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Message: 7525

Date: Thu, 02 Oct 2003 00:44:44

Subject: Re: TM reduction

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> what's the point of defining tenney height as p*q if you're only 
> going to use the log anyway, and tenney harmonic distance is 
already 
> log(p*q)?

It's simpler--it doesn't involve any transcendental functions. Aside 
from that logs do have advantages, because you get a norm.


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Message: 7526

Date: Thu, 02 Oct 2003 00:57:03

Subject: Re: hey gene

From: Carl Lumma

>> What's this:
>> 
>> ># h2 scale blocks
>> >
>> >cm1 := [9/7, 6/5, 8/7];
>> >c1 := [[-1, 0, 0], [-1, 0, 1], [-1, 1, 1], [0, 0, 0]];
>> >s1 := [1, 15/14, 6/5, 5/4, 9/7, 3/2, 12/7, 7/4];

What are cm1, c1, and s1?

>These are what I called "chord blocks", which are 7-limit scales 
>analogous to Fokker blocks. This works because 7-limit tetrads, 
>uniquely among prime limits, form a lattice. The resulting scales 
>have the nice property of having a lot of chords to work with.

I remember this stuff.  But I don't remember the bit about the
7-limit being unique in this.  What is it that makes, say, the
5-limit triads not form a "lattice"?

-Carl


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Message: 7528

Date: Fri, 03 Oct 2003 21:29:17

Subject: Re: hey gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> I remember this stuff.  But I don't remember the bit about the
> 7-limit being unique in this.  What is it that makes, say, the
> 5-limit triads not form a "lattice"?
> 
> -Carl

the major triads do, and the minor triads do, but if you want 1 point 
for each otonal *or* utonal chord, you only get a lattice in the 3-d 
case. in 2-d (usually 5-limit), you would get an array corresponding 
to the vertices of a hexagonal tiling -- not a lattice.


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Message: 7529

Date: Fri, 03 Oct 2003 21:30:21

Subject: [tuning] Re: Polyphonic notation

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> Oh, you're enforcing the 'new' definition of MOS. 
> >
> >Not on me, I hope. I don't like it and there are too many terms 
> >floating about as it is. We could just stick to Myhill's property.
> 
> Yes, I must admit I don't think we should change our usage of
> MOS, because of something Kraig said.

but the only reason i dragged it in in the first place was because of 
something else kraig said!!! it came in on false pretenses, and now 
it goes back out.


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Message: 7530

Date: Fri, 03 Oct 2003 21:31:45

Subject: Re: hey gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> i understand this, in a nutshell, to mean that the reduction
> process places the bounding vectors of the periodicity-block
> as close as possible to the center/origin, and as short as
> possible, thus ensuring that the entire block is compacted
> as much as possible towards the center/origin.
> 
> yes?

if you choose to use the center/origin as one of the vertices of your 
fokker periodicity block, then yes.


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Message: 7531

Date: Fri, 03 Oct 2003 21:34:28

Subject: Re: hey gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >>>5-limit triads can be represented as the vertices of a hexagonal 
> >>>tiling; this isn't a lattice in the sense of the word I use 
since 
> >>>it isn't a group.
> >> 
> >> Why isn't it a group?
> >
> >One step takes a C minor chord to a C major chord. Where does the 
> >next step go? It doesn't go to a chord at all--we don't have a 
group, 
> >since we don't have closure under addition.
> 
> Doesn't go to a chord?  Aren't you connecting the centers of the
> triangles?  Then I get Cm->CM->C#m.

no, you end up at the note E, because each step is supposed to be the 
same distance.

> If you connect the roots, or
> any of the vertices, I get Cm->CM-Am.

that doesn't make sense. how do you see that?


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Message: 7532

Date: Fri, 03 Oct 2003 21:38:04

Subject: Re: Please remind me

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 

> I take it that mapping generators to primes is 
> based on commas that are tempered out?

yes, each prime can then be expressed in terms of generators *and* 
periods -- see for example this table:

Yahoo groups: /tuning/database? * [with cont.] 
method=reportRows&tbl=10&sortBy=4

hopefully you understand what everything means there?

> How is this calculated easily?

not so easily, it seems. gene and graham have different methods of 
doing it . . .


