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Message: 7475 Date: Tue, 30 Sep 2003 18:10:36 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>no, scales with a period equal to a 1/N octave, where N is an integer >greater than 1, are distributionally even but not MOS.

Oh, you're enforcing the 'new' definition of MOS. Who came up with distrib. even? -Carl

Message: 7476 Date: Tue, 30 Sep 2003 18:51:06 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>>Who came up with distrib. even? >> >> -Carl

> >john clough and nora englesbrshmegegel . . . i forget her last name. >you can find the term in my 22 paper on your website.

Wow, was this added in a later rev? All I remember is "maximal evenness". -Carl

Message: 7477 Date: Tue, 30 Sep 2003 22:55:21 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>As Paul kindly said, at least with fifths it's a manageable sort of >mess. :-) And I would add: with many familiar landmarks, particularly >in the harmony.

This sort of attitude vastly: () Underestimates the effect of notation on music. Use ordinary notation, think up ordinary music. () Overestimates the difficulty of 'learning new nominals'. You've got new pitches, new rules, new fingerings, new sounds, new accidentals. The nominals matter so much?

>> >No. A little reflection allows me to explain that, as it stands now, >> >the semantic foundations of Sagittal notation have absolutely >> >nothing to do with any temperament.

>> >> I should have said, "good PBs" there. [I think of PBs as >> temperaments, which always gets me into trouble.]

> >So what's a _good_ PB for notational purposes?

The same kind that are good for composition purposes!

>That sounds even less >likely to be agreed upon than a good linear temperament. How about >we forget about this given our agreement below?

I thought it was well-agreed-upon: the simplicity of the commas vs. their size. There are different ways to calculate this, and the details of how to do so with planar and higher temperaments and raw PBs are not settled, but using any of the proposed methods is fine -- pick your fav.

>> With linear temperaments, you only need 1 accidental pair at a time, >> as I've pointed out.

> >But Carl, that's like saying you only need 6 pairs of accidentals to >notate 19-limit JI. One for each prime above 3. It becomes essentially >unreadable once you go past 2 accidentals per note.

How is saying you only need 1 like saying you only need 6?

>And even ignoring these "enharmonics", you need other accidentals when >you have multiple parallel chains, i.e. when the period is not the >whole octave.

Isn't this refuted by Paul's single-accidental decatonic notation?

>> If average use ("gimme 9 notes of such-and-such temperament in the >> 13-limit") turns out to require more commas than can fit on a list,

> >I don't understand how average-use could require "more commas than can >fit on a list". What could this mean except "an infinite number of >commas"?

I thought you said something about the list getting unwieldy. If you've come up with 600 symbols, I think that should be plenty!

>> you could try assigning (an) accidental(s) for each *temperament*, >> with the understanding that it/they would take on TM-reduced value(s) >> for the limit and scale cardinality being used.

> >Eek! So then we would have to learn not only new nominals for every >temperament, but new accidentals too?

Instruments don't read accidentals; people do. I'm not sure how learning 600 accidentals is any easier than learning tuning-specific interpretations of existing accidentals. In both cases, once the tonal system is learned, one should be able to hear the correct notes. And this proposal has the added benefit of not requiring any new fonts or eye training -- just use conventional sharps and flats. It's just a proposal. Drawbacks include: () Only works for linear temperaments. () It's kinda neat to not have to specify the temperament in advance. One could mix "temperaments" in the same bar just by using the appropriate accidentals from a master-list. Can't do this with the present proposal.

>There's definitely no need for this.

How do you know? Who can say what composers won't need?

>> >So the first part of my belief is that it is far better to have a >> >notation system whose semantics are based on precise ratios and then >> >use that to also notate temperaments, rather than trying to find the >> >ultimate temperament and then using a notation based on that to notate >> >both ratios and other temperaments.

>> >> Wow; this is exactly what I've been saying all along!!

> >Really? Then how have I managed to waste so much of my time answering >this thread?

Glad to see you have such a high opinion of peer review.

>> >Then if that's accepted, the second part is that it is best if the >> >simplest or most popular ratios have the simplest notations.

