Paul

I would believe naïvely that an interval like 64/63 is too

small to be seen as a step but I would never use that as an

argument since it concerns musical judgment rather than maths

and I am searcher in music-math relations, not musician.

However I mentionned (post 1000) a problem concerning linear

generator picking up unison vectors using convexity. That

suggests 64/63 could be a unison vector misinterpreted as a

step.

Besides you wrote precedently:

<<

Personally, I think Blackjack, let along Canasta, have

too many notes to be heard and conceptualized in their

entirety, the way diatonic scales and their Middle-Eastern

cousins are, and perhaps my decatonic scales can be.

>>

So rather than comment on receivability in gammier theory of

your detempered values, I would suggest a reinterpretation

of the Blackjack set using incidentally a decatonic gammier.

You have to judge if it has interest from musical viewpoint

or even may concern your decatonic scales. If not, it won't be

bad for it's not strictly derived from gammier theory. I had

to use ad hoc maths to insert anew consistency into a reduced

(so desorganized) set of 21 intervals out of a gammier having

41 intervals originally.

-----

As alternative to the Blackjack set seen as the scale 0 2

7 9 14 16 21 23 28 30 35 37 42 44 49 51 56 58 63 65 70 72

having alternatively steps 2 and 5, let us work on this

approach where the same set would be seen as

9 16 23 30 37 44 51 58 65

0 72

7 14 21 28 35 42 49 56 63

where 2 and 70 are missing for being considered as unison

vectors. The values 2 and 70 would keep sense only for the

transposition and would be stranger in the modes begining

with 0.

-----

In appearance, there is no problem to describe the melodic

relations here since the steps seem to be simply 7 and 9.

>From the tempered viewpoint it would be hard to go over

the following solution.

9---16---23---30---37---44---51---58---65

/ / / / / / / / / \

0 / / / / / / / / 72

\ / / / / / / / / /

7---14---21---28---35---42---49---56---63

However, I recall that the Blackjack set is supposed to

be very closed from just values in 11-limit and it's a

reduced set (after Canasta 31 and Miracle 41), so the

steps are not forcely all retained.

So I want to refine the problem seeking to detemper in

such way that the intervals would be both of minimal

sonance and melodically well-organized.

-----

I have a link to images (post 1000) showing that the gammier

ib1215 would be the simplest solution using 41 intervals and

where it is easy to see that the step relations would be

completely desorganized with the Blackjack set.

Starting from that it seemed clear that odd 11 couldn't be

used to reorganize the melodic relations (even if it keeps

sense vertically). Thus, seeking to reorganize the just

relations in the 7-limit I used the following relation

12/11 = 35/32 modulo 385/384

to find an optimal solution which requires to use also the

step 5 with 7 and 9. I don't give more details since it's not

a systematic approach.

-----

In the following graph the edges (--,\,---) correspond

to (5,7,9).

(2) seen as unison vector

\

0 --- 9

\ \

7 --- 16

\ \

14 --- 23 -- 28

\ \ \

21 --- 30 -- 35

\ \

37 -- 42 --- 51

\ \ \

44 -- 49 --- 58

\ \

56 --- 65

\ \

63 --- 72

\

(70) seen as unison vector

which is a regular lattice (in sense of "treillis") if

the unison vectors are removed. So, detempering as

(5,7,9) == (21/20, 16/15 or 15/14, 35/32)

we obtain this JI lattice having all low sonance excepted

the two using odd 105.

| 35/32 | 21/20 | 35/32 |

--- ( 1 )----35/32

| |

16/15 | |

| |

--- 16/15-----7/6

| |

15/14 | |

| |

--- 8/7------5/4-----21/16

| | |

16/15 | | |

| | |

--- 128/105----4/3------7/5

| |

15/14 | |

| |

--- 10/7------3/2----105/64

| | |

16/15 | | |

| | |

--- 32/21-----8/5------7/4

| |

15/14 | |

| |

--- 12/7-----15/8

| |

16/15 | |

| |

--- 64/35----( 2 )

So, rather than that

9---16---23---30---37---44---51---58---65

/ / / / / / / / / \

0 / / / / / / / / 72

\ / / / / / / / / /

7---14---21---28---35---42---49---56---63

we have that

9---16---23---30---37---44 51---58---65

/ / / \ / \ \ \ / / / \

0 / / \ \ \ \ / / 72

\ / / / \ \ \ / \ / / /

7---14---21 28---35---42---49---56---63

-----

I add simply that 9 as a step has to correspond to 35/32 here

to give consistency while it may appear as 12/11 in chord.

