Seven Limit Tonality Diamond and Hexachords?

More stuff:

The top part of a stellated hexany is J5 in my system. Here is
some theory with the SLTD:

Hexachord Theory and the Seven Limit Tonality Diamond

1. Take the Seven Limit Tonality Diamond: 10/7, 8/7, 12/7, 5/3, 5/4,
4/3, 1/1, 3/2, 8/5, 6/5, 7/6, 7/4, 7/5

2. Work with values so that the prime in the numerator is great than
that in the denominator (7>5>3>2)

3. Obtain the list 7/5, 5/3, 5/4, 1/1, 3/2, 7/6, 7/4. Drop 1/1.

4. This corresponds to J1 set. (The top of a stellated hexany is J5)

5. Put 1/1 back in. Now you have a septachord. Taking the six
hexachord subsets gives: J1, K, J5, G, H, I, H

6. Taking the remaining fractions and doing the same thing gives J1,
K, J5, G, H-, I-, H-

7. These are the second column of my hexachord system for 35 (7 X 5
grid)

8. The remaining hexachord, L, can be found by taking K and moving
one note: Either up 1 step from E to F: 16/15 or for the Z related
complement, move from E to A: 4/3

Here is the full 7 X 5 grid: There is one imperfection, P/Q.
The last row is irregular hexachords, they are all Z related and
have almost equal interval vector values. (Tie breaker rules
determine which columns they go in).

A G M1 M5 U
B1 H N1 N5 V1
B5 I O1 O5 V5
C J1 P/Q W
D J5 R1 R5 X
E K S1 S5 Y
F L T1 T5 Z

Column 2 is the lynchpin of the system. It is an interesting
column, because it is the only column where weight measure differs
from tritone fixing. (All of my ideas are in Paul Hj's Stuff) This
makes sense because it is based on minor thirds, which tie right into
tritone and weight class changes.

(Quick addendum: Weights are 0,1,2,3 because it is mod 6, and
absolute value. So for example, transposing a hexachord by
1 step adds 6 to the weight. Inverting reverses the weight. So
you obtain (0,1,2,3,2,1,0,1,2,3,2,1,0) for weights)

Paul Hj

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

> 3. Obtain the list 7/5, 5/3, 5/4, 1/1, 3/2, 7/6, 7/4. Drop 1/1.
>
> 4. This corresponds to J1 set. (The top of a stellated hexany is J5)
>
> 5. Put 1/1 back in. Now you have a septachord. Taking the six
> hexachord subsets gives: J1, K, J5, G, H, I, H
>
> 6. Taking the remaining fractions and doing the same thing gives J1,
> K, J5, G, H-, I-, H-

Are you using your group operations directly on 7-limit JI and not on
12-et? Becauase that strikes me as potentially interestong. Why is the
prime in the numberator larger than the one in the denomiantor?

-------- Original Message --------
Subject: Reading Hexametric Rhyme Supports Cardiac
Synchronization,
Date: Mon, 12 Mar 2007 07:32:45 -0700 (GMT-07:00)
From: rik3 <rik3@
just an article i ran into i thought i would share
with you
frederica

Reading Hexametric Rhyme Supports Cardiac
Synchronization, Especially After A Heart Attack

(July 14, 2004) - Bethesda, MD � According to new
findings from a team of European physiologists, you
might receive greater health benefits (and probably a
deeper appreciation of the classics) by forgoing the
movie �Troy,� and instead, reading The Iliad out loud.
The gist of this new research focuses on the
hexameter, the poetic format unique to classical Greek
and Roman epic poems like those found in the works of
Homer and Virgil.

Background

The effects of different breathing frequencies and
patterns found in poetry readings on cardiovascular
regulation have been investigated extensively in
recent years. Poetry recitation has been known to
cause a frequency adjustment of breathing oscillations
with endogenous blood pressure fluctuations (Mayer
waves) and even cerebral blood flow oscillations
during the saying of the Catholic Rosary and the �OM�
mantra. This effect is attributed to the breathing
frequency of approximately six breaths per minute
induced by the metric of both religious verses.
Researchers have also observed increased arterial
baroreflex sensitivity, which is a favorable long term
prognostic factor in cardiac patients. Thus, some have
endorsed recitation of specific poetry as a means to
control breathing patterns.