>> >> Right. And it's this aspect that makes the search more-or-less >> equivalent to the search for good PBs.

> >Nope. You've lost me there.

The simplest commas would be the most popular for a reason!

>> >I understand that you agree with this, and so it should be obvious >> >that the simplest accidental is no accidental at all and so the >> >simplest ratios should be represented by nominals alone. When we >> >agree that powers of 2 will not be represented at all, or will be >> >represented by an octave number, or by a distance of N staff positions >> >or a clef, then surely you agree that the next simplest thing is to >> >represent powers of three by the nominals.

>> >> Well, that's a weighted-complexity approach. But even with most >> weighted-complexity lists I've seen, non-rational-generator >> temperaments appear.

> >Huh? I thought you just agreed that we would first decide how to >_precisely_ notate ratios?

Yup. In fact, you can think of a PB/temperament *as* a notation in my scheme.

>Therefore we don't care about weighted >complexity, or any complexity (except at the 3-prime-limit), because >we know we are going to represent ratios of the other primes as being >_OFF_ the chain, by using accidentals.

Don't follow you here. But try to track me again. By saying you want to always keep the lowest primes the simplest ones in the map (by assuming 2-equiv. on the staff and by always using 3:2s for your nominals), you are effectively weighting your complexity measure. If you completely disallow temperaments like miracle (which do not have a 3:2 generator) from showing up in your notation search (think temperament search), it's a *very* strongly weighted function -- you're insisting that both generators be primes.

>Whether we use rational or irrational generators we can only represent >powers of _ONE_ ratio _EXACTLY_, _ON_ the chain, (modulo our interval >of equivalence).

"Ratio" obviously. Did you mean "prime"? Then your statement is false.

>> But certainly the project didn't start out this way, and even in >> the last few days I saw a blurb for George and/or you looking very >> confused about non-heptatonic systems.

> >I think we're only confused about how a notation whose nominals are >related by an irrational generator could be used notate ratios >precisely.

Just observe Paul's decatonic notation. It's the perfect embodiment of everything I've been saying. -Carl

Message: 7478 Date: Wed, 01 Oct 2003 06:59:21 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan I'm sorry Carl, I just don't have time to persue this any further. But it sure seems as though the ways we are misinterpeting each other are legion.

Message: 7479 Date: Wed, 01 Oct 2003 12:44:48 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>This has been taking me away from working on the explanations of the >sagittal system, that you agree are sorely needed, and from urgent >paying work. > >I'm really sorry I made that crack about "wasting my time". Please >accept my humble apologies. > >Maybe we'll communicate better when we've both had a chance to cool >off, and I'm not under so much time pressure.

Ok. Pls. reference: Yahoo groups: /tuning-math/message/6927 * [with cont.] -Carl

Message: 7480 Date: Wed, 01 Oct 2003 07:58:41 Subject: Re: hey gene From: monz hey Carl (and Gene, even moreso), --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Oh, and I don't see anything explaining TM reduction on > your website, or anywhere else for that matter. > > -Carl

i asked Gene for a good definition for TM reduction a long time ago ... and Gene, if you gave it to me and i lost it in the shuffle, i apologize. can you send it again? -monz

Message: 7481 Date: Wed, 01 Oct 2003 12:53:15 Subject: Re: hey gene From: Carl Lumma

>>i asked Gene for a good definition for TM reduction a >>long time ago ... and Gene, if you gave it to me and i >>lost it in the shuffle, i apologize. can you send it again?

> >Here's a definion of Minkowski reduction: > >Reduced Lattice Bases * [with cont.] (Wayb.) > >Manuel

I understand roughly what TM reduction is, and I even found this:

>>>In either case, could you please explain the mathematical >>>criterion that defines "Minkowski reduced", as you did for LLL?

>> >>Let p/q be reduced to lowest terms; then T(p/q) = pq. A pair of >>intervals {p/q, r/s} with p/q>1, r/s>1, T(p/q) < T(r/s) and p/q >>and r/s independent is Minkowski reduced iff the only numbers in >>the set {(p/q)^i (r/s)^j} such that T(t/u) < T(r/s) are powers >>of p/q.

