Gene,

I post here only a part of my answer to your questions. I

will continue as soon as I can. You wrote:

<< "Multilinear" as in tensor or wedge products, or what

exactly do you mean? >>

As you can see here

<http://www.aei.ca/~plamothe/sys72/re1215.gif>

the ib1215 set is convex in the lattice <3 5 7 11> z^4 and

that in the current sense (i.e. any segment between nodes

has no hole, the intermediate nodes are occupied). Multilinear

refer simply to the fact it's a 4D convex body (differ in R^4).

*** I note for anyone else that <3 5 7 11> == [42 23 58 33]

yellow == 3, red == 5, green == 7, blue == 11

so it's very easy to recognize the axis and to know the

JI corresponding values. Nodes have color of the minimal

N-limit.

You wrote:

<< It seems to me that a sequence defined by one generator

is trivially convex. >>

Sure. What is in question is the apparently obvious choice of

a convex segment centered on 0 (of a such linear sequence) to

obtain a pertinent object in musical sense.

That follows by simple imitation of the pure fifth generator.

There is no problem with the powers of the unique prime 3 for

there exist only two JI steps and a unique unison vector. In

each turn of the octave a new unison vector appears and the

precedent one replace a precedent step.

When a microgenerator like the Secor 7/72 is chosen for its

quality of very good approximation of the intervals in the

11-limit, if there is pretention to obtain more than a space

where possible tempered harmonies in 11-limit are very good, it

appears to me that one have to pay attention at what differs

from the one dimensional fifth generator.

Obviously, if someone believes JI reference is useless to build

scales in the melodic sense, I would say "Continue to imitate a

well-known simple schema and build long scales". I believe, for

my part, that if JI reference is important for harmony, it remains

important for melodic structuration since I discovered that the

two "logic" are inextricably related in algebraic structures.

The gammier theory indicates that the longest non-degenerated

structures (definition forward) within both the odd 21-limit and

the prime 7-limit or 11-limit (i.e. using lattices <2 3 5 7> Z^4

or <2 3 5 7 11> Z^5) have 10 degrees and there exist only two

such structures:

ib1183 in 7-limit

ib1215 in 11-limit.

Comparing the 41 intervals of ib1215 with the 41 intervals of

the convex linear Miracle sequence, it is easy to show there is

few differences in the following like-keyboard representation

<http://www.aei.ca/~plamothe/sys72/cl1215.gif>

Which of these two has more sense for melodical structuration?

First, I argue against the Miracle sequence that as long as one

see within it a reflect of an underlying decatonic structure one

have to admit that this sequence contains unison vectors, namely

2, 4, 68, 70. (What ask to distinguish tuning and structure).

For sure, imitating the fifth generator, one could consider that

sequence as a bivalent 41-tone scale 0 2 4 5 7 9 11 12 ... having

1 and 2 as steps and also 1 (whoops!) as unison vector (i.e.

step difference). Yet one could consider it as a bunch of bivalent

21-tone scales, having 2 and 5 as steps and also (whoops!) 3 as

unison vector.

68 70 72

61 63 65 67

54 56 58 60

47 49 51 53

40 42 44 46

33 35 37 39

26 28 30 32

19 21 23 25

12 14 16 18

5 7 9 11

0 2 4

Besides their so much logic quality one could ask Joseph Perhson

to verify their melodic practicability.

-----

Looking now at ib1215 as a simple set, it is easy to find first

it contains 6 atoms (i.e. intervals that can't be factorized

into lesser intervals in the set), namely

<22/21 21/20 16/15 15/14 12/11 10/9>

If ib1215 is coherent (as set), there exist a unique epimorphism

H : ib1215 --> Z

such that

H(22/21) = H(21/20) = ... = H(10/9) = 1

what is the case and then

H(2/1) = 10.

As any gammier structure ib1215 contains more than one periodicity

block whose intersection is the unison.

I add that the corresponding tempered values in 72-tET are

5 == <22/21> == <21/20>

7 == <16/15> == <15/14>

9 == <12/11>

11 == <10/9>

and that the only cases of ambiguity in detempering are 5, 7 (and

octave reverse).

PROBLEM

(a) Try to make that with the Miracle set whose atoms

have to correspond to 2 and 5.

(b) Where's the evidence that that may correspond to a

decatonic structure?

(c) What would be the atoms removing 2 and 4? and which

periodicity could be then possible?

-----

... to continue

Pierre

Gene,

I'll wait later to continue my answer. Hope all goes well for

your mother.

Pierre