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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.

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

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.

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.

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.

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?

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 . . .

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

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

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?

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

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

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

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

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

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

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.

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.

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.

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

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

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

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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