Finally, I could draw images later if this JI interpretation

has sense at musical viewpoint.

Pierre

--- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:

> I would believe naïvely that an interval like 64/63 is too

> small to be seen as a step but I would never use that as an

> argument since it concerns musical judgment rather than maths

> and I am searcher in music-math relations, not musician.

It's 1/44 of an octave, and in some temperings it is larger. It's a

single step in the 19,26,31,34 and 41 systems, and certainly a single

19-et step does sound very much like a scale step. I'll probably want

to comment further but I need to chase down your reference.

--- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:

>

> where 2 and 70 are missing for being considered as unison

> vectors. The values 2 and 70 would keep sense only for the

> transposition and would be stranger in the modes begining

> with 0.

Don't assume that the modes beginning with 0 are more important or

more special than other modes. Look at my suggestion with

superparticular steps. It is not based on the mode beginning with 0.

I thought you might like it because for my 22-tET-based decatonic

scales, the JI versions with superparticular steps were the ones you

found most suitable for your theory.

Paul wrote:

<< Don't assume that the modes beginning with 0 are more

important or more special than other modes. Look at my

suggestion with superparticular steps. It is not based on

the mode beginning with 0. >>

I don't assume that. I wrote my remark (that 2 and 70 keep

sense for the transposition) precisely for that.

<< The values 2 and 70 would keep sense only for the

transposition and would be stranger in the modes

beginning with 0. >>

In your case the modes are the 21 rotations and there exist

only one mode beginning by 0. In my case there exist 60

decatonic modes beginning with 0.

By transposition, (now in "pitch space"), these modes may begin

with 0, 7, 14, 21, etc. When a mode begin with 7n distinct

from 0 then 7n+2 is now stranger to the mode while 2 is used.

The "note" 7n+2 is stranger for being in the same class as the

the "tonic" 7n.

As I already said, (even if that concerned true gammier), all

possible melodic modes have not the same importance. The best

among them are the more compact in the lattice (in sense of

"réseau"). So the interval space (obtained by rotation) is

minimal and chords are maximal.

-----

Since I don't have time to write now on the last point I could

quote a french post to Antti Kartunnen in june 2001 about the

criteria used with modal classes. That contains figures helping

to understand my point about inequal importance of modes.

<<

- modalité

- transposabilité

- minimalité

- harmonicité

- simplicité

- connexité

La modalité réfère à la dualité majeure mineure. La transposabilité réfère

au nombre de modes possibles par changement de tonique (ça définit la

classe modale) et renversement. La minimalité réfère à l'espace des

intervalles générés par le mode. L'harmonicité réfère aux accords de base

possible. Bien que l'harmonie relève d'abord de la microtonalité, il y a

ici un aspect macrotonal. Voici comment caractériser l'harmonicité et la

transposabilité des 4 classes modales dans ib1051.

o - o

| \ | \ 10 modes (majeurs/mineurs) - 3 accords

o - o - o

o

| \

o - o - o 3 modes (symétriques) - 2 accords

\ |

o

o

| \

o - o - o 3 modes (symétriques) - 2 accords

\ |

o

o o

\ |

o - o 4 modes (majeurs/mineurs) - 1 accord

\ |

o

La simplicité réfère à la sonance d'un intervalle et la connexité au nombre

de connexions (conjointes) sur cet intervalle.

Mathématiquement, c'est liée à

la symétrie (modalité)

la convexité (transposabilité, minimalité, harmonicité)

la centralité (simplicité, connexité)

Pierre

--- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:

>As I already said, (even if that concerned true gammier), all

>possible melodic modes have not the same importance.

The relative importance they may have is related to the musical

grammar used, which lies outside the considerations of your model.

Before the 17th century, the major mode (what you call "Zarlino") of

the diatonic scale was considered one of the _least_ important

melodic modes. And it greater importance since the 17th century, I

don't believe is accounted for by anything in your theory.

>The best

>among them are the more compact in the lattice

I don't see how one melodic mode can be more compact in the lattice

than another. They are all identical lattice configurations.

>(in sense of

>"réseau").

So perhaps by "lattice" you don't mean "harmonic lattice". Can you

elucidate the "réseau" concept?

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

> >(in sense of

> >"réseau").