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Message: 7533

Date: Fri, 03 Oct 2003 14:59:09

Subject: Re: hey gene

From: Carl Lumma

>the major triads do, and the minor triads do, but if you want 1
>point for each otonal *or* utonal chord

http://lumma.org/tuning/doh.png - Type Ok * [with cont.]  (Wayb.)

I finally realized the length on the right is too much.  Sorry
Gene.

-Carl


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Message: 7534

Date: Fri, 03 Oct 2003 15:04:06

Subject: Re: [tuning] Re: Polyphonic notation

From: Carl Lumma

>but the only reason i dragged it in in the first place was because of 
>something else kraig said!!! it came in on false pretenses, and now 
>it goes back out.

If the pretenses don't matter, we can let history and/or sensicality
prevail.

-Carl


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Message: 7535

Date: Fri, 03 Oct 2003 22:07:13

Subject: [tuning] Re: Polyphonic notation

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >but the only reason i dragged it in in the first place was because 
of 
> >something else kraig said!!! it came in on false pretenses, and 
now 
> >it goes back out.
> 
> If the pretenses don't matter, we can let history and/or sensicality
> prevail.
> 
> -Carl

meaning?


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Message: 7537

Date: Fri, 03 Oct 2003 22:22:28

Subject: Re: [tuning] Re: Polyphonic notation

From: Carl Lumma

>> If the pretenses don't matter, we can let history and/or sensicality
>> prevail.
>> 
>> -Carl
>
>meaning?

Whatever we think it means.  I tend to see history as in, the
last few years on this list -- our contribution is substantial.
Given the term MOS, I don't see much of a role for sensicality,
but it seems the term applies just as well to the fractional-
octave case.

-Carl


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Message: 7538

Date: Sat, 04 Oct 2003 19:15:33

Subject: a whole bunch of posts without much math

From: Joseph Pehrson

I can't believe I'm reading (for the first time) all these posts here 
about *notation* and they don't even have all the much math in them!

This is the kind of material that should be on the MAIN Tuning List, 
in my opinion.  

Only if the materials gets excessively technical so that only a *few* 
people can follow it should it be on *Tuning Math...*

That's the way these forums have initially been set up and, I feel, 
Tuning Math has really wandered into general tuning theory.

When an entire *notation* is created over here, as in the case of 
Sagittal, that's a topic for the *general* list.  Mathematical 
details and specifics can be worked out over here, but that's not the 
sense of the last 100 posts (at least) or so I'm reading over here!

And who am *I* to proclaim an opinion on this over here?  Well, 
dunno, but I have a right to an opinion, and this is it!

J. Pehrson


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Message: 7539

Date: Sat, 04 Oct 2003 12:39:26

Subject: Re: a whole bunch of posts without much math

From: Carl Lumma

>I can't believe I'm reading (for the first time) all these posts here 
>about *notation* and they don't even have all the much math in them!
>
>This is the kind of material that should be on the MAIN Tuning List, 
>in my opinion.  

I think we should start a list for the duplicate posts that you like
to make.  Every time you get going, you just continue posting over
there.

>That's the way these forums have initially been set up and, I feel, 
>Tuning Math has really wandered into general tuning theory.

Saints preserve us!

>When an entire *notation* is created over here,

Bzzzz.

-Carl


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Message: 7540

Date: Sat, 04 Oct 2003 20:04:45

Subject: Re: a whole bunch of posts without much math

From: Joseph Pehrson

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

Yahoo groups: /tuning-math/message/6983 * [with cont.] 

> >I can't believe I'm reading (for the first time) all these posts 
here 
> >about *notation* and they don't even have all the much math in 
them!
> >
> >This is the kind of material that should be on the MAIN Tuning 
List, 
> >in my opinion.  
> 
> I think we should start a list for the duplicate posts that you like
> to make.  Every time you get going, you just continue posting over
> there.
> 
> >That's the way these forums have initially been set up and, I 
feel, 
> >Tuning Math has really wandered into general tuning theory.
> 
> Saints preserve us!
> 
> >When an entire *notation* is created over here,
> 
> Bzzzz.
> 
> -Carl


***Well, if people can complain about too much math over on the 
*main* list, why can't they complain about too *little* math on THIS 
list!  :)

I guess I'll just have to read this list more... but the character 
has changed somewhat from before it seems...

