fifths!
> >
> >
> >
> >
>
>
> __________________________________________________
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🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

Hi Mark,

--- In tuning@yahoogroups.com, Mark Rankin <markrankin95511@y...> wrote:

> What I was trying to say last night was that there is
> a quote from one of the ancient Greek philosophers
> (Aristoxenos?) who mentions certain philosophers "and
> their close-packed diagrams". Since the triangular or
> (tri-axial) grid is the "close-packed" one, as
> compared to the square one or the full hexagon one, I
> tend to consider this as evidence that the tri-axial
> matrix was known in ancient Greece.

Not likely.

I've always been intrigued by Aristoxenos's statement
about the "harmonists and their close-packed diagrams".
But most likely he was just talking about _katapyknosis_:

http://tonalsoft.com/enc/k/katapyknosis.aspx

Using string-lengths to measure ratios, and inserting
the arithmetic means between two diatonic notes (whole-tones)
to find a chromatic (semitone), and between two chromatic
(semitones) to find an enharmonic (quarter-tone), was
the standard means of defining the different scales of
the harmonists.

(And this approach persisted in the methods of many
later writers who struggled to incorporate Aristoxenos's
concepts within a rational framework.)

As i mentioned earlier today, Aristoxenos's distinctive
approach was to entirely avoid speaking of ratios at all,
and to speak instead only in terms of string *tension*,
and the various intervals empirically defined by musicians.

> Ernest McClain mentions "Sand Tables" which may have
> been used for making matrices.

Seeing that Gene has already replied to this post, i'm
surprised that he didn't mention the "yantra", the one
concept from McClain's work which he accepts:

http://tonalsoft.com/enc/y/yantra.aspx

http://tonalsoft.com/enc/y/yantgen.aspx

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

🔗Kraig Grady <kraiggrady@anaphoria.com>

It seems to me the Lambdoma diagrams of nicomachus are triangles turned on there sides and is also the way in which the chinese portrayed Pascals triangle/ Mt. Meru.
these are right angle triangles which are just as valid as equilateral triangles.
>
>Message: 1 > Date: Wed, 14 Dec 2005 13:55:57 -0800 (PST)
> From: Mark Rankin <markrankin95511@yahoo.com>
>Subject: Re: Re: help with lattice diagrams, explanations?
>
>The library closed last night at 9:00 PM, cutting my
>reply short. Now I'm back with 22 minutes before the
>library closes again (I should come earlier!).
>
>
>There are only 3 ways to fully cover a plane with
>tiles: with squares, with hexagons, and with
>triangles (which are biased, with one kind pointing up
>and a second group pointing down).
>
>What I was trying to say last night was that there is
>a quote from one of the ancient Greek philosophers
>(Aristoxenos?) who mentions certain philosophers "and
>their close-packed diagrams". Since the triangular or
>(tri-axial) grid is the "close-packed" one, as
>compared to the square one or the full hexagon one, I
>tend to consider this as evidence that the tri-axial
>matrix was known in ancient Greece.
>
>Ernest McClain mentions "Sand Tables" which may have
>been used for making matrices. >
> -- Mark
> >

Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, Guglielmo <gugliel@g...> wrote:
>
> >
> > There's the triaxial _Tonnetz_ on p 23 ("SEITE: 23" in German).
> >
> > This book was indeed published in 1915, so unless Riemann
> > used this type of diagram earlier, Tanaka did it before him.
> >
> > But Oettingen was earlier still.
> > Clark, can you find Oettingen's book?
>
> In a copy of "Tonal Functions" by Dirk Haagman, his preface
references
> Hugo Riemann's "Manual of Harmony", third edition, as published in
1898,
> and part of the text excerpts words by Riemann saying he first
published
> his theories in 1880.

But that might have been with the biaxial (rectangular) lattice, for
all I know . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, Mark Rankin <markrankin95511@y...>
wrote:
>
> The library closed last night at 9:00 PM, cutting my
> reply short. Now I'm back with 22 minutes before the
> library closes again (I should come earlier!).
>
>
> There are only 3 ways to fully cover a plane with
> tiles: with squares, with hexagons, and with
> triangles (which are biased, with one kind pointing up
> and a second group pointing down).

There are plenty of other ways to fully cover a plane with tiles! Did
you mean to add any conditions to the above?

But not necessarily with the same musical associations as Tanaka,
Wilson, etc., right?

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

It's worth noting that in these diagrams, a given vector doesn't
always correspond to the same interval. In Tanaka, Wilson, etc.'s
diagrams, they do.

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> It seems to me the Lambdoma diagrams of nicomachus are triangles
turned
> on there sides and is also the way in which the chinese portrayed
> Pascals triangle/ Mt. Meru.
> these are right angle triangles which are just as valid as
equilateral
> triangles.
>
>
> >
> >Message: 1
> > Date: Wed, 14 Dec 2005 13:55:57 -0800 (PST)
> > From: Mark Rankin <markrankin95511@y...>
> >Subject: Re: Re: help with lattice diagrams, explanations?
> >
> >The library closed last night at 9:00 PM, cutting my
> >reply short. Now I'm back with 22 minutes before the
> >library closes again (I should come earlier!).
> >
> >
> >There are only 3 ways to fully cover a plane with
> >tiles: with squares, with hexagons, and with
> >triangles (which are biased, with one kind pointing up
> >and a second group pointing down).
> >
> >What I was trying to say last night was that there is
> >a quote from one of the ancient Greek philosophers
> >(Aristoxenos?) who mentions certain philosophers "and
> >their close-packed diagrams". Since the triangular or
> >(tri-axial) grid is the "close-packed" one, as
> >compared to the square one or the full hexagon one, I
> >tend to consider this as evidence that the tri-axial
> >matrix was known in ancient Greece.
> >
> >Ernest McClain mentions "Sand Tables" which may have
> >been used for making matrices.
> >
> > -- Mark
> >
> >
>
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles
>