> So perhaps by "lattice" you don't mean "harmonic lattice". Can you

> elucidate the "réseau" concept?

"Réseau" is French for "lattice" in the sense we've been using

it; "lattice" has another meaning in English-language mathematics

which French sensibly gives another name to.

Paul,

In writing << all possible melodic modes have not the same

importance >> I was at hundred miles to think you could take

that in the sense of historical importance. In the Zarlino

gammier

1 5 2 6

6 3 0 4 1

1 5 2 6

there exist 16 modes. One of them is the non-convex

1 16/15 6/5 4/3 3/2 5/3 15/8 2

. 5 . 6

. 3 0 4 .

1 . 2 .

I wanted to say uniquely that a such mode is less important,

for instance, than the doristi mode

1 16/15 6/5 4/3 3/2 8/5 16/9 2

. . . .

6 3 0 4 .

1 5 2 .

for reason which appears clearly in the lattice. Like the

Zarlino mode itself, the second implies for instance

a) convexity and minimality (about the space obtained by

rotation), being

. o o o o .

. o o o o o .

. o o o o .

rather than the following (periodic but non-convex) for

the first one

. . .

o o o o .

o o o o o

. o o o o

. . .

b) the maximality for the number of triads (5 rather than 2

in the first case).

-----

I never thought that the reality was reduced at my poor

little considerations on a simple aspect.

Besides, I remember a time where I didn't resist to start

a discussion after your comments. Now, I could'nt.

-----

You wrote:

<< So perhaps by "lattice" you don't mean "harmonic

lattice". Can you elucidate the "réseau" concept? >>

In question for Gene I wrote:

<< We don't have this problem in French for we use

"treillis" in the first sense of partial ordering

where two elements have always inferior and superior

"bornes", and

"réseau" in the sense of discrete Z-module. >>

A module is like a vector space, If a vector V exists, then any

kV exists, where k is an element of

- a field like R (reals) for the vector space

- a ring like Z (integers) for the module

A Z-module is a module where the ring is Z.

A "réseau" (discrete Z-module) is discrete i.e. has all its

elements separate in R (at center of a ball where there not

exist another element).

What you name "harmonic lattice" is based on a represention

of a such lattice in which you represent some edges like,

for instance, edges 6/5, 5/4, 3/2 which are differences

between two vectors.

Z-module and vector space are suitable for interval space

while a pitch space is an "espace affine" or an "espace

affine muni d'une origine" when a tonic is defined. The

unison element don't exist in these "espaces affines" but

it's the neutral element of a "réseau" and a vector space.

Pierre

--- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:

> A Z-module is a module where the ring is Z.

To keep things from getting too confusing around here, I'll mention

again that a Z-module is the same as an abelian group.

> A "réseau" (discrete Z-module) is discrete i.e. has all its

> elements separate in R (at center of a ball where there not

> exist another element).

I think again we could fix our terminology around here and call this

a lattice. If we want the other sense of lattice, why not call

it "treillis"? If we are really lucky, the Anglo-Saxons will start

doing this too and we all will be happier.

> Z-module and vector space are suitable for interval space

> while a pitch space is an "espace affine" or an "espace

> affine muni d'une origine" when a tonic is defined.

I don't see that you need the concept, but if you think you do why

not just call it "affine space", since you are writing in English? If

a tonic is defined then it seems to me you are back with groups.

What is something acted on by a lattice, I wonder--it's a sort of

symmetric space?

Gene,

You give me a precious feedback about English and mathematics.

I will use simply the word lattice (to translate "réseau") and

the word treillis for the partial ordering structure I use so

abundantly.

-----

Now, I understand you want to avoid confusion precising that

a Z-module is simply an abelian group, but since Paul ask I

precise "my" (which is not mine) concept of lattice, I think

it would have been irrelevant to simply identify lattice with

abelian group. Paul is also a scientific and I believe he has

not only interest for simplified notion. If I would have asked

myself this question I would have been confused with a such

identification.

OK, a lattice is an abelian group, however how could we use

matrices if it was only a group? We use here clearly two

composition laws. What seemed to me important here was the

linearization process of the group and the lack of continuity

in the action group. It's why I talked about vectors and use

of integers rather than reals.

What would be interessant would be to show the pertinence of

the lattice representation (depending of the linear independance

of primes) which is remarkably suitable to focus on the local

part around unison, this only part concerning (in a modelization

viewpoint) the music, where distinction of sonance quality has

more sense than a simple aspect of the sensation.