Many features of the cardiorespiratory control during
recitation of poetry are still unknown. Recently, with
simultaneous recordings of an electrocardiogram and a
respiratory trace, new techniques for the analysis of
cardiorespiratory interaction were developed. They
unambiguously revealed that heart rate and respiration
may intermittently synchronize. The application of
these techniques promises new information about the
cardiorespiratory interaction, specifically after a
heart attack.

What is the Hexameter?

The ideal dactylic hexameter consists of six (hexa)
metrons or feet called dactyls (fingers). Each dactyl
consists of three syllables, the first long, the other
two short. Note that the last foot is not a real
dactyl, as it only consists of two syllables. The
following represents a hexameter:

Down in a | deep dark | hole sat an | old pig |
munching a | bean stalk |

A New Study

Now, European physiologists have investigated the
cardiorespiratory synchronization in healthy subjects
using a cross sectional study design: recitation of
hexameter verse, controlled breathing and spontaneous
breathing. They hoped to improve the understanding,
through poetry, of regulatory processes that maintain
stability and coherence between different
physiological functions since cardiorespiratory
interaction
seems to play a crucial role in this context.

The authors of �Oscillations of Heart Rate and
Respiration Synchronize During Poetry Recitation,�
are Henrik Bettermann, from the Department of Clinical
Research, Gemeinschaftskranke nhaus Herdecke
and Dirk Cysarz, at the Institute of Mathematics,
University of Witten/Herdecke, both in Germany;
Dietrich von Bonin and Peter Heusser at the Institute
for Complementary Medicine KIKOM, University of Berne,
Switzerland; Helmut Lackner at the Institute for
Noninvasive Diagnostics, Joanneum Research, Weiz,
Austria;
and Maximilian Moser with the Physiological Institute,
University of Graz, Graz, Austria.

Their findings appear in the Articles in Press section
of the American Journal of
Physiology � Heart and Circulatory Physiology. The
journal is one of 14 published each month by
the American Physiological Society (APS) (www.the-aps.
org).

Methodology

The researchers investigated the cardiorespiratory
synchronization in healthy subjects during
recitation of hexameter verse. Three different
exercises were compared using a cross sectional
study design: recitation of hexameter verse,
controlled breathing, and spontaneous breathing.

Some 20 healthy subjects without prior knowledge of
the hexameter text used for the recitation
were enrolled in the study. After an initial check 3
subjects had to be excluded due to frequent
ectopic heartbeats. The 20 subjects (10 female; age:
43 � 6.6 years, average � SD; 3 smokers) had
no history of cardiovascular diseases, especially no
hypo- or hypertension or anti-arrhythmical therapy.

All subjects were invited individually three times to
the therapy center at the same time of day.
In each of the three sessions the subjects performed a
different exercise (in random order):
hexameter recitation (H), controlled breathing (C) and
spontaneous breathing (S).
The researchers used a piece from Homers Odyssey in a
German translation, which did not alter
the rhythmic scheme of the verse.

During each session an electrocardiogram and the
nasal/oral airflow were recorded simultaneously.
The overall duration of each session was 50-60
minutes, divided into three successive measurements:
15 minutes quiet rest in a resting chair, 20 minutes
of exercise measurement, and 15 minutes quiet
rest in a resting chair. During S1 and S2 the subjects
were allowed to breathe spontaneously.
This procedure resulted in nine different measurements
of each subject. To ensure comparable
levels of physical activity during the three types of
exercises, the subjects walked through
the room at a pace of 50 steps per minute (given by an
electric metronome).
The three experiments had to be at least 24 hours
apart but within 14 days.

Results

With respect to cardiorespiratory interaction the
results of the analysis of the phase
difference and the coherence analysis revealed: (1)
during recitation of hexameter verse
the low frequency oscillations of the breathing
pattern were synchronized to a large extent with the
heart rate oscillations; (2) the cardiorespiratory
interaction was also synchronized during the
controlled breathing exercise, but to a slightly
lesser extent; (3) the resting periods before and
after the exercises showed a further reduction of
cardiorespiratory synchronization; and (4) during the
spontaneous breathing exercise, the cardiorespiratory
interaction was almost completely desynchronized.
Rhythmic speech thus has the strongest impact on
synchronization of low-frequency breathing
oscillations and heart rate fluctuations, whereas
cardiorespiratory interaction during everyday
activities is rarely synchronized.