> >That's astoundingly simple! Wouldn't it be quite reasonable >to further require that the only ratio t/u in the set >{(p/q)^i (r/s)^j} such that T(t/u) < T(r/s), is p/q itself? The >idea would be that otherwise, the two unison vectors are >"mismatched".

Aside from the fact that I don't know what elements are in a set notated like {(p/q)^i (r/s)^i}, and I can't fathom the function of t/u in this definition, my question was more along the lines of... What goes in? Ratios? Vals? What comes out? A list of commas that define a PB? -Carl

Message: 7482 Date: Wed, 01 Oct 2003 01:54:30 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>I'm sorry Carl, > >I just don't have time to persue this any further. But it sure seems >as though the ways we are misinterpeting each other are legion.

You don't have time to study Paul's decatonic notation, provide simple examples of a sagittal alternative, explain your reasoning or realize that you've made an unnecessary assumption about the generator, etc. Too bad. Expect me to continue to balk when I see ivory tower pronouncements about sagittal in the future. -Carl

Message: 7483 Date: Wed, 01 Oct 2003 20:37:55 Subject: Re: hey gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >>i asked Gene for a good definition for TM reduction a > >>long time ago ... and Gene, if you gave it to me and i > >>lost it in the shuffle, i apologize. can you send it again?

> > > >Here's a definion of Minkowski reduction: > > > >Reduced Lattice Bases * [with cont.] (Wayb.) > > > >Manuel

> > I understand roughly what TM reduction is, and I even found this: >

> >>>In either case, could you please explain the mathematical > >>>criterion that defines "Minkowski reduced", as you did for LLL?

> >> > >>Let p/q be reduced to lowest terms; then T(p/q) = pq. A pair of > >>intervals {p/q, r/s} with p/q>1, r/s>1, T(p/q) < T(r/s) and p/q > >>and r/s independent is Minkowski reduced iff the only numbers in > >>the set {(p/q)^i (r/s)^j} such that T(t/u) < T(r/s) are powers > >>of p/q.

> > > >That's astoundingly simple! Wouldn't it be quite reasonable > >to further require that the only ratio t/u in the set > >{(p/q)^i (r/s)^j} such that T(t/u) < T(r/s), is p/q itself? The > >idea would be that otherwise, the two unison vectors are > >"mismatched".

> > Aside from the fact that I don't know what elements are in a > set notated like {(p/q)^i (r/s)^i},

the last i should be a j. that's the set of ratios that can be expressed as (p/q)^i *times* (r/s)^j.

> and I can't fathom the > function of t/u in this definition,

the definition should have read, "the only numbers t and u in the set . . ." instead of "the only numbers in the set . . ."

> What goes in? Ratios? Vals?

ratios.

> What comes out? A list of commas that define a PB?

more generally, a set of ratios which are a basis for the same lattice as the ratios you put in are a basis for, but which are, if possible, simpler (in a certain sense) than the ones you put in. apparently minkowski is not the only reasonable reduction definition, so we could also have TKZ reduction, at least.

Message: 7484 Date: Wed, 01 Oct 2003 09:59:07 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >I'm sorry Carl, > > > >I just don't have time to persue this any further. But it sure seems > >as though the ways we are misinterpeting each other are legion.

> > You don't have time to study Paul's decatonic notation, provide simple > examples of a sagittal alternative, explain your reasoning or realize > that you've made an unnecessary assumption about the generator, etc. > Too bad. Expect me to continue to balk when I see ivory tower > pronouncements about sagittal in the future. > > -Carl

Carl, This has been taking me away from working on the explanations of the sagittal system, that you agree are sorely needed, and from urgent paying work. I'm really sorry I made that crack about "wasting my time". Please accept my humble apologies. Maybe we'll communicate better when we've both had a chance to cool off, and I'm not under so much time pressure. Regards, -- Dave Keenan

Message: 7485 Date: Wed, 01 Oct 2003 20:47:32 Subject: Re: hey gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> the definition should have read, "the only numbers t and u in the > set . . ." instead of "the only numbers in the set . . ."

let me try again -- it should have read "the only ratios t/u in the set . . ." there.

