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Message: 10403 Date: Sat, 28 Feb 2004 15:49:12 Subject: Re: DE scales with the stepwise harmonization property From: Carl Lumma >Augmented[9] >[28/25, 35/32, 15/14, 16/15] [1, 2, 3, 3] > >(28/25)/(35/32) = 128/125 >(15/14)/(16/15) = 225/224 Augmented[9], eh? How far is the 7-limit TOP version from... ! TOP 5-limit Augmented[9]. 9 ! 93.15 306.77 399.92 493.07 706.69 799.84 892.99 1106.61 1199.76 ! ...? -Carl
Message: 10404 Date: Sat, 28 Feb 2004 15:51:21 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma >> >> Rad. No 6- or 7-toners, eh? I wonder about the "9-limit". >> >> The 9-limit has the further advantage that you can hit more >> >> fifths, and thus improve omnitetrachordality. >> > >> >The 9-limit would be different, for sure. The simple symmetrical >> >lattice criterion wouldn't work, but it would be easy enough to >> >find what does. >> >> Nobody ever answered me if symmetrical is synonymous with unweighted. > >Probably no one was sure what the question meant. It means 3, 5, 7, >5/3, 7/3 and 7/5 are all the same size, however. As I thought then. -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service *
Message: 10407 Date: Sun, 29 Feb 2004 12:01:00 Subject: Re: Stepwise harmonizing property From: Carl Lumma >I was implicitly assuming that one of the chords harmonized to the >root, which doesn't make a lot of sense to assume. Dropping that makes >the analysis far easier--steps have this property iff they are >products (or ratios, but that adds nothing) of consonant intervals >(including 1 as a consonant interval.) How is this any different than a symmetric lattice distance of 2, which is what I thought you used in the first place. >This is obvious enough if you >think about it; the situation no longer depends on fine distinctions. >You get one consonant interval from the unison to the common note in >one chord, and from the common note to another interval in the next >chord; that interval, by definition one reachable by a chord with a >common note, is therefore a product of consonances of the system. I can't parse this. >In the 7-limit, this has the effect of adding 50/49 (= (10/7)(5/7)) to >the list of harmonizable intervals. I did, and also extended the size >up to 8/7, getting a considerably larger list this time, and including >some six and seven note scales for Carl. Well this is cool, but since many classic scales contain steps up to a minor third apart, perhaps 6/5 should be the cutoff. -Carl
Message: 10408 Date: Sun, 29 Feb 2004 12:07:59 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma >>> >> Rad. No 6- or 7-toners, eh? I wonder about the "9-limit". >>> >> The 9-limit has the further advantage that you can hit more >>> >> fifths, and thus improve omnitetrachordality. >>> > >>> >The 9-limit would be different, for sure. The simple symmetrical >>> >lattice criterion wouldn't work, but it would be easy enough to >>> >find what does. And why, pray tell, does symmetrical lattice distance not work in the 9-limit? -Carl
Message: 10412 Date: Sun, 29 Feb 2004 13:19:30 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma >> >>> >> Rad. No 6- or 7-toners, eh? I wonder about the "9-limit". >> >>> >> The 9-limit has the further advantage that you can hit more >> >>> >> fifths, and thus improve omnitetrachordality. >> >>> > >> >>> >The 9-limit would be different, for sure. The simple symmetrical >> >>> >lattice criterion wouldn't work, but it would be easy enough to >> >>> >find what does. >> >> And why, pray tell, does symmetrical lattice distance not work in >> the 9-limit? > >If you call something which makes 3 half as large as 5 or 7 >"symmetrical", it does. One of us is still misunderstanding Paul Hahn's 9-limit approach. In the unweighted version 3, 5, 7 and 9 are all the same length. If you prefer I think you can just use your product-of-two-consonances rule where the ratios of 9 have been included. -Carl
Message: 10415 Date: Mon, 01 Mar 2004 04:58:54 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma >> What you said was that symmetrical lattice distance won't work. > >It doesn't. > >> I asked why, and said Paul Hahn's version works. > >That's a symmetrical lattice, but it isn't a lattice of note-classes. If I didn't know better I'd say you were trying to BS me. What is a lattice of note classes? -Carl
Message: 10416 Date: Mon, 01 Mar 2004 23:42:32 Subject: Re: DE scales with the stepwise harmonization property From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > >> Augmented[9], eh? How far is the 7-limit TOP version > >> from... > >> > >> ! > >> TOP 5-limit Augmented[9]. > >> 9 > >> ! > >> 93.15 > >> 306.77 > >> 399.92 > >> 493.07 > >> 706.69 > >> 799.84 > >> 892.99 > >> 1106.61 > >> 1199.76 > >> ! > >> > >> ...? > >> > >> -Carl > > > >Just look at the horagram, Carl! > > > >107.31 > >292.68 > >399.99 > >507.3 > >692.67 > >799.98 > >907.29 > >1092.66 > >1199.97 > > Oh! Where are the 7-limit horagrams? > > -C. Some of them are in Yahoo groups: /tuning/files/perlich/ * some of them are in Yahoo groups: /tuning_files/files/Erlich/sevenlimit.zip * and this one, aug7.gif, is in both.