Conclusion

The special breathing pattern used for the recitation
of hexameter verse
produced a strong cardiorespiratory synchronization
with respect to low-frequency breathing oscillations
and heart rate variations. Controlled breathing showed
cardiorespiratory synchronization to a lesser extent.
The results of this study may improve our
understanding of regulatory processes that maintain
stability
and coherence between different physiological
functions since cardiorespiratory interaction seems to
play
a crucial role in this context.

-end-

Source: Articles in Press section of the American
Journal of Physiology � Heart and Circulatory
Physiology.
The journal is one of 14 published each month by the
American Physiological Society (www.the-aps. org).

The American Physiological Society (APS) was founded
in 1887 to foster basic and applied science,
much of it relating to human health. The Bethesda,
MD-based Society has more than 10,000 members
and publishes 3,800 articles in its 14 peer-reviewed
journals every year.

***

Editor�s Note: A copy of the research article is
available in pdf format to the press.
Members of the press are invited to obtain a pdf copy
of the study and to interview members
of the research team. To do so, please contact Donna
Krupa at (301) 634-7209 (direct dial),
(703) 967-2751 (cell) or mresnick@the- aps.org.

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--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@>
> wrote:
>
> > 3. Obtain the list 7/5, 5/3, 5/4, 1/1, 3/2, 7/6, 7/4. Drop 1/1.
> >
> > 4. This corresponds to J1 set. (The top of a stellated hexany is
J5)
> >
> > 5. Put 1/1 back in. Now you have a septachord. Taking the six
> > hexachord subsets gives: J1, K, J5, G, H, I, H
> >
> > 6. Taking the remaining fractions and doing the same thing gives
J1,
> > K, J5, G, H-, I-, H-
>
> Are you using your group operations directly on 7-limit JI and not
on
> 12-et? Becauase that strikes me as potentially interestong.

Interesting you should mention that. I just reread Erlich's "Forms
of Tonality" and realized he can describe Decatonic scales and
structures, for example, without specifying 22-tEt neccessarily.

So I guess, yes, once I learn how to do that. And get freed of
12-tET in a sense. (However, I guess Z12 is kind of big in
cosmology right now, per Dr. John Baez, it's the product
of the tiling of the space in equilateral triangles by the
tiling in squares, which kind of relates to D4 and S3 symmetries)

Why is the
> prime in the numberator larger than the one in the denomiantor?

It was about 3 am when I thought of that. I guess I just need
to choose one fraction to be otonal and that was the best way
I had to go about it. It seemed to work when I looked at the
7-limit tonality diamond today, but it might be kind of arbitrary.
(I know that is NOT the definition of otonal). At least there is a
kind of symmetry to it...

paul hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> More stuff:
>
> The top part of a stellated hexany is J5 in my system. Here is
> some theory with the SLTD:
>
> Hexachord Theory and the Seven Limit Tonality Diamond
>
> 1. Take the Seven Limit Tonality Diamond: 10/7, 8/7, 12/7, 5/3,
5/4,
> 4/3, 1/1, 3/2, 8/5, 6/5, 7/6, 7/4, 7/5
>
> 2. Work with values so that the prime in the numerator is great
than
> that in the denominator (7>5>3>2)
>
> 3. Obtain the list 7/5, 5/3, 5/4, 1/1, 3/2, 7/6, 7/4. Drop 1/1.
>
> 4. This corresponds to J1 set. (The top of a stellated hexany is J5)
>
> 5. Put 1/1 back in. Now you have a septachord. Taking the six
> hexachord subsets gives: J1, K, J5, G, H, I, H
>
> 6. Taking the remaining fractions and doing the same thing gives
J1,
> K, J5, G, H-, I-, H-
>
> 7. These are the second column of my hexachord system for 35 (7 X 5
> grid)
>
> 8. The remaining hexachord, L, can be found by taking K and moving
> one note: Either up 1 step from E to F: 16/15 or for the Z related
> complement, move from E to A: 4/3
>
> Here is the full 7 X 5 grid: There is one imperfection, P/Q.
> The last row is irregular hexachords, they are all Z related and
> have almost equal interval vector values. (Tie breaker rules
> determine which columns they go in).
>
> A G M1 M5 U
> B1 H N1 N5 V1
> B5 I O1 O5 V5
> C J1 P/Q W
> D J5 R1 R5 X
> E K S1 S5 Y
> F L T1 T5 Z
>
>
> Column 2 is the lynchpin of the system. It is an interesting
> column, because it is the only column where weight measure differs
> from tritone fixing. (All of my ideas are in Paul Hj's Stuff) This
> makes sense because it is based on minor thirds, which tie right
into
> tritone and weight class changes.
>
> (Quick addendum: Weights are 0,1,2,3 because it is mod 6, and
> absolute value. So for example, transposing a hexachord by
> 1 step adds 6 to the weight. Inverting reverses the weight. So
> you obtain (0,1,2,3,2,1,0,1,2,3,2,1,0) for weights)
>
> Paul Hj

