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Message: 10426 Date: Mon, 01 Mar 2004 18:52:18 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. > >How do you get that???? >Given a Fokker-style interval vector (I1, I2, . . . In): [-2 0 0 1] >1. Go to the rightmost nonzero exponent; add its absolute value >to the total. T=1 >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. [-1 0 0 0] T=1 >3. If any nonzero exponents remain, go back to step one, otherwise >stop. T=... whoops, I forgot an absolute value here. The correct value is 2. -Carl
Message: 10428 Date: Mon, 01 Mar 2004 12:25:59 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma >> If I didn't know better I'd say you were trying to BS me. What >> is a lattice of note classes? > >It's the kind of lattice I was talking about--for each octave >ewquivalence class, we have a lattice point. Hence there is a lattice >point representing 9,9/4,9/8... etc but only one. If you take the 2s out of the hanzos, we have that. -Carl
Message: 10429 Date: Mon, 01 Mar 2004 19:01:22 Subject: Re: Hanzos From: Carl Lumma >>>>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). I don't know where sqrt would be coming from. I thought everything would have to have whole number lengths. >>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. That was based on the rather simplistic idea that one steps out and two back leaves you one away from where you started. -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: 10433 Date: Mon, 01 Mar 2004 01:26:20 Subject: Re: Hanzos From: Carl Lumma >> 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. > >In this system you don't exactly, have 7-limit notes and intervals. >You do have "hanzos", with basis 2,3,5,7,9. The usual point of odd-limit is to get octave equivalence, and therefore I'd say the 2s should be dropped from the basis. >The hahnzo |0 -2 0 0 1> is a comma, 9/3^2, which obviously would >play a special role. Hahnzos map onto 7-limit intervals, but not >1-1. Are you happy with the idea that two scales could be >different, since they have steps and notes which are distinct as >hahnzos, even though they have exactly the same steps and notes >in the 7-limit? I think the answer here is yes, though I'm at a loss for why you're mapping hanzos to the 7-limit. >We've got three hahnzos corresponding to 81/80; My recollection is that Paul H.'s algorithm assigns a unique lattice route (and therefore hanzo) to each 9-limit interval. Certainly it can be used to find the set of lattice points within distance <= 2 of a given point. -Carl
Message: 10434 Date: Mon, 01 Mar 2004 23:04:55 Subject: Re: TOP and Tenney space webpage From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > > I have no idea how to reconcile this with > > > > Yahoo groups: /tuning-math/message/9797 * > > > > and with the fact that it's obvious that this linear operator, when > > acting on a monzo, is the only one returns its pitch (or interval > > size) in cents. > > I'm being contrary; it seems to me "measure" isn't really how we want > to look at it, since just intonation is now being viewed as one of an > infinite set of possible tuning maps. True. It only measures pitch (or interval size) in the untempered case.
Message: 10435 Date: Mon, 01 Mar 2004 01:18:50 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma >> One of us is still misunderstanding Paul Hahn's 9-limit approach. > >What in the world makes you think this has anything to do with me or >anything I've said? "" 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. If you call something which makes 3 half as large as 5 or 7 "symmetrical", it does. "" -Carl
Message: 10436 Date: Mon, 01 Mar 2004 23:05:58 Subject: Re: non-1200: Tenney/heuristic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > > Amazingly, Gene's page would have you believe you need to search > even > > in the codimension 1 case! > > You want I should derive the codimension 1 formula instead? Why not?
Message: 10438 Date: Mon, 01 Mar 2004 23:11:57 Subject: Re: Harmonized melody in the 7-limit From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > > > Also, could we screen based on which of the above > > combinations, and which orderings of those, produce the > > most low-numbered ratios in the scale? Or does such > > an approach fail on the grounds that it ignores temperament > > (aka TOLERANCE)? > > One approach would be to use tempering to simplify the problem. If we > pick linear temperaments which reduce the four sizes of scale step to > two, we also automatically enforce Myhill's property. That would seem to depend on the ordering, and if the period isn't an octave would seem to be impossible.
Message: 10440 Date: Mon, 01 Mar 2004 23:18:08 Subject: Re: DE scales with the stepwise harmonization property From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > >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 Just look at the horagram, Carl! 107.31 292.68 399.99 507.3 692.67 799.98 907.29 1092.66 1199.97
Message: 10443 Date: Mon, 01 Mar 2004 04:07:13 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma >> >> One of us is still misunderstanding Paul Hahn's 9-limit approach. >> > >> >What in the world makes you think this has anything to do with me or >> >anything I've said? >> >> "" >> 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. >> >> If you call something which makes 3 half as large as 5 or 7 >> "symmetrical", it does. >> "" > >Can you point out where in the above quote you found the words "Paul >Hahn?" What you said was that symmetrical lattice distance won't work. I asked why, and said Paul Hahn's version works. -Carl
Message: 10444 Date: Mon, 01 Mar 2004 23:20:44 Subject: Re: 9-limit stepwise From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > Here are some 9-limit stepwise harmonizable scales, with the same > bound on size of steps--8/7 is the largest. In order to keep the > numbers down, I also enforced that the size of the largest step (in > cents) is less than four times that of the smallest step--the > logarithmic ratio is the fourth number listed. > > In order I give scale type number on the list, scale steps, > multiplicities, largest/smallest, and number of steps in the scale. > > As you can see, the largest scale listed has 41 steps, which is > getting up there. (64/63)/(81/80)=5120/5103 and (49/48)/(50/49) = > 2401/2400; putting these together gives us hemififths, and hence > Hemififths[41] as a DE for this. Hemififths is into the > microtemperament range by most standards; it has octave-generator with > TOP values [1199.700, 351.365] and a mapping of > [<1 1 -5 -1|, <0 2 25 13|]. I've never tried to use it, and so far as > I know neither has anyone else, but this certainly gives a motivation. > Ets for hemififths are 41, 58, 99 and 140. We also get Schismic[29] > out of scale 4, Diaschismic[22] out of scale 5, I thought Diaschismic had no unique definition in the 7-limit.
Message: 10446 Date: Mon, 01 Mar 2004 23:22:48 Subject: Re: 9-limit stepwise From: Paul Erlich --- 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.
Message: 10447 Date: Mon, 01 Mar 2004 15:23:55 Subject: Re: DE scales with the stepwise harmonization property From: Carl Lumma >> 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.
Message: 10448 Date: Mon, 01 Mar 2004 23:23:55 Subject: Re: 9-limit stepwise From: Paul Erlich --- 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? I'm confused -- I thought the largest step was supposed to be less than 200 cents?
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