Message: 7486 Date: Wed, 1 Oct 2003 12:05:09 Subject: Re: Please remind me From: Manuel Op de Coul

>Is SCALA your creation?

Yes.

>I am confused when you say >to include 2/1 as a unison vector. Isn't the point to temper out the >unison vectors that you give?

Right, I should have said "as an interval in the list of unison vectors". The largest interval in that list will not be tempered if there's more than one, and it doesn't need to be a 2/1. Manuel

Message: 7488 Date: Wed, 1 Oct 2003 12:07:32 Subject: Re: hey gene From: Manuel Op de Coul

>i asked Gene for a good definition for TM reduction a >long time ago ... and Gene, if you gave it to me and i >lost it in the shuffle, i apologize. can you send it again?

Here's a definion of Minkowski reduction: Reduced Lattice Bases * [with cont.] (Wayb.) Manuel

Message: 7489 Date: Wed, 01 Oct 2003 14:45:06 Subject: Re: hey gene From: Carl Lumma

>> >>Let p/q be reduced to lowest terms; then T(p/q) = pq. A pair of >> >>intervals {p/q, r/s} with p/q>1, r/s>1, T(p/q) < T(r/s) and p/q >> >>and r/s independent is Minkowski reduced iff the only numbers in >> >>the set {(p/q)^i (r/s)^j} such that T(t/u) < T(r/s) are powers >> >>of p/q.

>> > >> >That's astoundingly simple! Wouldn't it be quite reasonable >> >to further require that the only ratio t/u in the set >> >{(p/q)^i (r/s)^j} such that T(t/u) < T(r/s), is p/q itself? The >> >idea would be that otherwise, the two unison vectors are >> >"mismatched".

>> >> Aside from the fact that I don't know what elements are in a >> set notated like {(p/q)^i (r/s)^i},

> >the last i should be a j.

Whoops, finger failure.

>that's the set of ratios that can be expressed >as (p/q)^i *times* (r/s)^j.

I figured it was a multiply, but didn't realize this was what it meant.

>> and I can't fathom the >> function of t/u in this definition,

> >the definition should have read, "the only numbers t and u in the >set . . ." instead of "the only numbers in the set . . ."

Uh...

>> What goes in? Ratios? Vals?

> >ratios. >

>> What comes out? A list of commas that define a PB?

> >more generally, a set of ratios which are a basis for the same >lattice as the ratios you put in are a basis for, but which are, if >possible, simpler (in a certain sense) than the ones you put in.

Ok. So how can we get this into monz' dictionary?

>apparently minkowski is not the only reasonable reduction definition, >so we could also have TKZ reduction, at least.

Reduced Lattice Bases * [with cont.] (Wayb.) What's bit here...

>In the definition of Minkowski reduction, successive basis vectors >b_i are added to the lattice basis only if b_i is the shortest vector >in the lattice which will allow the basis to be extended. In >Korkin-Zolotarev reduction, though, successive basis vectors b_i are >chosen based on their length in the orthogonal complement of the space >spanned by the previous basis vectors b_1, ..., b_(i-1).

I wonder if this has anything to do with straightness. Oh, and I don't follow this...

>> >That's astoundingly simple! Wouldn't it be quite reasonable >> >to further require that the only ratio t/u in the set >> >{(p/q)^i (r/s)^j} such that T(t/u) < T(r/s), is p/q itself? The >> >idea would be that otherwise, the two unison vectors are >> >"mismatched".

-Carl

Message: 7490 Date: Wed, 01 Oct 2003 21:46:54 Subject: Re: Please remind me From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> > once you know the mapping from generators to primes, it's no > > different, you can use rms or whatever in exactly the same way. > > i've done a huge number of such calculations for the case of

50:49

> > and 64:63 vanishing, on the main tuning list . . .