Okay, now to extend the Seven Limit Lattice. Using the rule
of Prime(num)> Prime(denom) for the top seven (From which
I get J1, K, J5, G, H, I, H as subsets), let's assign fractions
to the remaining 5: These and the missing note in the first
seven give complementary hexachords to those above.

Top seven: 1/1, 7/6, 7/5, 5/3, 5/4, 3/2, 7/4

Remaining five: Use 8/7, 4/3, 8/5 (Inverses). Now there
are two that are not in the 7LTD, these are extensions. Use
15/14 and 40/21. Now the tritones land in an interesting pattern.
Let's label a note "7" if it occurs as the 7 part of a tritone,
otherwise "5". In terms of an I-Ching Diagram:

5 7
7 5
5 7
7 5
5 7
7 5

In terms of C4 X C3

5 7 7 5
5 7 7 5
7 5 5 7

You get three kinds of minor thirds: 6/5 (twice) 7/6 and 25/21
in this pattern 7/6, 6/5, 25/21, 6/5. A 7/5 tritone will be over
C,D,E and a 10/7 over Db, Eb, F. Then they both flip.

Way am I doing this? Well, G, H, I, J1, J5 and K form a thread
in the I-Ching system. All but I use 2 tritones. I uses 3.

By these operations:

A) Swap tritone(s) with complements
B) Reverse tritones with non-tritone
C) (May not be needed) Flip A and B in a tritone pair

You can cover all hexachord types. There are 35, which is 7 X 5.
A hexachord type (I-Ching diagram) represents both the hexachord
and it's complement. Also represents it's inverse, this kind of
comes out in the wash.

The symbols:

- - 0 in A 6 in B
--- 0 and 6 in A
-o- 0 in B 6 in A
-e- 0 and 6 in B

For each row, like this

0 6
1 7
2 8
3 9
4 10
5 11

B is A'

Now we have some fractions to use also, based on the 7-limit

My hunch is that using these fractions (and possibly 1/1, 6/5,
10/6, 12/7 if needed) will represent the 7 X 5 hexachords
in some kind of pattern.