Message: 10417 Date: Mon, 01 Mar 2004 04:02:32 Subject: Re: Hanzos From: Carl Lumma >> My recollection is that Paul H.'s algorithm assigns a unique >> lattice route (and therefore hanzo) to each 9-limit interval. > >So what? You still get an infinite number representing each interval, >since you can multiply by arbitary powers of the dummy comma 9/3^2. An infinite number from where? If you look at the algorithm, that dummy comma has zero length. >> Certainly it can be used to find the set of lattice points >> within distance <= 2 of a given point. > >Hahn's alogorithm can, or hahnzos can, or what? The algorithm can. -Carl
Message: 10418 Date: Mon, 01 Mar 2004 23:44:21 Subject: Re: 9-limit stepwise From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > > wrote: > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > > > wrote: > > > > > > On the other end of the size scale we have these. Paul, have you > > ever > > > considered Pajara[6] as a possible melody scale? > > > > Seems awfully improper, but descending it resembles a famous > > Stravisky theme. > > What I should have asked was if you've tried 443443 as a melody scale > in 22-et. Right; I thought you were talking about 2 2 7 2 2 7, but then you clarified. See Yahoo groups: /tuning-math/message/9886 *.
Message: 10420 Date: Mon, 01 Mar 2004 23:47:10 Subject: Re: 9-limit stepwise From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" > <gwsmith@s...> > > > wrote: > > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" > <gwsmith@s...> > > > > wrote: > > > > > > > > On the other end of the size scale we have these. Paul, have > you > > > ever > > > > considered Pajara[6] as a possible melody scale? > > > > > > Seems awfully improper, but descending it resembles a famous > > > Stravisky theme. > > > > What I should have asked was if you've tried 443443 as a melody > scale > > in 22-et. > > Right; I thought you were talking about 2 2 7 2 2 7, but then you > clarified. See Yahoo groups: /tuning- * math/message/9886. Oh yeah, the answer is yes.
Message: 10424 Date: Mon, 01 Mar 2004 18:45:24 Subject: Re: Hanzos From: Carl Lumma >>>>My recollection is that Paul H.'s algorithm assigns a unique >>>>lattice route (and therefore hanzo) to each 9-limit interval. >>> >>>So what? You still get an infinite number representing each >>>interval, since you can multiply by arbitary powers of the dummy >>>comma 9/3^2. >> >>An infinite number from where? If you look at the algorithm, that >>dummy comma has zero length. > >If it has length zero then we are not talking about a lattice at all, >though a quotient of it (modding out the dummy comma) might be. In a >symmetrical lattice it necessarily has the same length as, for >example, 11/3^2, which is of length sqrt(1^2+2^2-1*2)=sqrt(3). It >does *not* have the same length as 11/9, which is of length one, of >course. Yes, you're onto something here. In the unweighted lattice there is a point for 9/3^2, which lies on the diameter-1 hull. >> >> Certainly it can be used to find the set of lattice points >> >> within distance <= 2 of a given point. >> > >> >Hahn's alogorithm can, or hahnzos can, or what? >> >> The algorithm can. > >Why do you think Hahn's definition of "distance" would work for this >problem? If you tell me what it is, we could check and see. However, >you've just told me it does not have the basic properties of a >metric, so I'm inclined to object to calling it a "distance". I thought you acknowledged the receipt of the algorithm... >Given a Fokker-style interval vector (I1, I2, . . . In): > >1. Go to the rightmost nonzero exponent; add its absolute value >to the total. > >2. Use that exponent to cancel out as many exponents of the opposite >sign as possible, starting to its immediate left and working right; >discard anything remaining of that exponent. > >3. If any nonzero exponents remain, go back to step one, otherwise >stop. As for the problem, let's start over. Call the position occupied by 1/1 in 1/1,8/7,4/3,8/5 the root of utonal tetrads. Now 7-limit tetrads sharing a common dyad (pair of pitches) with an otonal tetrad rooted on 1/1 will have roots... 5/4, 3/2, 15/8, 7/4, 35/32, 21/16 ...and those sharing a common tone (single pitch) will have roots... ack, this is visually exhausting... am I correct that they are the 1- and 2-combinations of: 1, 3, 5, 7, 1/3, 1/5, 1/7 ? If so, can you say why adding 9 and 1/9 to this list will not produce an equivalent 9-limit result? -Carl
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