> > Paul, would you be so kind as to reference a message on "tuning" > of the case of 50/49 & 64/63 vanishing, so I can look at the > calculations? Thanks! > > Paul

i did the calculations numerically, but it would be no more difficult to do symbolically than the meantone case. like there, you have to first know the mapping from generators (and periods) to primes (unless you've got a trick to get around that that i'm unaware of). i used the following matlab program to calculate the 7-limit error for a given choice of generator (use p=2 for the 'rms' case): function err=decaton(inp,p); err1=(abs(log(3/2)/log(2)-inp)).^p; err2=(abs(log(7/4)/log(2)-2+2*inp)).^p; err3=(abs(log(7/6)/log(2)-2+3*inp)).^p; err4=(abs(log(5/4)/log(2)-1.5+2*inp)).^p; err5=(abs(log(5/3)/log(2)-2.5+3*inp)).^p; err6=(abs(log(7/5)/log(2)-.5)).^p; err=(err1+err2+err3+err4+err5+err6).^(1/p); and then used matlab's optimization toolbox to find the value of 'inp' (that is, of the generator) that minimized this error. nonetheless, at least for p=2, it would be straightforward to minimize this error function using calculus. how about another case, 225/224 and 2401/2400 vanishing? just to make things interesting, let's optimize for 9-limit consonances rather than just 7-limit consonances. here's all the calculus worked out: Yahoo groups: /tuning/message/22626 * [with cont.]

Message: 7491 Date: Wed, 01 Oct 2003 21:51:40 Subject: Re: hey gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> I wonder if this has anything to do with straightness.

yes, a reduced basis will have good straightness, because the set of basis vectors is, in some sense, as short as possible. and, as we discussed before, shortness implies straightness. the "block" always has the same "area", so if the vectors are close to parallel, they'll have to be long to compensate. remember that whole confusing discussion?

> Oh, and I don't follow this... >

> >> >That's astoundingly simple! Wouldn't it be quite reasonable > >> >to further require that the only ratio t/u in the set > >> >{(p/q)^i (r/s)^j} such that T(t/u) < T(r/s), is p/q itself? The > >> >idea would be that otherwise, the two unison vectors are > >> >"mismatched".

> > -Carl

that would mean that otherwise, r/s is much longer than p/q, and the basis itself is an odd one to choose because it necessarily involves two unison vectors of such different proportions.

Message: 7492 Date: Wed, 01 Oct 2003 14:57:28 Subject: Re: hey gene From: Carl Lumma

>> the definition should have read, "the only numbers t and u in the >> set . . ." instead of "the only numbers in the set . . ."

> >let me try again -- it should have read "the only ratios t/u in the >set . . ." > >there.

Now it makes sense...

> >>Let p/q be reduced to lowest terms; then T(p/q) = pq. A pair of > >>intervals {p/q, r/s} with p/q>1, r/s>1, T(p/q) < T(r/s) and p/q > >>and r/s independent is Minkowski reduced iff the only ratios t/u > >>in the set {(p/q)^i (r/s)^j} such that T(t/u) < T(r/s) are powers > >>of p/q.

So IOW, if you have a pair of unison vectors for a PB, you shouldn't be able to stack them both in some way to get an interval that's simpler than the more complex of the pair is by itself. That is simple. How does this work when there are more than 2 vectors in the basis?

> >That's astoundingly simple! Wouldn't it be quite reasonable > >to further require that the only ratio t/u in the set > >{(p/q)^i (r/s)^j} such that T(t/u) < T(r/s), is p/q itself? The > >idea would be that otherwise, the two unison vectors are > >"mismatched".

So you want to rule out cases where one uv is way longer than the other? -Carl

Message: 7493 Date: Wed, 01 Oct 2003 15:02:13 Subject: Re: hey gene From: Carl Lumma

>> I wonder if this has anything to do with straightness.

> >yes, a reduced basis will have good straightness, because the set of >basis vectors is, in some sense, as short as possible. and, as we >discussed before, shortness implies straightness. the "block" always >has the same "area", so if the vectors are close to parallel, they'll >have to be long to compensate. remember that whole confusing >discussion?