-----

Now, about the affine space. I agree we have no need for that.

I mentioned that only in relation with a precedent discussion

where Paul had difficulty to understand I talked always about

intervals without reference to pitch space.

I wanted to say here that, mathematically speaking, interval

space (representable by module and vector space) is a more

fundamental object than a pitch space (representable by affine

space with origin or not). Sure, we can identify space of

bipoints in affine space with vector space but we absolutely

not need to refer at a such pitch space to treat fundamentally

the interval space.

-----

As you may know, I'm not a mathematician (and decidely not a

fan of calculation) so it may seem paradoxal I could appear more

fussy than you about mathematical definitions. It's simple. That

is the principal matter of my thoughts under the mathematical

angle. Besides, it's very hard for me to speak about other

angles in English.

I never try to apply mathematical tools a priori without a

deep motivation in term of modelization. Moreover I learn

only the parts I need. In my research, I approach always with

instinctive representation, formulating first some intuitive

notions, and when I need to refine, then I find generally in

the mathematics the pretty concepts filling well my need.

Now I begin to look at relation between topologic, metric and

normologic concepts to precisely describe what I'm discovering

and exploring in the musical interval space.

-----

I'm closing here my actual short session of posts.

Pierre

--- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:

> Now, I understand you want to avoid confusion precising that

> a Z-module is simply an abelian group, but since Paul ask I

> precise "my" (which is not mine) concept of lattice, I think

> it would have been irrelevant to simply identify lattice with

> abelian group.

I agree we should not identify lattice with abelian group, we should

also require it to be a discrete subgroup of R^n. Two examples are:

(1) Note classes in the p-limit, symmetrically arranged. For example

we can define Q(3^a 5^b 7^c) = a^2+b^2+c^2+ab+ac+bc, and thereby put

a metric on the 7-limit note classes, making it into a lattice.

(2) Notes under an L1 (taxicab) norm; if we set

||2^a 3^b 5^c 7^d|| = ln(2)|a| + ln(3)|b| + ln(5)|c| + ln(7)|d|

we exhibit it as a discrete subgroup of R^4, and hence a lattice.

> I wanted to say here that, mathematically speaking, interval

> space (representable by module and vector space) is a more

> fundamental object than a pitch space (representable by affine

> space with origin or not). Sure, we can identify space of

> bipoints in affine space with vector space but we absolutely

> not need to refer at a such pitch space to treat fundamentally

> the interval space.

I agree, and I haven't seen anyone get any milage out of affine

concepts, with or without a distinguished point.

--- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:

> Paul,

>

> In writing << all possible melodic modes have not the same

> importance >> I was at hundred miles to think you could take

> that in the sense of historical importance. In the Zarlino

> gammier

>

> 1 5 2 6

> 6 3 0 4 1

> 1 5 2 6

>

> there exist 16 modes. One of them is the non-convex

>

> 1 16/15 6/5 4/3 3/2 5/3 15/8 2

>

> . 5 . 6

> . 3 0 4 .

> 1 . 2 .

>

> I wanted to say uniquely that a such mode is less important,

> for instance, than the doristi mode

>

> 1 16/15 6/5 4/3 3/2 8/5 16/9 2

>

> . . . .

> 6 3 0 4 .

> 1 5 2 .

Pierre, "mode" in musicians' parlance means rotation -- a different

scale degree becomes the first scale degree, and all others rotated

by the same amount. All modes of a scale have identical lattice

configurations . . . differing only by a translation.

My claim is that, in this sense of "mode", one cannot

determine "better" or "worse" modes without some reference to the

particulars of the musical style they are to be used for. Sonance of

the degrees against the 1/1 is irrelevant unless the musical style

uses a 1/1 drone, for example.

Question:

Has anyone here studied any of these lattices, in relationship to

musical tuning?

1. A2.12 sublattice of the Leech Lattice (one of 23 Neimeier lattices)

2. A1.24 sublattice of the Leech Lattice

3. K12 Lattice

4. Barnes-Walls Lattice

5. Coxeter-Todd Lattice

There first two are based on M12 and M24 respectively, and of course

Leech is just a lattice of S(5,8,24) blown up, K12 I think is the

same as Coxeter-Todd?

Finally, is the A4 or D4 lattice discussed here based on the same

nomenclature as the Dynkin diagrams? (Used for Lie Algebras, et c)

Thanks

PGH