JP


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Message: 7541

Date: Sat, 04 Oct 2003 13:19:49

Subject: Re: a whole bunch of posts without much math

From: Carl Lumma

>***Well, if people can complain about too much math over on the 
>*main* list, why can't they complain about too *little* math on THIS 
>list!  :)

:)

>I guess I'll just have to read this list more... but the character 
>has changed somewhat from before it seems...

The thing is, we have no way to predict what you or anyone else
will or won't understand.

-Carl


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Message: 7542

Date: Sat, 04 Oct 2003 20:45:06

Subject: Re: a whole bunch of posts without much math

From: Joseph Pehrson

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

Yahoo groups: /tuning-math/message/6985 * [with cont.] 

> >***Well, if people can complain about too much math over on the 
> >*main* list, why can't they complain about too *little* math on 
THIS 
> >list!  :)
> 
> :)
> 
> >I guess I'll just have to read this list more... but the character 
> >has changed somewhat from before it seems...
> 
> The thing is, we have no way to predict what you or anyone else
> will or won't understand.
> 
> -Carl


***Well, truly I'm sick of the "carping" on the other list... (and I 
can say this, because those people never come over here... :)

Maybe this list is turning out more like the *old* tuning list used 
to be...

I'll just have to follow it more:  "so many lists, so little 
time..."  :)

JP


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Message: 7543

Date: Sat, 04 Oct 2003 21:27:55

Subject: Re: hey gene

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> So CM and C#m aren't connected then?  Why wouldn't you connect
> them?

They don't have a common interval. You can get to C#m by a common-
interval path, of course--CM-Am-C#m

> >This gives hexagons,
> 
> So does connecting the centers of all the triangles.

The two are equivalent.

> >where you have a line from Cm to CM, and lines *in 
> >different directions* from CM to Am and CM to Em. If you head in 
the 
> >*same* direction, you end up in the center of a hexagon, which 
does 
> >not correspond to either a major or a minor triad.
> 
> If you continue in the same direction and distance as from Cm -> CM,
> you wind up at C#m.  

No you don't. You end up at the *note* E; C#m is twice as far away.

Is there some sort of reasoning behind not
> including it because it only shares one pitch (instead of two) with
> CM?

See above. If you like, you can join both notes and chords into an 
equilateral triangular lattice, but it seems a little weird to do so. 
However, it might be a way to construct scales.


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Message: 7544

Date: Sat, 04 Oct 2003 21:29:37

Subject: Re: Please remind me

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
<paul.hjelmstad@u...> wrote:

> Now I finally get it. I take it that mapping generators to primes 
is 
> based on commas that are tempered out? How is this calculated 
easily?

I do all of this stuff via wedgies and using Maple.


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Message: 7545

Date: Sat, 04 Oct 2003 21:51:33

Subject: Re: hey gene

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Just one more question.  Can you prove that the 7-limit is the
> only one that works?  What a strange thing.

Yes, but it hardly matters, since higher limit versions make 
progressively less sense. This sort of strange thing happens often in 
math, where two infinite classes of thing have an isomorphism between 
two of the things in each class. So, for example, the group 
PSL2(7) ~ PGL3(2). Here we have the lattices An (triangles, 
tetrahedra, etc) and the lattices Dn (which can be described as the 
cubic lattice Zn colored checkerboard fashion, and then taking only 
the red lattice points.) The note lattices are An, but in the 7-limit 
case it happens that A3 ~ D3, and so the 7-limit note lattice is the 
face-centered cubic lattice, where the centers of the tetrads give us 
a Z3 (integer-coordinate cubic lattice.) Actually showing the 7-limit 
case is unique is best done algebraically.