Yes, but I meant does the difference between KZ and M have to do with straightness? -Carl

Message: 7494 Date: Wed, 01 Oct 2003 15:03:09 Subject: Re: hey gene From: Carl Lumma

>> >That's astoundingly simple! Wouldn't it be quite reasonable >> >to further require that the only ratio t/u in the set >> >{(p/q)^i (r/s)^j} such that T(t/u) < T(r/s), is p/q itself? The >> >idea would be that otherwise, the two unison vectors are >> >"mismatched".

> >So you want to rule out cases where one uv is way longer than >the other?

I see you've just answered affirmatively. -C.

Message: 7495 Date: Wed, 01 Oct 2003 22:04:43 Subject: Re: hey gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> I wonder if this has anything to do with straightness.

> > > >yes, a reduced basis will have good straightness, because the set

of

> >basis vectors is, in some sense, as short as possible. and, as we > >discussed before, shortness implies straightness. the "block"

always

> >has the same "area", so if the vectors are close to parallel,

they'll

> >have to be long to compensate. remember that whole confusing > >discussion?

> > Yes, but I meant does the difference between KZ and M have to do > with straightness? > > -Carl

dunno, haven't thought about it . . .

Message: 7496 Date: Wed, 01 Oct 2003 15:07:17 Subject: Re: hey gene From: Carl Lumma

>> Yes, but I meant does the difference between KZ and M have to do >> with straightness? >> >> -Carl

> >dunno, haven't thought about it . . .

In case you missed it, it's this bit here, which I don't quite follow...

>In the definition of Minkowski reduction, successive basis vectors >b_i are added to the lattice basis only if b_i is the shortest vector >in the lattice which will allow the basis to be extended. In >Korkin-Zolotarev reduction, though, successive basis vectors b_i are >chosen based on their length in the orthogonal complement of the space >spanned by the previous basis vectors b_1, ..., b_(i-1).

-Carl

Message: 7497 Date: Wed, 01 Oct 2003 00:14:55 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> At 04:57 PM 9/30/2003, you wrote:

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

> >> >> What gets wrecked if one's basis isn't distrib. even?

> >> > > >> >you yourself said that one begins a set of nominals forming a > >> >periodicity block (on the tuning-math list; maybe you should

reply

> >to

> >> >this over there).

> >> > >> What does distrib. evenness have to do with PBs?

> > > >a fokker periodicity block, when all but one of the unison vectors > >are tempered out, becomes a distributionally even scale. that's my > >Hypothesis, anyway.

> > So distrib. even and MOS are equivalent?

no, scales with a period equal to a 1/N octave, where N is an integer greater than 1, are distributionally even but not MOS.

> >i mentioned *altered* versions of DE scales because one may wish

to

> >start with a "non-fokker" periodicity block.

> > What's a non-fokker PB -- a shape other than a parallelepiped?

right, like the hexagons in the gentle introduction, or more radically, something like a harmonic minor scale.

Message: 7498 Date: Wed, 01 Oct 2003 22:13:54 Subject: Re: hey gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> Yes, but I meant does the difference between KZ and M have to do > >> with straightness? > >> > >> -Carl

> > > >dunno, haven't thought about it . . .

> > In case you missed it, it's this bit here, which I don't quite > follow... >

> >In the definition of Minkowski reduction, successive basis vectors > >b_i are added to the lattice basis only if b_i is the shortest

vector

> >in the lattice which will allow the basis to be extended. In > >Korkin-Zolotarev reduction, though, successive basis vectors b_i

are

> >chosen based on their length in the orthogonal complement of the

space

> >spanned by the previous basis vectors b_1, ..., b_(i-1).

> > -Carl

i'm confused about that, because wouldn't b_2, b_2 + b_1, b_2 + 2*b_1, b_2 - b+1, etc., all have the same length in the orthogonal complement of the space spanned by b_1?

Message: 7499 Date: Wed, 01 Oct 2003 00:15:45 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> no, scales with a period equal to a 1/N octave, where N is an

integer

> greater than 1, are distributionally even but not MOS.

i should have said "may be" instead of "are" and "but cannot be" instead of "but not".

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