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > More stuff:
> >
> > The top part of a stellated hexany is J5 in my system. Here is
> > some theory with the SLTD:
> >
> > Hexachord Theory and the Seven Limit Tonality Diamond
> >
> > 1. Take the Seven Limit Tonality Diamond: 10/7, 8/7, 12/7, 5/3,
> 5/4,
> > 4/3, 1/1, 3/2, 8/5, 6/5, 7/6, 7/4, 7/5
> >
> > 2. Work with values so that the prime in the numerator is great
> than
> > that in the denominator (7>5>3>2)
> >
> > 3. Obtain the list 7/5, 5/3, 5/4, 1/1, 3/2, 7/6, 7/4. Drop 1/1.
> >
> > 4. This corresponds to J1 set. (The top of a stellated hexany is
J5)
> >
> > 5. Put 1/1 back in. Now you have a septachord. Taking the six
> > hexachord subsets gives: J1, K, J5, G, H, I, H
> >
> > 6. Taking the remaining fractions and doing the same thing gives
> J1,
> > K, J5, G, H-, I-, H-
> >
> > 7. These are the second column of my hexachord system for 35 (7 X
5
> > grid)
> >
> > 8. The remaining hexachord, L, can be found by taking K and
moving
> > one note: Either up 1 step from E to F: 16/15 or for the Z
related
> > complement, move from E to A: 4/3
> >
> > Here is the full 7 X 5 grid: There is one imperfection, P/Q.
> > The last row is irregular hexachords, they are all Z related and
> > have almost equal interval vector values. (Tie breaker rules
> > determine which columns they go in).
> >
> > A G M1 M5 U
> > B1 H N1 N5 V1
> > B5 I O1 O5 V5
> > C J1 P/Q W
> > D J5 R1 R5 X
> > E K S1 S5 Y
> > F L T1 T5 Z
> >
> >
> > Column 2 is the lynchpin of the system. It is an interesting
> > column, because it is the only column where weight measure
differs
> > from tritone fixing. (All of my ideas are in Paul Hj's Stuff)
This
> > makes sense because it is based on minor thirds, which tie right
> into
> > tritone and weight class changes.
> >
> > (Quick addendum: Weights are 0,1,2,3 because it is mod 6, and
> > absolute value. So for example, transposing a hexachord by
> > 1 step adds 6 to the weight. Inverting reverses the weight. So
> > you obtain (0,1,2,3,2,1,0,1,2,3,2,1,0) for weights)
> >
> > Paul Hj
>
> Okay, now to extend the Seven Limit Lattice. Using the rule
> of Prime(num)> Prime(denom) for the top seven (From which
> I get J1, K, J5, G, H, I, H as subsets), let's assign fractions
> to the remaining 5: These and the missing note in the first
> seven give complementary hexachords to those above.
>
> Top seven: 1/1, 7/6, 7/5, 5/3, 5/4, 3/2, 7/4
>
> Remaining five: Use 8/7, 4/3, 8/5 (Inverses). Now there
> are two that are not in the 7LTD, these are extensions. Use
> 15/14 and 40/21. Now the tritones land in an interesting pattern.
> Let's label a note "7" if it occurs as the 7 part of a tritone,
> otherwise "5". In terms of an I-Ching Diagram:
>
> 5 7
> 7 5
> 5 7
> 7 5
> 5 7
> 7 5
>
> In terms of C4 X C3
>
> 5 7 7 5
> 5 7 7 5
> 7 5 5 7
>
> You get three kinds of minor thirds: 6/5 (twice) 7/6 and 25/21
> in this pattern 7/6, 6/5, 25/21, 6/5. A 7/5 tritone will be over
> C,D,E and a 10/7 over Db, Eb, F. Then they both flip.
>
> Way am I doing this? Well, G, H, I, J1, J5 and K form a thread
> in the I-Ching system. All but I use 2 tritones. I uses 3.
>
> By these operations:
>
> A) Swap tritone(s) with complements
> B) Reverse tritones with non-tritone
> C) (May not be needed) Flip A and B in a tritone pair
>
> You can cover all hexachord types. There are 35, which is 7 X 5.
> A hexachord type (I-Ching diagram) represents both the hexachord
> and it's complement. Also represents it's inverse, this kind of
> comes out in the wash.
>
> The symbols:
>
> - - 0 in A 6 in B
> --- 0 and 6 in A
> -o- 0 in B 6 in A
> -e- 0 and 6 in B
>
> For each row, like this
>
> 0 6
> 1 7
> 2 8
> 3 9
> 4 10
> 5 11
>
>
> B is A'
>
> Now we have some fractions to use also, based on the 7-limit
>
> My hunch is that using these fractions (and possibly 1/1, 6/5,
> 10/6, 12/7 if needed) will represent the 7 X 5 hexachords
> in some kind of pattern.

Forgot to add:

Symbol manipulations allowed:

A) --- <-> -e- (must end up with equal numbers)
B) - - <-> --- together with -o- <-> -e- (must do both, and all)
C) - - <-> -o- (must end up with equal numbers)

All other permutations are illegal.

The next step is to find the best representatives of the 35
so that we can assigned fixed lattice values. (with 15/14 20/21
added). 35/24 is an alternate value for 3/2, too.

A is the tritone swap, used in lower realm
B changes from 3<->0, or 2<->1 tritones (change of realm)
C is the non-tritone swap, used in upper realm

Reducing 35 to 26 hexachords (D4 X S3) (A-Z letters) produces exactly
13 in upper realm and 13 in lower realm

ABCDELMNRSTXY in upper realm
FGHIJKOPQUVWZ in lower realm

AEMMY

BNT CSX DLR

JOV HQW G/K F/Z

UPI

G, H, I, J, K with A) send to OV, QW, K, Z, PI depending
with B) we go to the upper realm
with C) we move around upper realm in the same manner

The M5 relation is contained within each grouping

> Paul Hj
>