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Message: 7546

Date: Sat, 04 Oct 2003 21:43:25

Subject: Re: hey gene

From: Carl Lumma

>> Just one more question.  Can you prove that the 7-limit is the
>> only one that works?  What a strange thing.
>
>Yes, but it hardly matters, since higher limit versions make 
>progressively less sense. This sort of strange thing happens often in 
>math, where two infinite classes of thing have an isomorphism between 
>two of the things in each class. So, for example, the group 
>PSL2(7) ~ PGL3(2). Here we have the lattices An (triangles, 
>tetrahedra, etc) and the lattices Dn (which can be described as the 
>cubic lattice Zn colored checkerboard fashion, and then taking only 
>the red lattice points.) The note lattices are An, but in the 7-limit 
>case it happens that A3 ~ D3, and so the 7-limit note lattice is the 
>face-centered cubic lattice, where the centers of the tetrads give us 
>a Z3 (integer-coordinate cubic lattice.) Actually showing the 7-limit 
>case is unique is best done algebraically.

Thanks Gene!!

-Carl


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Message: 7547

Date: Sat, 04 Oct 2003 01:15:35

Subject: Re: hey gene

From: Carl Lumma

>>the major triads do, and the minor triads do, but if you want 1
>>point for each otonal *or* utonal chord
>
>http://lumma.org/tuning/doh.png - Type Ok * [with cont.]  (Wayb.)
>
>I finally realized the length on the right is too much.  Sorry
>Gene.

Just one more question.  Can you prove that the 7-limit is the
only one that works?  What a strange thing.

-Carl


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Message: 7548

Date: Sat, 04 Oct 2003 10:21:37

Subject: Re: Please remind me

From: Graham Breed

Paul Erlich wrote:

>not so easily, it seems. gene and graham have different methods of 
>doing it . . .
>  
>
This is unison vectors to mapping?  I have two different methods, using 
either matrix or exterior algebra.  The latter I got from Gene.  It's 
really the same method formalized two different ways.  And it's fairly 
easy as long as you understand the math it's based on.

I don't have time to write it up now -- and as it seems the more basic 
things aren't understood either I need to look to them first.  So I'm 
going to take the easy way out and say to search back through the 
archives, or inspect the Python code (temper.py only uses exterior 
algebra (wedge products)).

Oh, and this page might help:

Linear temperaments from matrix formalism * [with cont.]  (Wayb.)

It's out of date now, and only covers octave and fifth generators, but 
it's the same basic idea.

Going the other way, and getting unison vectors from a mapping, is more 
difficult because of torsion.


                   Graham


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Message: 7549

Date: Sun, 05 Oct 2003 13:37:07

Subject: Re: Ekmelische Musik

From: Joseph Pehrson

--- In tuning-math@xxxxxxxxxxx.xxxx pitchcolor@a... wrote:

Yahoo groups: /tuning-math/message/3708 * [with cont.] 


> >Ekmelic is a generic German term used to describe prime harmonics 
greater
> >than 5.
> 
> You mean ekmelisch; I don't know if that's exactly true. If I
> remember correctly it comes from the Greek words ek=out and
> melos=series so it means "out of the normal range". So in that
> sense it can be seen as the equivalent of Ivor Darreg's term
> "xenharmonic". The opposite term is emmelisch. 
> 
> Manuel>>
> 
> Hi and thanks for the info; I meant to point out that the term is 
widely used 
> in reference to just intonation rather than equal tunings, as your 
note 
> verifies.  "Ekmelisch" may be roughly equivalent to "xenharmonic" 
but I think 
> it is important to point out that "ekmelisch" does actually refer 
to 
> "harmonics;" hence "series," rather than to some arbitrary "unusual 
tuning."  
> More specifically, it refers to harmonics which are above those 
associated 
> with traditional Western music - those of 7 and beyond.  Martin 
Vogel used 
> the term to describe prime harmonics 7 and beyond in his books "the 
future of 
> Music", "the number 7 in music", and "on the relations of tone" 
(all in 
> German of course)  The use of the term "ekmelisch" in other texts 
(Ernst 
> Bindel et al) is consistent with this, however, the international 
conferences 
> which were hosted by the late Herf-Richter were titled "Musik miot 
> Mikrotönen, Ekmelische Musik," which would suggest that it was 
being used as 
> a broader umbrella, especially since Ezra Sims was there and 72-
equal played 
> a major role in those Salzburg conferences.  Anyone know what's 
going on over 
> there nowadays?
> 
> Aaron


***I'm assuming, then, that the Sims notation was used in these 
Salzburg conferences??

J. Pehrson


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